Local Lyapunov exponents: Sublimiting growth rates of linear random differential equations

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1963 Wolfgang Siege...

Author: Wolfgang Siegert

182 downloads 402 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1963

Wolfgang Siegert

Local Lyapunov Exponents Sublimiting Growth Rates of Linear Random Differential Equations

123

Wolfgang Siegert Allianz Lebensversicherungs - AG Reinsburgstrasse 19 70178 Stuttgart Germany [email protected]

ISBN 978-3-540-85963-5 e-ISBN 978-3-540-85964-2 DOI: 10.1007/978-3-540-85964-2 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008934460 Mathematics Subject Classification (2000): 60F10, 60H10, 37H15, 34F04, 34C11, 58J35, 91B28, 37N10, 92D15, 92D25 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

Establishing a new concept of local Lyapunov exponents, two separate theories are brought together, namely Lyapunov exponents and the theory of large deviations. Specifically, for the stochastic differential system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt the new concept is introduced. Due to stationarity, the Lyapunov exponents of Ztε (which by Oseledets’ Multiplicative Ergodic Theorem describe the exponential growth rates of Ztε ) do not depend on the initial position x of X ε . Now the goal of this work is to provide a Lyapunov-type number for each regime of the drift b. As this characteristic number shall depend on the domain in which X ε , a dynamical system perturbed by additive white noise, is starting, it yields a concept of locality for the Lyapunov exponents of Ztε . Furthermore, the locality of such local Lyapunov exponents is to be understood as reflecting the quasi-deterministic behavior of X ε which asserts that in the limit of small noise, ε → 0, the process X ε has metastable states depending on its initial value as well as on the time scale chosen (Freidlin-Wentzell theory). Up to now local Lyapunov exponents have been defined as finite time versions of Lyapunov exponents by several authors, but here we target at investigating the large time asymptotics t → ∞. So the goal is to connect the large parameters t and ε−1 in the customary definition of the Lyapunov exponents in order to approach the sublimiting distributions (Freidlin) which are supported by the metastable states of X ε . The local Lyapunov exponent is then understood to be the exponential growth rate of Z ε on the time scale chosen, subject to convergence in probability as ε → 0. Notably, the system itself changes in the sense that the noise intensity converges to zero with the time horizon depending on the noise intensity parameter. In contrast to this new concept the Lyapunov exponents as obtained by the Multiplicative Ergodic Theorem reflect the information of limit distributions, i.e. of invariant

v

vi

Preface

measures, as time increases to infinity for the system parameter ε > 0 being fixed. As a result we prove that the local Lyapunov exponent is bounded from above by the largest real part of the spectrum of the matrix A evaluated at the metastable state corresponding to the time scale; the respective bound from below holds true with the smallest real part of an eigenvalue of A at the corresponding metastable state. Assuming that A takes its values in the diagonal matrices, it is shown that its eigenvalues at the respective metastable state are precisely the possible local Lyapunov exponents. Moreover, in a “strongly” hypoelliptic situation it can be proved that only the largest eigenvalue is observed under convergence in probability. The latter result is regarded as sublimiting Furstenberg-Khasminskii formula, since the resulting limit is obtained as a (trivial) integral which produces the top eigenvalue. For the above tasks the prerequisites which fundamentally consist of knowing the exit probabilities of all the stochastic systems involved will be provided in detail: For this purpose, an integrated account of the theory for nondegenerate stochastic differential systems (Freidlin and Wentzell) and of the exit probabilities for degenerate stochastic systems (Hernández-Lerma) is given in chapters 2 and 3. The subsequent final chapter is the heart of the book. Here, all the results are proven and discussed. Acknowledgements Foremost, I would like to thank Prof. Peter Imkeller for posing the problem considered in this book as well as for his support during my doctoral studies. Furthermore, I would like to thank Prof. Ludwig Arnold and Prof. Peter Kloeden for their interest in my work. I greatly acknowledge many fruitful discussions with Dr. Ilya Pavlyukevich and his precious comments on the draft. Moreover, I am grateful to Prof. Onesimo Hern´ andez-Lerma and Prof. Wolfgang Kliemann for their valuable comments and for sharing my enthusiasm. The financial support by the German Research Foundation (Deutsche Forschungsgemeinschaft) through the Research Training Group “Stochastic Processes and Probabilistic Analysis” as well as through the Collaborative Research Centre “SFB 649 Economic Risk” is gratefully acknowledged. Last but not least, I thank my family, notably my wife Barbara and my parents, for their permanent backing.

Berlin/Stuttgart July 2008

Wolfgang Siegert

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Linear differential systems with parameter excitation . . . . . 1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spherical coordinates for linear systems . . . . . . . . . . . . . . . . . . . 1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents . . . 1.4 The deterministic case: Lyapunov exponents for asymptotically constant linear systems . . . . . . . . . . . . . . . . . 1.5 Sample systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 12 20

1

2

Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory . . . . . . . . . . . . . . . 2.1 Preliminaries and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The limiting distribution (stationary measure) . . . . . . . . . . . . . 2.3 The large deviations principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Exit probabilities for non-degenerate systems . . . . . . . . . . . . . . 2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Sample systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 44 53 55 60 68 72 91 108

3

Exit probabilities for degenerate systems . . . . . . . . . . . . . . . . . 125 3.1 Exit probabilities for degenerate systems depending on a small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2 Uniform consequence for the exit probability . . . . . . . . . . . . . . . 140

4

Local Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Local Lyapunov exponents: upper and lower bound . . . . . . . . . 4.2 The local growth rate of the determinant . . . . . . . . . . . . . . . . . . 4.3 Local Lyapunov exponents in the diagonal case . . . . . . . . . . . . .

143 144 156 157

vii

viii

Contents

4.4 Local Lyapunov exponents in the two-dimensional, general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Qualitative theory of nonlinear real noise systems on time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The local Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 178 186 226

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

List of Figures

1

A prototypical potential function U : Rd → R with two wells . .

1.1 1.2

Geometrical interpretation of the quantity vt := (cos αt )2 . . . . . 42 The switching curve and tendencies of ¯ α) = − x + 1 (λ2 − λ1 ) sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . 48 h(x, 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Sketch of the potential function U1 . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the density pε ∼ e−2U1 /ε . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of ϕ and ϕδ within the domains D and D∗ . . . . . . . . . . . ε,x ε,x Sketch of the stopping times τm and θm+1 . ................ Visualization of the 1-graphs on {1, 2, 3} . . . . . . . . . . . . . . . . . . . A two well potential function U : Rd → R (cf. figure 1) . . . . . . . Schematic depiction of the thermohaline circulation . . . . . . . . . .

64 65 74 86 93 110 116

4.1 4.2

The tendencies of the Verhulst drift vector field . . . . . . . . . . . . . The tendencies of the comparison drift vector field given the sublimiting statistics of X ε,x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drift vector field F with one attracting switching curve and the sublimiting statistics of X ε,x . . . . . . . . . . . . . . . . . . . . . . . . . . The tendencies and switching curves of the sample drift vector field F (x, y) = 12 − y + y(1 − y)x on R2 . . . . . . . . . . . . . . Strictly separated switching surfaces on Rd . . . . . . . . . . . . . . . . . ¯ α) = − x + 1 (λ2 − λ1 ) sin 2α and The drift vector field h(x, 2 the sublimiting statistics of X ε,x0 . . . . . . . . . . . . . . . . . . . . . . . . . .

164

4.3 4.4 4.5 4.6

3

168 180 184 185 224

ix

Introduction

During the last decades intense research has been devoted to dynamical systems subject to random perturbations: Considerable effort has been dedicated to investigate exit times and exit locations from given domains and how they relate to the respective deterministic dynamical system. Building upon considerations in physics and chemistry (see e.g. the classical paper by Kramers [Kr 40]) the theory of large deviations by Freidlin and Wentzell ([Fr-We 98], [Fr 00]) provides the correct mathematical framework for tackling these problems in case of Gaussian perturbations. This theory sets up the precise time scales for transitions of non-degenerate stochastic systems between certain regimes. The behavior of such systems is called metastable. The theory of random dynamical systems, on the other hand, considers stochastic processes which satisfy a certain flow property, the cocycleproperty. The main cornerstone here is the Multiplicative Ergodic Theorem by Oseledets [Os 68]; also see Arnold [Ar 98]: This theorem assigns Lyapunov exponents to linear random dynamical systems. These are the exponential growth rates as time grows large for fixed intensities of the underlying noise. The following work now attempts to close the gap between these two stochastic disciplines: It does not study the exponential growth rate for a fixed noise intensity and large time (resulting in the Lyapunov exponents), but considers the exponential growth, if the time horizon depends on the noise intensity. Thus one considers the Lyapunov characteristics on time scales. Since these time scales correspond to metastable points, the Lyapunov exponents are localized by connecting the large time asymptotics to the limit of vanishing noise intensity. What one usually does when dealing with “local Lyapunov exponents” is to replace the infinite time limit (characterizing the Lyapunov exponents) by a large, but finite time horizon; see Abarbanel et al. [Ab-Brw-Ke 91] and [Ab-Brw-Ke 92], Wolff [Wo 92], Pikovsky [Pk 93], Pikovsky and Feudel [Pk-Fe 95] and Bailey et al. [Ba-El-Ny 97]. A similar discussion in the same spirit is undertaken by Monahan [Mo 02]: In the case of the Maas model he describes the concept of a “local Lyapunov exponent” for which the infinite W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

1

2

Introduction

time limit is replaced by a large, but finite time horizon; this time then needs to be large enough for the system to sample the local attractor, but smaller than the average escape time of the regime. This rationale is also applied when calculating the respective exponents numerically. Now generally speaking, the problem in the case of an elliptic stochastic differential system is that switches to other regimes occur with strictly positive probability—no matter how small the time horizon is chosen. On the other hand, e.g. as computers can necessarily work with finite calculation horizons only, our concept of localizing Lyapunov exponents by means of time scales justifies the above finite-time procedure, if time scales are chosen appropriately. Let us further describe this rationale: The base systems under consideration are dynamical systems with additive white noise perturbations; for simplicity let the process X ε be defined by the stochastic differential equation (SDE) of gradient type √ dXtε = − ∇U (Xtε ) dt + ε dWt for the moment. It describes the motion of a particle in a potential landscape which is derived from the real-valued, differentiable function U defined on Rd . The linearization of X ε is then given as the solution of the linear, real noise driven differential equation dZtε = − HU (Xtε ) Ztε dt , the so called variational equation, in which HU (x) denotes the Hesse matrix of second derivatives of U at x. The variational equation governs the evolution of “infinitesimal disturbances” of X0ε . The Lyapunov exponents of the system are now defined as the exponential growth rates lim

t→∞

1 log | Ztε ( . , x, z) | . t

Their existence is assured by Oseledets’ [Os 68] Multiplicative Ergodic Theorem and their number does not exceed the dimension d of the state space of X ε . Due to the stationarity of the flow X ε in Oseledets’ theorem, the Lyapunov exponents do not depend on the initial condition x = X0ε,x . Suppose that the potential function U has the qualitative shape as depicted in figure 1. Moreover, let Λ01 (x) ≥ . . . ≥ Λ0d (x) denote the (decreasingly indexed) eigenvalues of − HU (x). The goal of this contribution is to provide a Lyapunov-type number for each regime of the potential function U . As this characteristic number shall depend on the initial point (more precisely, on the well in which the stochastic solution X ε is starting), it shall yield a concept of locality for Lyapunov exponents. The

Introduction

3 U

v V

K1

K3

K2

Fig. 1 A prototypical potential function U : Rd → R with two wells

main motivation for the concept thus defined is that in the small-noise-limit the particle X ε stays in the initial shallow well near K1 for an exponentially long time (Kramers’ law, Freidlin-Wentzell theory) during which we can “see” the shallow well; afterwards the particle finally overcomes the barrier at K3 and the deeper well around K2 dominates the picture; mathematically this is made precise by observing that K1 and K2 are the supports of the sublimiting distribution on the corresponding time scales (Freidlin [Fr 77] and [Fr 00]). Furthermore, K1 and K2 are the metastable points of the potential. In order to capture this metastable behavior we connect the parameters t and ε by ε(t) :=

ζ log t

for a scaling parameter ζ > 0 for approaching the sublimit distributions. Hence, we consider ε(t) 1 log Zt ( . , x, z) t and we then conjecture that this random variable converges in probability to Λ0j (Kk ) as t → ∞, where k ∈ {1, 2} depends on the initial position x and on the time scale parameter ζ of Xtε ; more precisely, if x is in the K1 -regime and if also ζ < 2v, then k = 1; otherwise, k = 2. This conjecture is proven in section 4.3 under the additional assumption that HU ( . ) only takes its values in the diagonal matrices; the index j ∈ {1, . . . , d} then, of course, depends on the initial direction z of Z. We call these limit numbers the local Lyapunov exponents of Z. In the general case, i.e. abstaining from the diagonality condition, section 4.4 gives conditions under which the above

4

Introduction

exponential growth rate converges in probability to the top eigenvalue of −HU (Kk ), Λ01 (Kk ) , where Kk again denotes the metastable state for the time scale chosen; this limit then is the local Lyapunov exponent of Z. In other words, defining T (ε) := eζ/ε , the previous discussion concerns the convergence in probability of 1 log ZTε (ε) ( . , x, z) T (ε) as ε → 0. In comparison with the previously mentioned concept by Abarbanel et al. [Ab-Brw-Ke 91] and [Ab-Brw-Ke 92], Wolff [Wo 92], Pikovsky [Pk 93], Pikovsky and Feudel [Pk-Fe 95] and Bailey et al. [Ba-El-Ny 97] who would take the finite time growth rate 1 log ZTε ( . , x, z) T for fixed T and ε as “local Lyapunov exponent”, one therefore obtains a rigorous explanation for how to correctly choose the time horizon depending on the underlying noise intensity ε. Furthermore, we would like to comment on a second type of “local Lyapunov exponents” which can be found in the literature: Let d = 1, then the drift of Z ε in the variational equation is − HU (x) ≡ −U (x) , the negative curvature of U at x. Several authors then call this number −U (x) the “local” or “local in phase space” or “instantaneous” Lyapunov exponent; see Fujisaka [Fu 83], van den Broeck and Nicolis [vB-Ni 93], Witt et al. [Wt-Ne-Kt 97] and Pikovsky and Feudel [Pk-Fe 95]. However, this is not in accordance with our understanding, since the corresponding stochastic system X ε does not stay near an arbitrary initial point x, but is confined to one of the metastable points K1 , K2 by the drift −U . The system Z ε mainly samples −U (Kk ) for k ∈ {1, 2} , but neglects contributions of some other −U (x). Therefore our result for the one-dimensional situation (see remark 4.1.4) only has the values −U (K1 )

and

− U (K2 )

as local Lyapunov exponents in the sense of our definition in contrast to Fujisaka [Fu 83], van den Broeck and Nicolis [vB-Ni 93], Witt et al. [Wt-Ne-Kt 97] and Pikovsky and Feudel [Pk-Fe 95]. In other words, this expresses the fact that the instantaneous rates −U (x) do not have equal

Introduction

5

rights, but the Dirac measures δK1 and δK2 are the sublimiting distributions reflecting the preferences of X ε on the time scales. A third, deterministic concept of locality of Lyapunov exponents different from ours has been introduced by Eden, Foias and Temam [Ed-Fo-Tm 91]. It has already been indicated above that Kramers’ law, made precise by Freidlin and Wentzell, plays a dominant role in the following considerations. It is interesting that already the classical Eyring-Kramers formula for the exit times of X ε contains the eigenvalues of the Hesse matrix of U at K1 , K2 ε ε and K3 . Namely let τ12 (and τ21 ) denote the time at which X ε enters the K2 -well when started in K1 (and vice versa). Then in the limit as ε → 0, under non-degeneracy assumptions, the following asymptotic expressions are known to hold for the mean exit times, d 2π i=1 Λ0i (K3 ) 2v/ε ε E τ12 ≈ 0 e d Λ1 (K3 ) Λ0 (K1 ) i=1

and ε E τ21

i

d 2π i=1 Λ0i (K3 ) 2V /ε ≈ 0 e d 0 Λ1 (K3 ) i=1 Λi (K2 )

which are cited from Bovier at al. [Bv-Ec-Gd-Kn 04] in the above notation. Abstracting the previous considerations one detects that the underlying diffusion does not have to stem from an SDE of gradient type; the Freidlin-Wentzell theory of large deviations and metastability also admits more general drift functions b and also state dependent noise coefficients σ under certain assumptions. Overall, the two characteristic features of the above stochastic differential system (X ε , Z ε ) are the following: Firstly, it is degenerate in the sense that in the equation for Z ε there is no noise component but only a drift coefficient depending on X ε ; in other words, the differential equation of Z ε is random, driven by the real noise process X ε . Secondly, the differential system is linear with respect to Z ε . Further abstracting the system matrix A is by no means restricted to be the negative of the Hesse matrix of some potential function; it can be any matrix valued function defined on the state space of X ε . Moreover, we will also admit that the state spaces of X ε and Z ε can have different dimensions. Hence, the general object of the subsequent considerations is the real-noise driven, linear stochastic differential system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ,

(1)

where ε ≥ 0, W is a Wiener process on Rd , A ∈ C(Rd , Rn×n ) (or A ∈ C(Rd , Cn×n )), the drift b : Rd → Rd does not necessarily stem from a

6

Introduction

potential, but exhibits several regimes and the values of σ : Rd → Rd×d are close to idRd . All in all, X ε is a diffusion in Rd with small noise intensity and Z ε is a linear cocycle in Rn . Generalizing the previously sketched idea the local Lyapunov exponents are the possible limits of ε(t) 1 log Zt ( . , x, z) t as t → ∞, where ε(t) ≡

ζ log t

for a scaling parameter ζ > 0, depending on the initial condition x of X ε and the initial direction z of Z ε . The goal of proving convergence in probability for this exponential growth rate on a time scale is organized as follows: In the first chapter we collect known results on linear, real noise driven differential systems such as the Multiplicative Ergodic Theorem, their decomposition in spherical coordinates and the deterministic Hartman-WintnerPerron theorem. The latter two subjects yield coordinate systems which will decisively come into play in chapter 4. Beforehand, chapter 2 gives an account of the Freidlin-Wentzell theory as needed for describing the locality, metastability and sublimiting distributions. Since this theory is based on the fundamental exit time law for non-degenerate stochastic systems, this result is recalled in detail. The system Z ε of (1) is a degenerate stochastic differential system by definition, since there is no stochastic differential in the Z-component. This is in particular also true for the angle of Z ε . As the behavior of the latter process needs to be investigated in chapter 4 chapter 3 recalls known results on its exit probabilities; more precisely, we give an account of the theorems by Hern´ andez-Lerma concerning exit probabilities of degenerate systems which are not covered by Freidlin-Wentzell theory. Chapter 4 finally investigates the exponential growth rates on the time scales: Firstly, it is proven that the top real part of an eigenvalue at the metastable state is an upper bound for the local Lyapunov exponent; likewise, the smallest real part of an eigenvalue at the metastable state is a lower bound for the local Lyapunov exponent; see section 4.1. Since det Z ε is a process in R and the situation is quite tractable for a one-dimensional state space, a consequence for the exponential growth rate of det Z ε can be drawn which is subject to section 4.2. The coordinates from the Hartman-Wintner-Perron theorem allow to explicitly calculate the exponential growth rate under the additional assumption that A solely takes diagonal values; see section 4.3. Finally, section 4.4 gives criteria under which one can obtain convergence in the general, two-dimensional case: Here the result by Hernández-Lerma comes into play as the decomposition of |Ztε | by means of spherical coordinates demands an assertion on the angle process of Ztε . More precisely,

Introduction

7

a statement concerning the Lebesgue measure of the times which the angle process spends at the switching curves of its drift can be deduced; this result which is analogous to Freidlin’s [Fr 00] metastability theorem then allows to calculate a sublimiting version of the Furstenberg-Khasminskii formula. A final remark is necessary concerning the use of the notion of “time scales”: There is now a very elaborate theory of “dynamic equations on time scales”; this theory has been invented by Hilger [Hi 88]; also see Bohner and Peterson [Boh-Pet 01]. This concept understands “time scales” (also called “measure chains”) as certain time sets which are underlying to the systems under consideration. Hence, the investigation of an ordinary differential equation (ODE) means to work with the “time scale” T = R+ or R; an ordinary difference equation is understood as dynamic equation on the “time scale” T = N0 or Z. However, in this paper here the physical time set is always R+ and a time scale in our context is understood as a time horizon T ≡ T (ε) which depends on the parameter ε > 0 of the base SDE of X ε . The symbol 2 will be used to mark the end of a proof. In order to avoid latent ambiguities, it will also be employed sporadically to finish remarks and examples where necessary.

Chapter 1

Linear differential systems with parameter excitation

This chapter intends to introduce to the general setting of linear stochastic systems Z˙ t = A(t) Zt , d Zt denotes where (A(t))t≥0 is a matrix valued stochastic process and Z˙ t ≡ dt the derivative with respect to the “time” variable t. Such differential systems are called parametrically excited (perturbed) or real noise linear systems; in the engineering literature the terminology rheo linear system is also used. The system matrix is assumed to be a continuous mapping defined on the state space of a Markov process which serves as stochastic input for the system. Hence, the above matrix process is of the form

A(t) = A(Xt ), where (Xt )t≥0 is the input process. Note for preciseness that some authors further differentiate the nature of the noise by distinguishing “real noise” on the one hand which is defined on the two-sided time set R and Markovian noise on the other hand which is a Markov process with time set R+ ; see Arnold and Kliemann [Ar-Kl 83, p.4]. In this book both terms will be used interchangeably in the latter sense. Linear real noise systems with statedependent coefficient matrix A(Xt ) and Markovian input noise Xt as above have for example been investigated by Frisch [Fs 66]. In this reference it is argued heuristically how a Fokker-Planck equation might be obtained for (Xt , Zt )t , if Xt is stationary. Real noise systems with Markovian input noise are also subject to the considerations of Kats and Krasovskii [Ka-Kv 60]; however, these authors consider the case that (Xt )t≥0 is a Markov chain with finite state space. Now the setting of our work also assumes that the input process has finitely many “states of preference”, but is a continuous process defined by a stochastic differential equation with respect to Brownian motion. These preferential (“metastable”) states correspond to certain time scales which will be made precise in the next chapter.

W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

9

10

1 Linear differential systems with parameter excitation

Since the real noise process (Xt )t≥0 will be defined by some SDE, the coupled system (X, Z) is also given by the resulting SDE. Note that this differential equation is degenerate by definition in the sense that no noise term, i.e. no summand of the type g(Xt , Zt ) dBt containing the differential of a Brownian motion (or some more general stochastic process) B, is visible in the Z-direction, but solely in the X-variable; see e.g. Bunke [Bu 72, Ch.6] for standard results on such equations. A different type of randomly perturbed linear SDE is the so called white noise case; see e.g. Khasminskii [Kh 67], [Kh 80] and [Kh 60]. Here, there is no real noise input but stochastic integration with respect to Brownian motion in the SDE for Z itself. The latter type of systems is not subject to our investigations, but also will be commented on in this chapter. From an applications’ point of view real noise stochastic systems are often considered as more realistic for describing “real”-world problems. The reason is that most processes to be modeled are concerned with variables of a specified magnitude or even restricted to a bounded interval such as concentrations (e.g. in chemistry), population fluctuations (in life sciences) or strictly positive parameters in technical systems; see e.g. Kliemann [Kl 80, App.IV: p.3] and [Kl 83b], Arnold et al. [Ar-Hh-Lf 78], Ahmadi and Morshedi [Ah-Mr 78] and Griesbaum [Gb 99] for further examples and discussions. In contrast using white noise instead is mostly considered as too drastic an idealization; see Kliemann [Kl 80, App.IV: p.3] and Wihstutz [Wh 75, p.3].

1.1 The model The system which lies at the heart of these investigations is the real-noise driven system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

where A ∈ C(Rd , Kn×n ) is a continuous matrix function (K = R or C); d, n ∈ N denote the dimensions of the state spaces of X ε and Z ε , respectively; ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which denotes a Brownian motion in Rd , defined on a complete probability space (Ω, F , P), and X ε is a diffusion, defined by the above SDE with coefficient functions b ∈ C ∞ (Rd , Rd ) and σ ∈ C ∞ (Rd , Rd×d ). A detailed study of the SDE for X ε will follow in the next chapter specifying the assumptions; see (2.1) and the corresponding set of assumptions 2.1.1. For the moment, just assume that there exists a unique non-exploding solution1 X ε,x , where the superscript x ∈ Rd denotes its initial value. 1 For standard terminology concerning SDEs, such as solution, uniqueness and nonexplosiveness, we refer to Hackenbroch and Thalmaier [Hb-Th 94, Ch.6], Khasminskii [Kh 80, Ch.I-III] and Freidlin and Wentzell [Fr-We 98, Ch.1] among many others.

1.1 The model

11

Since the stochastic process X ε,x is non-explosive and has (P-almost surely) continuous paths, the SDE (1) defines a continuous stochastic process (X ε , Z ε ) in Rd × Kn ; more precisely, the differential equation for Z ε , dZtε = A Xtε,x (ω) Ztε dt , Z0ε = z ∈ Kn , (1.1) is a random differential equation (RDE) which ω-wise, i.e. for any realization Xtε,x (ω), is solved as an ODE; hence, the continuity of t → A(Xtε,x (ω)) guarantees existence and uniqueness of the solution path Ztε (ω, x, z) in Kn (see e.g. Coppel [Cp 65, p.42]) which together with X ε,x forms the solution (X ε , Z ε ) of (1) in Rd × Kn , starting in (x, z). Collecting these solution paths (possibly defined as 0 on an exceptional subset of Ω of zero P-measure) yields a well-defined mapping Z ε : R+ × Ω × Rd × Kn −→ Kn , (t, ω, x, z) → Z ε (t, ω, x, z) := Z ε (t, ω, x) z := Ztε (ω, x) z := Ztε (ω, x, z) and these notations will be used equivalently. Being linear in the z-variable the mappings Z ε (t, ω, x) form a matrix process which (as the corresponding fundamental matrix) solves the random matrix differential equation dZtε = A Xtε,x (ω) Ztε dt , Z0ε = idKn . Furthermore, as the fundamental solution of a linear equation, Z ε (t, ω, x) takes its values in the set of invertible matrices, its inverse being governed by the matrix differential equation d(Z −1 )t = − (Z −1 )t A(Xtε,x ) dt . In particular, the Wronski-determinant det(Ztε (ω, x)) does not vanish and differentiating pathwise with respect to t yields the random differential equation d det(Ztε (ω, x)) = trace A Xtε,x (ω) det(Ztε (ω, x)) dt, det(Z0ε (ω, x)) = 1, and therefore one obtains the Jacobi equation (Liouville equation) t

det(Ztε (ω, x)) = det(Z0ε (ω, x)) exp trace A Xuε,x (ω) du (1.2) 0

t trace A Xuε,x (ω) du ; = exp 0

see e.g. Coppel [Cp 65, p.44] for the pathwise calculations. Note that for ε = 0 all objects remain well-defined as the solutions of the resulting ODE.

12


Under appropriate conditions it can be proved that X ε and hence also Z ε depend continuously on the parameter ε > 0; see Blagovescenskii and Freidlin [Bla-Fr 61]. However, this will not be used in the sequel. In this work we are interested in Lyapunov exponents, i.e. exponential growth rates 1 log | Ztε (ω, x, z) | , t which are realized for large times t = T (ε) ; such growth rates will then be called local Lyapunov exponents. In the following section spherical coordinates will be introduced in order to get an integral decomposition for the growth rate mentioned previously. This representation which is also an ingredient of the Furstenberg-Khasminskii formula (see p.27) provides quantitative information concerning |Z ε | and will play a fundamental role in subsection 4.4.2.

1.2 Spherical coordinates for linear systems In the following the system Z ε will be decomposed by means of spherical coordinates; for this section we take K = R, i.e. Ztε (ω, x, z) is an element of Rn where n ∈ N. In the literature on stochastic systems the use of such coordinates (also called the projection method ) is accredited to Khasminskii [Kh 67] and Infante [In 68]; see e.g. Kliemann [Kl 79, p.465] and [Kl 80, p.144] and Crauel [Cra 84, p.13]. It is, however, interesting to note that this use of spherical coordinates is by no means restricted to the stochastic case. In fact, this method had been a well established tool in investigating deterministic linear differential systems before; see e.g. Levi-Civita [LC 11] and Wintner [Wi 50] and [Wi 57]. Since the matrix Z ε (ω, x) is invertible, the process Ztε (ω, x, z) is nonzero for z = 0 and can hence be characterized by its radial and spherical components via εt (ω, x, z) := | Ztε (ω, x, z) | ∈ (0, ∞) and ψtε (ω, x, z) :=

Ztε (ω, x, z) ∈ S n−1 , εt (ω, x, z)

respectively, if z = 0 ; here, S n−1 := {y ∈ Rn : |y| = 1} denotes the unit sphere in Rn . For the system Z ε as given by (1), the defining pathwise differential equation (1.1) is equivalent to the system of the two RDEs d εt = Q(Xtε , ψtε ) εt dt ,

(1.3)

dψtε = h(Xtε , ψtε ) dt ,

(1.4)

1.2 Spherical coordinates for linear systems

13

where

Q(x, ψ) := Q A(x), ψ := A(x)ψ , ψ

and

h(x, ψ) := h A(x), ψ := A(x)ψ − A(x)ψ , ψ ψ ;

here, . , . denotes the standard scalar product of Rn . This follows in a d straightforward manner by calculating the pathwise derivative dt for fixed (ω, x, z) where z = 0, since d d ε ≡ Z ε , Z ε 1/2 dt t dt t t 1 1 d Z ε , Z ε = 2 εt dt t t

1 1 d ε ε Z ,Z = 2 2 εt dt t t 1 = ε A(Xtε ) Ztε , Ztε ≡ A(Xtε ) ψtε , ψtε εt t and hence d d ε εt dt Ztε − Ztε dt t d Ztε d ε ψt ≡ = dt dt εt ( εt )2 εt A(Xtε ) Ztε − Ztε A(Xtε ) ψtε , ψtε εt = ( εt )2 ε ε ≡ A(Xt ) ψt − A(Xtε ) ψtε , ψtε ψtε

due to (1.1); conversely, these differential equations for ( ε , ψ ε ) imply that d ε ε d ε Z ≡ ( ψ ) dt t dt t t d ε d ε t + εt ψ = ψtε dt dt t = ψtε A(Xtε ) ψtε , ψtε εt + εt A(Xtε ) ψtε − εt A(Xtε ) ψtε , ψtε ψtε ≡ A(Xtε ) Ztε . Note that h(A, ψ) ≡ Aψ − Aψ, ψ ψ vanishes at an element ψ ∈ S n−1 , if and only if ψ is an eigenvector of the matrix A. The above vector field h(A, . ) is considered as the projection of the linear vector field ψ → Aψ onto S n−1 for a fixed A ∈ Rn×n . Due to the multiplicative structure of the RDE for ε , it can be integrated pathwise by separation of variables, thus giving t

| Ztε (ω, x, z) | ≡ εt (ω, x, z) = | z | exp Q Xuε,x (ω), ψuε (ω, x, z) du 0

14


and therefore log |z| 1 1 log |Ztε (ω, x, z)| = + t t t

t

Q Xuε,x (ω), ψuε (ω, x, z) du.

(1.5)

0

Now since h(x, −ψ) = −h(x, ψ) for all x and ψ, the SDE for ψtε is symmetric with respect to the origin; more precisely, h(x, . ) can be viewed as a vector field on the projective space P n−1 , i.e. on S n−1 where opposite points are identified; therefore, ψ ε is also considered as a P n−1 -valued process. The SDE for ε and formula (1.5) remain unaffected, since Q(x, −ψ) = Q(x, ψ) for all x and ψ. Furthermore, the projective process ψ ε is decoupled from the radial process ε ; therefore, (X ε , ψ ε ) is a Markov process in Rd × S n−1 or Rd × P n−1 , respectively, whose generating partial differential operator is given by ∂ Lε := G ε + h , ∂ψ where G ε :=

d

bi

i=1

d ε ∂ ∂2 + aij ∂xi 2 i,j=1 ∂xi ∂xj

ε

is the generator of X to be discussed later; see section 2.2 . Remark 1.2.1 (n = 2). Consider the two-dimensional case, i.e. Kn = R2 . Here, we can canonically identify ψ ∈ S 2 with its angle α ∈ [0, 2π) via

cos α = ψ , sin α for which we will use the abbreviation . ψ = α in the following. In particular, the spherical process ψtε on S 1 defines the angle process αεt ∈ [0, 2π) by

cos αεt := ψtε . sin αεt The previous SDE (1.4), dψtε = h(Xtε , ψtε ) dt , implies the following equation for αεt : ¯ ε , αε ) dt , dαεt = h(X t t ¯ being defined by for the drift h ¯ ¯ A(x), α , h(x, α) := h

(1.6)


15

where ¯ A, α := − a12 sin2 α + a21 cos2 α + a22 − a11 sin α cos α h

for matrices A =

a11 a12 a21 a22

∈ R2×2 ;

. see Arnold and Kliemann [Ar-Kl 83, p.59]. More precisely, let α = also cos α sin α =: ψ(α) and define

d − sin α ψ(α) = ; ψ (α) := dα cos α then one directly calculates that ¯ h(A, α) =

Aψ(α) , ψ (α)

for any A ∈ R2×2 ; furthermore, (ψ(α), ψ (α)) is an orthonormal basis of R2 for any α and one obtains altogether for the pathwise derivatives that d d ε ψ ≡ ψ(αεt ) dt t dt d ε α = ψ (αεt ) dt t ε ε ¯ = ψ (αεt ) h(A(X t ), αt ) ε ε = A(Xt )ψ(αt ) , ψ (αεt ) ψ (αεt ) = A(Xtε )ψ(αεt ) − A(Xtε )ψ(αεt ) , ψ(αεt ) ψ(αεt ) ≡ h(A(Xtε ), ψtε ) which proves that the RDEs for ψ ε and αε are equivalent, since this calculation can be read in both directions. ¯ that h(A ¯ Note that it follows from the above definition of h + c I2 , . ) = 2×2 ¯ h(A, . ), for all A ∈ R and c ∈ R, where I2 denotes the two-dimensional unit matrix; an example for such a system is given by Ahmadi and Morshedi [Ah-Mr 78, Sec.IV]; also see Kliemann and R¨ umelin [Kl-Rm 81, p.17]. Also note that if A ∈ R2×2 is symmetric, then ¯h reads ¯ h(A, α) ≡ − a12 sin2 α + a21 cos2 α + (a22 − a11 ) sin α cos α 1 = a12 cos 2α + (a22 − a11 ) sin 2α . 2 In any case the RDE (1.3) for ε now reads ¯ tε , αεt ) εt dt , d εt = Q(X

16


¯ being defined by for Q ¯ α) := Q(A(x), ¯ Q(x, α) , where ¯ Q(A, α) := a11 cos2 α + a22 sin2 α + (a12 + a21 ) sin α cos α

(1.7)

is just designed such that Q(A, ψ) ≡ Aψ, ψ

cos α cos α a11 a12 , ≡ a21 a22 sin α sin α = a11 cos2 α + a22 sin2 α + (a12 + a21 ) sin α cos α ¯ ≡ Q(A, α) , . if α = ψ. Formula (1.5) hence becomes log |z| 1 1 log |Ztε (ω, x, z)| = + t t t

t

¯ X ε,x (ω), αε (ω, x, z) du. Q u u

(1.8)

0

Furthermore, the drift function ¯ h(x, . ) is π-periodic which corresponds to the fact that h(x, −ψ) = −h(x, ψ) for ψ ∈ S 1 ; αε is the solution of the above real noise SDE modulo π. Thus (Xtε , αεt ) is a Markov process in R2 × [0, π) with generator ¯ ∂ . L¯ε := G ε + h ∂α Another way of describing the motion of the angle is by means of ξtε := tan αεt whenever αεt ∈ / {(2k + 1)π/2 : k ∈ Z}; defining F (A, ξ) := a21 − a12 ξ 2 + [ a22 − a11 ] ξ for A ∈ R2×2 and ξ ∈ R, the dynamics of ξ ε is described by the differential equation dξtε = F (A(Xtε ), ξtε ) ≡ a21 (Xtε ) − a12 (Xtε ) (ξtε )2 + a22 (Xtε ) − a11 (Xtε ) ξtε dt ,

(1.9)


17

since 1 d ε ¯ ξt = h(Xtε , αεt ) dt cos2 αεt 1 − a12 (Xtε ) sin2 αεt + a21 (Xtε ) cos2 αεt ≡ cos2 αεt + a22 (Xtε ) − a11 (Xtε ) sin αεt cos αεt = a21 (Xtε ) − a12 (Xtε ) (ξtε )2 + a22 (Xtε ) − a11 (Xtε ) ξtε . Remark 1.2.2. The above pathwise considerations can be rewritten for any linear differential equation with time dependent system matrix, Z˙ t = A(t) Zt

(t ≥ 0)

in Rn , as ˙ t = Q(t, ψt ) t , ψ˙ t = h(t, ψt ) , where t := | Zt | ∈ (0, ∞) and ψt := Ztt ∈ S n−1 whenever Z0 = 0. Here, the notation is adapted as Q(t, ψ) := Q A(t), ψ ≡ A(t)ψ , ψ and

h(t, ψ) := h A(t), ψ := A(t)ψ − A(t)ψ , ψ ψ .

In particular, 1 1 log | Z0 | log | Zt | = + t t t

t

Q(u, ψu ) du . 0

For the two-dimensional case n = 2,

a11 (t) a12 (t) A(t) := , a21 (t) a22 (t) . and the then well defined angle process αt = ψt , i.e.

cos αt := ψt , sin αt it follows that ¯ αt ) , α˙ t = h(t,

(1.10)

18


where ¯ ¯ h(t, α) := h(A(t), α) ≡ − a12 (t) sin2 α + a21 (t) cos2 α + (a22 (t) − a11 (t)) sin α cos α ≡ A(t)ψ(α) , ψ (α) . The formula (1.10) becomes 1 1 log | Z0 | log | Zt | = + t t t

t

¯ αu ) du , Q(u,

(1.11)

0

¯ α) := Q(A(t), ¯ where Q(t, α). Furthermore, ξt := tan αt is driven by the differential equation ξ˙t = F (A(t), ξt ) ≡ a21 (t) − a12 (t) ξt2 + [ a22 (t) − a11 (t) ]ξt . The following definition is taken from Arnold and Kliemann [Ar-Kl 83, p.13] and from Kliemann [Kl 79, p.464]. Definition 1.2.3 (Switching surfaces of a drift function). Let F : Rd × R → R be a continuous function. The points (x, y) ∈ Rd × R with F (x, y) = 0 define connected surfaces in Rd ×R. These are called the switching surfaces (of F ). If d = 1, these sets will also be called the switching curves (of F ). This terminology will now be used for investigating the behavior of ξ ε and αε , respectively. Here, we follow Arnold and Kliemann [Ar-Kl 83, p.59f.]; also see Kliemann [Kl 80, p.148f.] and [Kl 79, p.465f.]. Definition-Remark 1.2.4 (Switching surfaces corresponding to (1.1)). Consider the RDE (1.1) which is the pathwise differential system defined by the SDE (1) and suppose that Kn = R2 , i.e. the system matrix is given by a mapping A ∈ C(Rd , R2×2 ). By (1.9) A( . ) induces the drift vector field for ξ ε ≡ tan αε , F (x, ξ) := F (A(x), ξ) ≡ a21 (x) − a12 (x) ξ 2 + [ a22 (x) − a11 (x) ] ξ , where x ∈ Rd and ξ ∈ R. According to the previous definition 1.2.3 we look for the zeros, F (x, ξ) = F (A(x), ξ) = 0 ;


19

from the solution formula for quadratic equations it follows that one gets no switching curve, if a12 = 0 and (a22 − a11 )2 < −4a12 a21 ; one switching curve ξ1 (A(x)) =

a22 − a11 , if a12 = 0 and (a22 − a11 )2 = −4a12 a21 , 2a12

and ξ1 (A(x)) = ∞ , if a12 = 0 and a11 = a22 and a21 = 0 , respectively; two switching curves ξ1,2 (A(x)) =

(a22 − a11 )2 + 4a12 a21 , 2a12 0 and (a22 − a11 )2 > −4a12 a21 , =

(a22 − a11 ) ± if a12

and ξ1 (A(x)) = ∞ , ξ2 (A(x)) =

a21 , if a12 = 0 and a11 = a22 , a11 − a22

respectively; infinitely many switching curves ξ ∈ R ∪ {∞} , if a12 = 0 and a11 = a22 and a21 = 0 . The above switching curves for F , i.e. for the motion of ξ ε ≡ tan αε , directly translate into switching curves of the drift ¯ h(A(x), α) ≡ − a12 (x) sin2 α + a21 (x) cos2 α + (a22 (x) − a11 (x)) sin α cos α ¯ of the angle motion; since h(A(x), . ) is π-periodic, we can restrict our investigations on an interval of that length, e.g. (− π2 , π2 ] ; then we get the following ¯ switching curves of h: Ai (x) := Ai (A(x)) := arctan ξi (A(x)) , if there are i ∈ {1, 2} many zeros ξi (A(x)) of F (A(x), ξ). In the last case of infinitely many switching curves any α is a zero of ¯ h(A(x), . ). It follows from the above that a12 (x) = 0 implies that π2 is a switching curve; furthermore, if a21 (x) = 0, then 0 is a switching curve.

20


In case that there are exactly two switching curves A1,2 (A(x)), the previous notation is redefined such that A1 (A(x)) is attracting and A2 (A(x)) is repelling, in the sense that ¯ α → h(A(x), α) changes its sign from + to − at A1 (A(x)) and changes from − to + at A2 (A(x)). Remark 1.2.5. The above considerations have been accomplished in the real noise situation of the differential equation (1). Actually, this technique of projecting the linear system onto the sphere using spherical and angular coordinates can also be transferred to the white noise case; the Itô-formula then provides the SDEs for the radial, spherical and angular component, respectively. We do not give details here, since this goes beyond the framework of this book, but refer to the literature instead: See Khasminskii [Kh 67] and [Kh 80, Sec.VI.7-VI.9], Nishioka [Nk 76], B¨ ohme [Bm 80], Auslender and Milshtein [Al-Mi 82], Arnold et al. [Ar-Oe-Pd 86], Arnold and Kliemann [Ar-Kl 87a], Pinsky and Wihstutz [Pi-Wh 88], Pardoux and Wihstutz [Pd-Wh 88] and [Pd-Wh 92] and Imkeller and Lederer [Im-Ld 99] and [Im-Ld 01].

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents In this section the “classical” Lyapunov exponents are to be discussed as the result of Oseledets’ [Os 68] Multiplicative Ergodic Theorem. Since this result and its arguments are not to be used in the sequel, we refer to Arnold [Ar 98] for details. Here, the solution of the RDE (1.1) is modeled as random dynamical system (RDS) over the canonical metric dynamical system. More precisely, let Ω := C(R+ , Rd ) denote the path space of X ε , dXtε = b (Xtε ) dt +

√ ε σ (Xtε ) dWt

(t ≥ 0) ,

where ε > 0. Ω is fixed as canonical probability space in this section. If endowed with the topology of uniform convergence on compacts, Ω is a Polish space which can then be equipped with its Borel-σ-algebra; the latter is the trace σ-algebra of B(Rd )R+ in Ω, i.e. F := B(Ω) = Ω ∩ B(Rd )R+ ;

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents

21

see Arnold [Ar 98, p.544f.] and Hackenbroch and Thalmaier [Hb-Th 94, 1.24]. For all t ∈ R+ , there is the canonical shift transformation on Ω defined by θ(t) : Ω → Ω ,

ω → θ(t)ω := ω(t + • ) ≡ s → ω(t + s) .

Here, the mapping (t, ω) → θ(t)ω is continuous, hence measurable with respect to the underlying Borel-σ-algebras. Let (Ptε )t denote the Markov transition probabilities of the Markov process X ε . Furthermore, suppose that there exists a unique stationary probability distribution for X ε , i.e. a unique probability measure2 ρε on Rd which is invariant with respect to the Markov transition probabilities (Ptε )t in the sense that ρε ( . ) = Ptε (x, . ) ρε (dx) (t ≥ 0) ; Rd

conditions assuring existence and uniqueness of such a measure ρε will be given in section 2.2 where the SDE of X ε is investigated in detail; see p.62f. Then there is a unique probability Pρε on (Rd )R+ , B(Rd )R+ such that the coordinate process is a (time-homogeneous) Markov process with transition semigroup (Ptε )t and initial distribution ρε ; see e.g. Arnold [Ar 98, p.548] and Hackenbroch and Thalmaier [Hb-Th 94, 2.5]. Since the outer measure of Ω is full, Pρ∗ε (Ω) = 1, due to the continuity of the paths of X ε , Pρε also induces a probability distribution on F ; see e.g. Hackenbroch and Thalmaier ε [Hb-Th 94, p.43f.]. It will be denoted by the same symbol Pρε . Let X ε,ρ denote the process with distribution Pρε on the path space Ω, i.e. the system X ε with initial distribution ρε . Note that the above stationarity of the measure ρε (stationarity of the canonical Markov process), i.e. the (Ptε )t -invariance of ρε , is equivalent to Pρε being (θ(t))t -invariant in the sense that Pρε ◦ θ(t)−1 = Pρε

for all t ∈ R+ ;

this fact is also expressed by stating that all shifts θ(t) preserve the measure Pρε ; see Arnold [Ar 98, p.545,549]. Altogether, the above system (Ω, F , Pρε , (θ(t)t∈R+ ) satisfies 1) (ω, t) → θ(t)ω is F ⊗ B(R+ )/F -measurable, 2) θ(0) =idΩ and θ(s + t) = θ(s) ◦ θ(t) for all s, t ∈ R+ , 3) Pρε ◦ θ(t)−1 = Pρε for all t ∈ R+ ,

which are the characterizing properties for calling Ω, F , Pρε , (θ(t)t∈R+ a metric (measure preserving) dynamical system; see Arnold [Ar 98, p.536f.]. 2

This measure ρε is not to be confused with the radial component process εt of the previous section.

22


Furthermore, this metric dynamical system is ergodic meaning that Pρε (A) ∈ {0, 1} on all sets A ∈ F which are (θ(t))t∈R+ -invariant, i.e. θ(t)−1 A = A for all t ∈ R+ . This is due to the fact that the measures which are ergodic in this sense are the extreme points of the convex set of (θ(t))t -invariant measures; however, due to the postulated uniqueness of ρε , there is only one such measure which is Pρε ; see Arnold [Ar 98, p.539]. ε Let X ε,ρ denote the stationary Markov process which realizes Pρε as defined above. This process is now specified as stochastic input for the differential system (1); more precisely, consider the RDE ε dZtε = A Xtε,ρ (ω) Ztε dt ,

Z0ε = z ∈ Kn

and let Φε : R+ × Ω × Kn −→ Kn , (t, ω, z) −→ Φε (t, ω, z) ≡ Φε (t, ω)z t ε =z + A Xuε,ρ (ω) Φε (u, ω, z) du 0

denote its unique (up to indistinguishability) solution. Then the following holds true: 1) Φε is B(R+ ) ⊗ F ⊗ B(Kn )/B(Kn )-measurable, 2) Φε is a cocycle over (θ(t))t∈R+ in the sense that for all ω ∈ Ω and s, t ∈ R+ : Φε (0, ω) = idKn

and

Φε (t + s, ω) = Φε (t, θ(s)ω) ◦ Φε (s, ω) ,

3) Φε ( . , ω, . ) is continuous for any ω ∈ Ω, 4) Φε (t, ω) ≡ Φε (t, ω, . ) is linear for any t ∈ R+ and ω ∈ Ω ; see Arnold [Ar 98, 2.2.12]. These properties are summarized by calling Φε a linearrandom dynamical system (RDS) on Kn over the metric dynamical system Ω, F , Pρε , (θ(t)t∈R+ with time R+ ; see Arnold [Ar 98, p.5f.]. In particular, since Φε is defined by the characterizing RDE (1), it will be called the RDS generated by (1). Being the solution of a linear differential equation the linear operator Φε (t, ω) is even invertible, i.e. Φε (t, ω) ∈ GL(n, K) for all t ∈ R+ and ω ∈ Ω. The pathwise Jacobi (Liouville) equation (1.2) now reads t

ε det(Φε (t, ω)) = exp trace A Xuε,ρ (ω) du . (1.12) 0


23

The adjoint matrix of Φε (t, ω) will be denoted by Φε (t, ω)∗ in the sequel. Then the Multiplicative Ergodic Theorem states the following in the above setting: Theorem 1.3.1 (Multiplicative Ergodic Theorem). Suppose that X ε has a unique stationary distribution ρε and let Φε be the linear, invertible RDS generated by (1), where ε > 0. Assume that β + ∈ L1 (Pρε ) and β − ∈ L1 (Pρε ) which are defined by β ± (ω) := sup log+ Φε (t, ω)±1 , 0≤t≤1

where log+ a := max(0, log a) and . denotes the operator norm. Then ε ∈ F of full measure, Pρε (Ω ε ) = 1, there exists a (θ(t))t∈R+ -invariant set Ω such that the following holds: 1/2t lim Φε (t, ω)∗ Φε (t, ω) =: Ψε (ω)

t→∞

ε and is non-negative definite, its eigenvalues being given exists for any ω ∈ Ω by n values ε ε ε eΛ1 ≥ eΛ2 ≥ · · · ≥ eΛn > 0 , ε ; writing the where Λε1 ≥ Λε2 ≥ · · · ≥ Λεn > −∞ do not depend on ω ∈ Ω distinct numbers in this list of eigenvalues of Ψε (ω) as ε

ε

ε

eλ1 > eλ2 > · · · > eλpε > 0 and defining dεi as the counting multiplicity of λεi in the list {Λε1 , . . . , Λεn }, it follows that dεi = dim Uiε (ω)

ε and i ∈ {1, . . . , pε } , for all ω ∈ Ω ε

where Uiε (ω) denotes the eigenspace of Ψε (ω) corresponding to eλi . Further define Upε (ω) ⊕ · · · ⊕ Uiε (ω) , i ∈ {1, . . . , pε } ε Vi (ω) := {0} , i = pε + 1 ε , entailing the flag for any ω ∈ Ω ε {0} ≡ Vp+1 (ω) ⊂ Vpε (ω) ⊂ · · · ⊂ V1ε (ω) = Kn .

Then the Lyapunov exponent λε (ω, z) := lim

t→∞

1 log | Φε (t, ω)z | t

24


ε and z ∈ Kn \ {0} and it holds that exists for all ω ∈ Ω ε z ∈ Viε (ω) \ Vi+1 (ω) ⇐⇒ λε (ω, z) = λεi .

The latter fact is equivalent to the following characterization, Viε (ω) = z ∈ Kn : λε (ω, z) ≤ λεi . ˜ ε and t ∈ R+ it holds that For all ω ∈ Ω λε (θ(t)ω, Φε (t, ω)z) = λε (ω, z)

(z ∈ Kn \ {0})

and hence Φε (t, ω) Viε (ω) = Viε (θ(t)ω)

(i ∈ {1, . . . , pε }) .

The Multiplicative Ergodic Theorem has initially been proved by Oseledets [Os 68]. The one-sided version considered above is taken from Arnold [Ar 98, 3.4.1]. The fact that the proof there is presented for the case K = R is no restriction: According to Arnold [Ar 98, 3.4.10.(ii)], all arguments hold true for the complex case, K = C, as can be also read off from Ruelle’s [Ru 82] generalization to real or complex Hilbert spaces. The numbers Λε1 , Λε2 , . . . , Λεn are the possible exponential growth rates of ε Φ due to the above theorem and are called the Lyapunov exponents of Φε (under ρε ). Furthermore the proof of the Multiplicative Ergodic Theorem (via the Furstenberg-Kesten Theorem) and the above version (1.12) of the pathwise Jacobi (Liouville) equation yield together with the ergodic theorem that n i=1

ε

Λεi

≡

p

dεi λεi

i=1

1 log | det Φε (t, . ) | Pρε -a.s. t t

ε 1 trace A Xuε,ρ ( . ) du = lim t→∞ t 0 ≡ trace A(x) ρε (dx) Pρε -a.s. , = lim

t→∞

(1.13)

Rd

provided that A ∈ L1 (ρε ). This formula is called the trace formula for the sum of the Lyapunov exponents; see Arnold [Ar 98, 3.4.15, 3.3.11 & 3.3.4] and Oseledets [Os 68, p.203]. Further note that the previous condition A ∈ L1 (ρε )

(1.14)


25

is sufficient for the integrability assumption β ± ∈ L1 (Pρε ) of the Multiplicative Ergodic Theorem, since by the defining RDEs for Φε and (Φε )−1 , ε Φ (t, ω)±1 ≤ 1 +

t

ε A Xuε,ρ (ω) Φε (u, ω)±1 du

0

and hence by the general version of the Gronwall lemma (see e.g. Arnold [Ar 98, p.557]), t

ε ε,ρε Φ (t, ω)±1 ≤ exp A Xu (ω) du 0

which implies that ± β (ω) Pρε (dω) ≡

sup log+ Φε (t, ω)±1 Pρε (dω)

Ω 0≤t≤1

Ω

t

ε,ρε sup log exp A Xu (ω) du Pρε (dω)

≤

+

Ω 0≤t≤1 1

ε,ρε A Xu (ω) du Pρε (dω)

= Ω 1

=

0

0

1

= 0 = Rd

0

ε,ρε A Xu (ω) Pρε (dω) du Ω ε −1 (dx) du A(x) Pρε ◦ Xuε,ρ Rd

A(x) ρε (dx)

λ2 > · · · > λp consisting of the distinct real parts of eigenvalues of A, where Ei is the sum of the generalized eigenspaces to eigenvalues with real part equal to λi and is dynamically characterized as lim

t→±∞

1 log | zt | = λi ⇐⇒ z0 ∈ Ei \ {0} . t

In this terminology, the above spaces Eλ are given as the direct sums of the corresponding spaces E1 , . . . , Ep . The solution flow can also be described in the Riesz calculus as 1 ( χ In − A )−1 etχ dχ , etA = 2πi ∂D where the interior of the domain D now contains all eigenvalues of A and ∂D is supposed to be rectifiable. For the above statements see e.g. Coppel [Cp 65, Sec. II.1 & III.2], Riesz and Sz¨ okefalvi-Nagy [Ri-Na 55, Ch.XI], Kato [Kt 80, §I.5] and Arnold [Ar 98, 3.2.3]. Now we discuss the result by Hartman and Wintner [Ha-Wi 55] and Perron [Pe 29]. As the above cited Oseledets-splitting and the corresponding Lyat punov spectrum for the constant coefficient differential equation dz dt = A zt is derived from the Jordan canonical form of the matrix A, it does not come as a surprise that the same strategy is used for the perturbed equation (1.19). The following lemma will turn out to be crucial in this argumentation. In the sequel let t0 be an arbitrary initial time of the system and let . denote the operator-norm. Furthermore, d/dt and the dot “ · ” will be used interchangeably for the derivative with respect to the “time” variable t.

1.4 The deterministic case: Asymptotically constant linear systems

31

Lemma 1.4.2 (Riccati-type differential inequality). Let v C 1 ([t0 , ∞), R) satisfy the differential inequality v(t) ˙ ≥ β v(t) − γ(t) (t > t0 ) ,

∈

where γ ∈ C 0 ([t0 , ∞), R+ ) fulfills

t+1

t→∞

γ(u) du −−−−→ 0 t

and β is a real-valued function; let v ∗ ∈ R be a subsequential limit point of v and assume that β is continuous at v ∗ . Then β(v ∗ ) ≤ 0 . In particular, if v satisfies the Riccati-type differential inequality v(t) ˙ ≥ b v(t) 1 − v(t) − γ(t) (t > t0 ) , where b > 0 is constant and γ is as above, then either lim sup v(t) ≤ 0

or

t→∞

lim inf v(t) ≥ 1 . t→∞

Proof. 1) In order to obtain a contradiction we assume that β(v ∗ ) > 0. As v ∗ is a continuity point of β, this implies that one can choose constants η1 , η2 > 0 such that β(v) ≥ η1 > 0

v ∈ Bη2 (v ∗ ) ,

for all

where the latter set Bη2 (v ∗ ) denotes the closed ball with center v ∗ and radius η2 . t+1

Due to the convergence assumption on t γ(u)du , one can fix a positive constant η3 < min(η1 , η2 ) and a time t1 ≥ t0 such that

t+1

γ(u) du ≤ η3

(t ≥ t1 ) ;

t

in particular, this implies that

t+T

γ(u) du ≤ (T + 1) η3

(t ≥ t1 , T > 0) .

t

Let [τ, τ + T ] denote some time interval such that τ ≥ t1 , T > 0 and v(t) ∈ Bη2 (v ∗ )

for all

t ∈ [τ, τ + T ] .

32


Applying the proposed differential inequality v(t) ˙ ≥ β(v(t)) − γ(t) over this τ +• time interval [τ, τ + T ] and using the previous estimate of τ γ(u) du now yields that τ +s τ +s v(τ + s) − v(τ ) = v(u) ˙ du ≥ β v(u) − γ(u) du τ

τ

≥ η1 s − (s + 1) η3 > η3 s − (s + 1) η3

(1.20)

= − η3 for all s ∈ [0, T ]. Next we use this inequality (1.20) to prove that t1 can be enlarged such that also for all t ≥ t1 ; |v(t) − v ∗ | < η2 for this purpose let [ τ, τ + T ] denote some interval such that τ ≥ t1 , T > 0 and ( τ ≤ t ≤ τ + T) ; |v(t) − v ∗ | < η2 − η3 as v ∗ is a subsequential limit of t → v(t), such time intervals exist for any τ) ∈ previous choice of t1 ; due to the above reasoning (1.20) and since v( [v ∗ − (η2 − η3 ), v ∗ + (η2 − η3 )], v ( τ + s )

(1.20)

>

v( τ ) − η3 ≥ v ∗ − η2

(0 ≤ s ≤ T)

so that t → v(t) cannot exit the interval [v ∗ − η2 , v ∗ + η2 ] via the lower boundary; on the other hand, the same reasoning, since also v( τ + T) ∈ ∗ ∗ [v − (η2 − η3 ), v + (η2 − η3 )], yields that v ( τ)

(1.20)

0, hence implying that T →∞

v(τ + T ) ≥ [v(τ ) − η3 ] + (η1 − η3 ) T −−−−→ ∞ in contradiction to the previously proven boundedness |v(t) − v ∗ | < η2 for large t.


33

2) Now we consider the Riccati-part of the statement, i.e. the case that β(v) := b v (1 − v) . Since b > 0, β is a downward sloped quadratic function which is strictly positive on (0, 1), strictly negative on R \ [0, 1] and vanishes at 0 and 1. Now let v ∗ be some subsequential limit point of v. Then the first part of the lemma, shown above, yields that β(v ∗ ) ≤ 0 which — according to the special choice of β — is only possible for v ∗ ∈ / (0, 1). But this is just a reformulation of the above claim. Theorem 1.4.3 (Hartman-Wintner-Perron). Let K denote the real or complex number field, K = R or C, and consider the linear ODE dzt = [ A + G(t) ] zt dt

(1.19)

in Kn , where A ∈ Kn×n is constant and the continuous map G : [t0 , ∞) → Kn×n satisfies t+1 t→∞ G(u) du −−−−→ 0 . (1.21) t

Then either zt = 0 for all large t or the Lyapunov exponent of zt exists and is equal to the real part of one of the eigenvalues Λj of A, i.e.3 : λ(zt ) := lim

t→∞

1 log | zt | ∈ { −∞ , Re(Λ1 ) , . . . , Re(Λn ) } . t

Proof. We first choose an enumeration of the eigenvalues such that Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) . After a constant, invertible coordinate transformation we can assume that A is given in (complex) Jordan canonical form ⎛ ⎞ Λ1 0 0 ⎜ a Λ2 0 ⎟ ⎜ ⎟ ⎜ 0 a Λ3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Λ 4 A = ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ Λn−1 0 ⎠ a Λn 3 Here we adopt to the common convention that the Lyapunov exponent of the trivial solution zt = 0 is defined as λ(0) := −∞. Furthermore, Re( . ) denotes the real part operation as usual.

34


where in this example Λ1 = Λ2 = Λ3 and Λn−1 = Λn , so that the boxes depict lower Jordan blocks of the following qualitative shape: ⎞ ⎛ Λ 0 ⎟ ⎜ a Λ 0 ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎟ ⎜ a . . . ⎟ ; ⎜ .. .. .. J := ⎜ ⎟ ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎝ a Λ 0 ⎠ a Λ here, a > 0 is some positive parameter and all matrix entries not mentioned are zero. Note that we have taken the “ordinary” lower Jordan blocks J 1 and carried out another transformation z˜t := Czt , where C := diag(1, a, . . . , an−1 ) ; then d d z˜t = C zt = CAzt + CG(t)zt ≡ CAC −1 z˜t + CG(t)C −1 z˜t ; dt dt hence, we can redefine A as CAC −1 and G(t) as CG(t)C −1 ; after this transformation the Jordan-blocks of A are of the form Jja and G(t) still enjoys the prescribed assumption. The possible limits of 1t log |zt | are invariant under such invertible transformation of variables (or more generally under so-called Lyapunov-transformations). More precisely, the norm |C . | on Kn is equivalent to the “old” norm | . | as Kn is finite dimensional; since t → ∞, factors from converting these norms into each other, end up as vanishing summands; therefore, we can redefine zt as z˜t for the following considerations without loss of generality. d zt = [ A + G(t) ] zt , is of the form Hence, the underlying ODE (1.19), dt d 1 z = Λ1 z 1 + (Gz)1 dt d i z = ai−1 z i−1 + Λi z i + (Gz)i dt

(i = 2, . . . , n) ,

where the superscript i indicates the i-th coordinate and ai := Ai+1,i ∈ {0, a}

(i = 1, . . . , n − 1)

denotes the n − 1 entries of A below the diagonal. Putting a0 := 0 as well as z 0 := 0 for definiteness and defining


35

it := zti , this implies that

d i i d i 2 d i i d i i z¯ z = z z + z¯ z = dt dt dt dt

d i z = 2 Re z¯i dt i = 2 Re z¯ ai−1 z i−1 + Λi z i + (Gz)i = 2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re(¯ z i (Gz)i )

(1.22)

for all i = 1, . . . , n , where “ ¯ ” denotes complex conjugation; hence, we can estimate d i 2 i 2 z i z i−1 ) + 2 Re(¯ z i (Gz)i ) dt − 2 Re(Λi ) = 2ai−1 Re(¯ (1.23) i i−1 i i ≤ 2ai−1 + 2 |(Gz) | . Now define λ1 > λ2 > · · · > λp as the distinct numbers in the list Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) . Note in particular that by definition, for all k = 1, . . . , p i := min{ i : Re(Λi ) = λk }

implies that

ai−1 = 0 .

(1.24)

We now define auxiliary processes describing the norms of the projection of zt onto the subspaces belonging to the different real parts of eigenvalues; more precisely: Lkt := ( it )2 , k = 1, . . . , p {i : Re(Λi )=λk }

Mtk :=

Ntk

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

( it )2

,k = 1 , k = 2, . . . , p

{i : Re(Λi )>λk }

⎪ n ⎪ ⎪ ⎪ ⎪ ( it )2 ≡ | zt |2 ⎩ ⎧ ⎪ ⎨

:=

i=1

( it )2

,k = p + 1 , k = 1, . . . , p

{i : Re(Λi )≤λk }

⎪ ⎩0

,k = p + 1 .

36


Then | zt |2 = Mtk + Ntk = L1t + · · · + Lpt . Now it follows that k dL k dt − 2λk L + , i 2 d( ) = − 2λk ( i )2 dt {i : Re(Λi )=λk } ≤2 ai−1 i i−1 + 2 {i : Re(Λi )=λk }

⎛

⎞1/2 ⎛

≤2⎝

( i )2 ⎠

{i : Re(Λi )=λk }

⎛

i |(Gz)i |

{i : Re(Λi )=λk }

⎝

⎞1/2

a2i−1 ( i−1 )2 ⎠

{i : Re(Λi )=λk }

⎞1/2 .1/2 n i 2⎠ i 2 ( ) |(Gz) |

{i : Re(Λi )=λk }

i=1

+ 2⎝ ⎛

⎞ 1/2

= 2 (Lk )1/2 ⎝

a2i−1 ( i−1 )2 ⎠ + 2 (Lk )1/2 |(Gz)|

{i : Re(Λi )=λk , i−1 ≥ i }

⎛

≤ 2 a (Lk )1/2 ⎝

(1.25)

⎞1/2

( i )2 ⎠

+ 2 (Lk )1/2 G |z|

{i : Re(Λi )=λk }

1/2 = 2 a Lk + 2 G (Lk )1/2 M k + N k . / Mk + Nk k 2 a + 2 G =L Lk

(1.26)

for all k = 1, . . . , p , where we used in consecutive order the definition of Lkt , the calculation (1.23), the Cauchy-Schwarz inequality, (1.24), the fact that ai ∈ {0, a} and (1.25). Similarly, it follows from the definition of Mtk , (1.22), the estimate that Re(a b) ≥ −|a| |b| for complex numbers a and b, (1.24) and (1.25) that d Mk = dt =

{i : Re(Λi )>λk }

{i:Re(Λi )>λk }

0

d i 2 ( ) dt 1 2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re z¯i (Gz)i


≥ −2

ai−1 i i−1 + 2λk−1

{i : Re(Λi )>λk }

( i )2

{i : Re(Λi )>λk }

− 2

37

| (Gz) | i

i

{i : Re(Λi )>λk }

≥ − 2 a M k + 2 λk−1 M k − 2 G (M k )1/2 |z| √ = 2 (λk−1 − a) M k − 2 G M k M k + N k

(1.27)

for all k = 2, . . . , p + 1. In the same manner it follows from the definition of Ntk , (1.22), the estimate that Re(a b) ≤ +|a| |b| for complex numbers a and b, (1.24) and (1.25) that d Nk = dt


=

d i 2 ( ) dt

2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re(¯ z i (Gz)i )

{i:Re(Λi )≤λk }

≤2


+ 2

ai−1 i i−1 + 2λk

( i )2


| (Gz) | i

i


≤ 2 a N k + 2 λk N k + 2 G (N k )1/2 |z| √ = 2 (λk + a) N k + 2 G N k M k + N k .

(1.28)

for all k = 1, . . . , p. From (1.27) and (1.28) one gets for |z|2 ≡ M p+1 ≡ N 1 that d |zt |2 ≤ 2 λ1 + a + G(t) |zt |2 , 2 λp − a − G(t) |zt |2 ≤ dt so that integration over t ≥ t0 yields that t G(s) ds |zt0 | ≤ |zt | exp (λp − a)(t − t0 ) − t0

t ≤ exp (λ1 + a)(t − t0 ) + G(s) ds |zt0 | . t0

This proves again (besides the invertibility of the fundamental matrix solution) that if zt1 = 0 for some time t1 , then also zt = 0 for all times t ≥ t1 . Furthermore, due to the standing assumption (1.21) it follows that 1 t

t

t→∞

G(s) ds −−−−→ 0 t0

38


and hence the previous string of inequalities implies the statement about the Lyapunov exponent for a non-vanishing solution in case p = 1: 1 t→∞ log | zt | −−−−→ λ1 t since a can be chosen arbitrarily small. Thus it is left to show the claim concerning the Lyapunov exponents in case p ≥ 2; again a > 0 serves as the small parameter and since p ≥ 2, we can choose it such that 2a
0 due to (1.29) and γ := 2 G due to (1.21). Therefore, as v ≡ v k takes its values in [0, 1], lim vtk ∈ {0, 1}

t→∞

(k = 1, . . . , p + 1) .

Since v 1 ≡ 0 and v p+1 ≡ 1, the following quantity is thus well-defined: J := max k = 1, . . . , p : lim vtk = 0 . t→∞

This implies, as M k+1 = L1 + · · · + Lk , that for k = 1, . . . , p Mtk+1 − Mtk Lkt 0 , k = J ; k+1 k t→∞ = = vt − vt −−−−→ 1 , k = J . |zt |2 |zt |2

(1.31)

In particular one also gets Lkt | zt |2 t→∞ Lkt = −−−−→ 0 J | zt |2 LJ Lt t

(k = J ).

The importance about (1.31) is that it reduces the study of |zt | to large t, namely 2 log

LJ t − log |zt | =

(1.32) 2 LJ t for

LJ t→∞ 1 log t 2 −−−−→ 0 2 |zt |

and hence the claim (1.30) concerning the Lyapunov exponent λ(zt ) reduces to proving that lim

t→∞

1 log LJ t = 2 λJ . t

40


Considering (1.26) for k = J after dividing by LJ t (= 0) now yields: t

1 1 d J J log Lu − 2 λJ du lim sup log Lt − 2λJ = lim sup t t du t→∞ t→∞ t0 t 1 d log LJ ≤ lim sup − 2λ J du u t→∞ t t0 du 3 (1.26) 1 t MuJ + NuJ ≤ 2a + 2 lim sup G(u) du LJ t→∞ t t0 u t (1.31) 1 ≤ 2 a + 4 lim sup G(u) du t t0 t→∞ (1.21)

= 2a.

Since a can be chosen arbitrarily small, this shows that the Lyapunov exponent λ(zt ) exists and is equal to λJ . Remark 1.4.4 (Asymptotic growth of the projections). Due to the structure of the differential equation in Jordan canonical form in the proof of theorem 1.4.3, it follows in particular from remark 1.4.1 that for any k = 1, . . . , p 2 Lkt = Pλk zt . Thus (1.32) implies that by choosing λ := λJ ≡ λ(zt ), the Lyapunov exponent of one particular solution zt , one gets that P≶λ zt t→∞ −−−−→ 0 . | Pλ zt | Remark 1.4.5. The proof of the above theorem 1.4.3 shows that the linearity of the perturbation G(t)z is not crucial. The argument also goes through unchanged for the nonlinearly perturbed equation dzt = [ A zt + g(t, zt ) ] dt , where the continuous map g : [t0 , ∞) × Rn → Rn satisfies | g(t, z) | ≤ γ(t) |z| together with

t+1

t→∞

γ(u) du −−−−→ 0 . t


41

Remark 1.4.6 (Kn = R2 ). As in theorem 1.4.3 again consider the linear ODE (1.19) dzt = [ A + G(t) ] zt dt under the same assumptions as imposed there; let Kn = R2 for simplicity. The exponential growth rate limt→∞ 1t log | zt | has been calculated as Re(Λ1 ) or Re(Λ2 ) in the non-trivial cases zt ≡ 0. Despite these numerically exact results, the formulas (1.10) or (1.11), 1 log | z0 | 1 log | zt | = + t t t

t

¯ αu ) du , Q(u, 0

which evaluate the growth rate of zt by a functional of its angle αt , have not been used in the course of the proof. It is the purpose of this remark to bridge this gap in the illustrative two-dimensional situation. If p = 1, i.e. if Re(Λ1 ) = Re(Λ2 ), we get in the notion of the previous proof (see p.35) that the norms of the relevant projections are L1t = Mt2 = Nt1 = |zt |2 and Mt1 = Nt2 = 0; the proof then estimates |zt |2 and does not contain a conclusion concerning the angle. If p = 2, i.e. if Re(Λ1 ) > Re(Λ2 ), the situation changes. Both eigenvalues are real, λ1 := Λ1 > Λ2 =: λ2 , each of which has an eigenvector. After the transformation mentioned at the beginning of the previous proof,

λ1 0 A = 0 λ2 and the canonical unit vector ei is the eigenvector corresponding to λi , i = 1, 2. Then it follows from the notion of p.35 that Lkt = ( kt )2 ⎧ ⎪ ⎨0 k Mt = ( 1t )2 ⎪ ⎩4 2

, k ∈ {1, 2} ,

i 2 2 i=1 ( t ) ≡ | zt | ⎧4 2 i 2 2 ⎪ ⎨ i=1 ( t ) ≡ | zt | Ntk = ( 2t )2 ⎪ ⎩ 0

,k = 1 ,k = 2 ,k = 3 , ,k = 1 ,k = 2 ,k = 3 ,

where as before it ≡ | zti |. The auxiliary processes v 1 , v 2 and v 3 as defined M1 M3 on p.38 read in this case as vt1 = | ztt|2 ≡ 0 , vt3 = | ztt|2 ≡ 1 and vt := vt2 :=

Mt2 ( 1t )2 ≡ = (cos αt )2 , 1 | zt |2 ( t )2 + ( 2t )2

42


.................. .. ......... ........ . t ............. .. . . . . . . . . . . . ......... ...... . . . . . . . . . . . ....... t .. ..... . . . . . . . . . .... .... ..... ... ........ ...... ..... ... ................ ........ ..... . .... .. . ... . ............. . . . . . ... .. . ... . . . . . . ... .. .. . . . . . ... .. .. . . . . . ... .. ... . . . . ... .. ... ... ... t .. ... ... .. ..... .. ... .. ....... .. . . .... ... ... ... .. . . .. ... . . ... ... . .. .. ... . . .. . . t . ... ... . .. ... .. ... .. 1 . . ... . . . t ... ... ... ... ... ... ... t ... .. . ... ... ... ... ... ... ... ... ... . . . ... .... .... ... ..... ..... ...... ..... ...... ...... . ....... . . . . ... ........ ........ .......... ................ .......... ...................................

e2 ............. .................r

z

s |z |

sin α

α

| cos α | √ =: v

r

e

Fig. 1.1 Geometrical interpretation of the quantity vt := (cos αt )2

where αt denotes the angle of zt as measured canonically with respect to the e1 -coordinate axis. As figure 1.1 illustrates, vt = cos2 αt quantifies the distance between |zztt | and e2 and hence induces a metric on the projective space S 1 . Another popular choice to obtain a metric on S 1 is to work with | sin αt | which measures the distance between the projective lines |zztt | and e1 ; the latter metric is commonly used in the proof of the Multiplicative Ergodic Theorem; see Arnold [Ar 98, Prop. 3.2.8 & Lem. 3.4.6]. The system vt then satisfies the ODE d 1 2 d 2 2 ( 2t )2 dt ( t ) − ( 1t )2 dt ( t ) v˙ t = 4 |zt | from which the following Riccati-type differential inequality follows (see p.38f.), v˙ t ≥ 2 (λ1 − λ2 ) vt ( 1 − vt ) − 2 G(t) . Lemma 1.4.2 applies with b := 2 (λ1 − λ2 ) > 0 and γ := 2 G due to (1.21). Therefore, lim vt ∈ {0, 1} t→∞

and the following quantity is thus well-defined, 1 , if limt→∞ vt = 1 , J := 2 , if limt→∞ vt = 0 .


43

. It displays the direction eJ to which the projective line ψt := |zztt | = αt converges; see figure 1.1; due to continuity the angle also converges, α∞ := limt→∞ αt ∈ {0, π2 , π, 3π 2 }. Therefore, it follows altogether that t→∞ ¯ ¯ A + G(t), αt ≡ Q A + G(t), ψt − α∞ ) −−− → Q A, eJ ≡ Q(A, Q and hence the formulas (1.10) and (1.11) imply that 1 log |z0 | 1 log |zt | = + t t t

t

t→∞ ¯ ¯ A + G(u), αu du − −−→ Q(A, α∞ ) ∈ {λ1 , λ2 }, Q

0

¯ (see (1.7)) as well as the diagonal where in the last step the definition of Q form of A have been used. Thus the gap between theorem 1.4.3 and the formulas (1.10) and (1.11) is closed. The following theorem shows that every real part of an eigenvalue can indeed be seen as a Lyapunov exponent. Since it will not be used in the sequel, the arguments will not be provided here. Instead, see Coppel [Cp 65, p.100–102] for the proof which consists of using remark 1.4.1, defining an appropriate integral transformation and applying the fixed point theorem by Schauder and Tychonoff then. Theorem 1.4.7. Suppose that the assumptions of the Hartman-WintnerPerron theorem 1.4.3 hold, i.e. that we consider dzt = [ A + G(t) ] zt dt ,

(1.19)

where A is a constant matrix and G is a continuous matrix function again satisfying t+1 t→∞ G(u) du −−−−→ 0 . (1.21) t

Then for any real part λ of an eigenvalue Λi of A, λ ∈ { Re(Λ1 ) , . . . , Re(Λn ) } , there exists an η ≡ η(A) > 0 and a time T ≡ T (A, G) > t0 such that for all t1 ≥ T and for all χ 0, but not on ε ; it is the rate function for the large deviation principle xy x as the restriction of I0T to for X ε,x as we will later see. We also define I0T all paths f such that f (T ) = y. The quasipotential for the SDE (2.1) is then defined as V : Rd × Rd −→ R+ , xy (.) V (x, y) := inf I0T T >0 x ≡ inf I0T (f ) : f ∈ C([0, T ], Rd) , f0 = x , fT = y , T > 0 ;

it describes the “minimal cost” of forcing X ε to connect x and y eventually. Now the assumptions are listed under which (2.1) is investigated: Assumption 2.1.1 (on the coefficients of SDE (2.1)). The drift b ∈ C ∞ (Rd , Rd ) and the diffusion term σ ∈ C ∞ (Rd , Rd×d ) are supposed to further satisfy: (S) For all ε > 0 and x ∈ Rd , there exists a unique strong, non-exploding solution X ε,x to the SDE (2.1). (E) σ takes its values in the invertible matrices, σ −1 ∈ C ∞ (Rd , Rd×d ), and a ≡ σσ ∗ is strictly positive definite, i.e. there is a constant c > 0 such that c |x2 |2 ≤ a(x1 )x2 , x2 ≤ c−1 |x2 |2

(x1 , x2 ∈ Rd ) .

(K) There exist finitely many points K1 , . . . , Kl ∈ Rd such that (K1) any trajectory Xt0,x (except those in a finite union of lower dimensional submanifolds of Rd ) of the deterministic system dXt0 = b Xt0 dt is attracted to one of the points Ki as t → ∞ ; (K2) each Ki is stable, i.e. there exist open sets Ui ⊃ Ki such that Ui is attracted to Ki under the deterministic motion X 0 . (V)

lim V (0, y) = ∞ .

|y|→∞

(G) The system is generic in the sense that there are no symmetries in the function V . More precisely, it is assumed that the minima and maxima in equations (2.11)–(2.17) below are attained at only one point; this means (in the terminology coined below) that the main state, the rotation rate and the exit rate of cycles of any order shall be well defined.

2.1 Preliminaries and assumptions

57

Remark 2.1.2 (on the set of assumptions 2.1.1). on (S): If σ(x) = idRd for all x ∈ Rd , a sufficient condition for (S) is e.g. that for some constant c > 0 and all x ∈ Rd , x, b(x) ≤ c 1 + |x|2 ; see Stroock and Varadhan [Str-Vdh 97, Th. 10.2.2]. Since (K) forces all trajectories to converge to one of the finitely many attractors Ki , one might think of a drift b for which x, b(x) < 0 outside some sufficiently large ball. on (E): Condition (E) (“ellipticity” of (2.1) ) makes sure that the generator G ε is an elliptic differential operator and also bounds the covariance a from above. (E) makes sure that the Markov transition probabilities Ptε (x, . ) have Lebesgue densities; together with assumptions (B) defined below, (E) guarantees the existence of an invariant probability measure ρε with Lebesgue density pε (x) > 0 to which the Markov transition probabilities converge; see the next section. The crucial point here is that we will use time scales T (ε) “below which” this limiting distribution ρε is observed, so this invariant measure does not play a decisive role in the subsequent analysis; however, we will use it to emphasize the different behaviors of the transition probabilities in the limiting and the sublimiting cases, respectively; see sections 2.2 and 2.5. From an applications’ point of view the existence of an invariant probability is a reasonable property; see section 2.6. on (E): Due to assumption (E), V is indeed a well defined function in R+ . on (K): This assumption, taken from Freidlin [Fr 77], allows to approximate the behavior of X ε by the one of a certain Markov chain whose finite state space is given by the union of small spheres around the points Ki . It is this construction which backs the exit time law 2.5.3 for cycles. For those initial conditions x for which X 0,x does not converge to one of the points K1 , . . . , Kl , there exist unstable equilibria Kl+1 , . . . , Kl to which the respective trajectories converge; these trajectories separate the domains of attraction D1 , . . . , Dl of the respective points K1 , . . . , Kl and therefore form separatrices of the deterministic dynamical system X 0 ; hence, these trajectories (and their initial values) constitute a finite union of lower dimensional submanifolds of Rd separating the domains D 1 , . . . , Dl . Conversely, one could start with all equilibria K1 , . . . , Kl of the deterministic dynamical system X 0 and then single out the stable ones K1 , . . . , Kl , where l ≤ l ; see Freidlin [Fr 00, p.337].

58

2 Locality and time scales of the underlying non-degenerate system

on (K): The sets Ki need not be one point sets: Instead one can also allow K 1 , . . . , K l ⊂ Rd to be just compact sets. However, since the concepts of the current chapter will be applied to the case where the Ki ’s are one point sets, we formulate this assumption as it will be used. In this more general situation, where the Ki ’s are compact sets, the statements of this chapter would hold true, if in (K) we add two more subassumptions: (K3) if x, y ∈ Ki , then x ∼ y (1 ≤ i ≤ l); (K4) if x ∈ Ki and y ∈ Kj , then x y (i = j) ; thereby we made use of the following equivalence relation which is induced by the quasipotential V : x ∼ y :⇐⇒ V (x, y) = V (y, x) = 0 . In this general situation these sets Ki are for example the limit cycles of the ODE x˙ = b(x) ; consider e.g. the drift

−x2 − x1 (x21 + x22 )2 − 3(x21 + x22 ) + 2 , b(x) := x1 − x2 (x21 + x22 )2 − 3(x21 + x22 ) + 2 which is taken from Jetschke [Je 89, p.55]; note that this drift decomposes orthogonally as b = −∇U + L (see 2.4.5 on the implications for the quasipotential), where

1 3 −x2 U (x) := |x|6 − |x|4 + |x|2 and L(x) := x1 6 4

;

in polar coordinates, := |x| and α := arctan xx21 , this ODE x˙ = b(x) can be rewritten as ˙ = − 5 + 3 3 − 2 ≡ − U ( ) = − ( 2 − 1) ( 2 − 2) α˙ = 1 ; since U (x) = U ( ) only depends on the absolute value of the position, this implies that b satisfies (K1)–(K4) with the stable sets K1 := {0} and K2 := { x ∈ R2 : |x| = 2 }. The invariant set { x ∈ R2 : |x| = 1 } violates the stability criterion (K2) and hence constitutes the saddle for the radial motion. An even simpler example for this phenomenon of a limit cycle is the twodimensional family of drifts

2.1 Preliminaries and assumptions

59

−x2 + x1 η − (x21 + x22 ) bη (x) := x1 + x2 η − (x21 + x22 ) which represents the normal form of the Hopf-bifurcation, where η ∈ R denotes the bifurcation parameter; cf. Guckenheimer and Holmes [Gu-Hl 83, p.146f.], Andronov et al. [And-Pon-Vi 33, §5] and Jetschke [Je 89, p.158;52,226]; a slight generalization of this system is considered by Leng et al. [Lg-SN-Tw 92, Ex.3]. Also in this case the drift decomposes orthogonally as bη = −∇Uη + L , where 1 η Uη (x) := |x|4 − |x|2 4 2

and

−x2 ; L(x) := x1

in polar coordinates this ODE can be rewritten as ˙ = − 3 + η ≡ − Uη ( ) = (η − 2 ) α˙ = 1

;

again Uη (x) = Uη ( ) only depends on the absolute value of the position, and bη satisfies (K) with the only stable set {0} , η≤0 K(η) := √ 2 { x ∈ R : |x| = η } , η > 0 . If η ≤ 0, the qualitative behavior of xt resembles the motion in example 2.6.1; if η > 0, then the origin is a repelling fixed point. on (K): A sample drift which violates the stability assumption (K) is the following “mock” Van der Pol-ODE given in polar coordinates by ˙ = (1 − ) α˙ = 2 − cos α ; cf. Zeeman [Ze 88, p.126]. on (V): This assumption assures that the sets {y : V (x, y) ≤ c} which exhaust the state space Rd are all compact for any x and c ≥ 0. on (E) and (V): Instead of considering a diffusion on Rd , we can also take a diffusion X ε on a d-dimensional compact manifold M . In this case (E) and (V) are redundant except from assuming a to be positive definite on M . The (unique) stationary measure is already finite in this case; see example 2.6.11.

60


The compact setting is for example assumed by Wentzell and Freidlin [We-Fr 69] and Freidlin and Wentzell [Fr-We 98, §§ 6.4, 6.6]. One could equally well consider the “mixed” case with state space M × Rd for a compact manifold M under suitably adopted assumptions. on (G): As already mentioned, this assumption will be made precise below. In order to illustrate what might go wrong here, note that (G) is violated for example, if the potential function U in SDE (2.2) has wells of equal depth.

2.2 The limiting distribution (stationary measure) In this section we collect some facts on the connection between SDEs and partial differential equations (PDEs) to illustrate the behavior of the transition probabilities and of invariant measures. Since the contents of this section is well documented in textbooks, we refrain from giving proofs here and refer instead to standard monographs as e.g. Friedman [Fri 75, Ch. 6], Friedman [Fri 76, Sec. 14.4], Khasminskii [Kh 80, Ch. III & IV], Stroock and Varadhan [Str-Vdh 97], Wentzell [We 79, §11.2] and Fleming and Rishel [Fl-Ris 75, Ch.V]. Also see the expositions by Freidlin and Wentzell [Fr-We 98, Ch.1 & 4], Khasminskii [Kh 60], Ichihara and Kunita [Ic-Ku 74], Andronov et al. [And-Pon-Vi 33], Bernstein [Bs 33], Horsthemke and Lefever [Hh-Lf 85, Sec. 4.4] and Hackenbroch and Thalmaier [Hb-Th 94, Sec. 6.6]; recent results are due to Pardoux and Veretennikov [Pd-Ver 01] for example. In order to emphasize the difference between the “forward variables” and the “backward variables” in the “forward equation” and the “backward equation” to come, we allow for time-dependent coefficients in this section, i.e. we consider the following SDE in Rd , √ dXtε,x = b (t, Xtε,x ) dt + ε σ (t, Xtε,x ) dWt (t ≥ s ≥ 0) , (2.3) ε,x d Xs = x ∈ R , where the coefficients are functions b ∈ C ∞ (R+ × Rd , Rd ) and σ ∈ C ∞ (R+ × Rd , Rd×d ) and where s ≥ 0 denotes some initial time; assume that any σ(t, . ) satisfies the ellipticity condition (E) for some universal constant c > 0. For simplicity it is furthermore assumed that all coefficient functions b and a ≡ (σσ ∗ ) as well as their first derivatives are bounded (with respect to the respective norms). The latter assumption can be relaxed (see Friedman [Fri 75, p.147]); however, since the main focus of this paper lies on the behavior on bounded domains and during time scales, this assumption does not restrict the scope of this section for illustrative purposes.

2.2 The limiting distribution (stationary measure)

61

The partial differential operator associated with (2.3) is G ε (s, x) :=

d

bi (s, x)

i=1

d ∂ ε ∂2 + aij (s, x) . ∂xi 2 i,j=1 ∂xi ∂xj

In the sequel let Ps,x denote the probability measure P conditioned on events concerning the process X ε starting in x at time s ≥ 0 ; Es,x then denotes the corresponding expected value operator. ε {Xtε ∈ . } then possess densities with The transition probabilities Ps,x respect to the Lebesgue measure, ε {Xtε ∈ B} = pε (s, x; t, y) dy (s < t , B ∈ B(Rd ) ) . Ps,x B

These densities solve the backward (parabolic) equation −

d d ∂ pε (s, x; t, y) ∂ pε (s, x; t, y) ε ∂ 2 pε (s, x; t, y) = bi (s, x) + aij (s, x) ∂s ∂xi 2 i,j=1 ∂xi ∂xj i=1

≡ G ε (s, x) pε (s, x; t, y) ,

a PDE with respect to the backward (i.e. past) variables (s, x). Equivalently, the densities solve the forward (parabolic) equation d ∂ pε (s, x; t, y) ∂ = − bi (t, y) pε (s, x; t, y) ∂t ∂yi i=1 d ε ∂2 aij (t, y) pε (s, x; t, y) 2 i,j=1 ∂yi ∂yj ∗ =: G ε (t, y) pε (s, x; t, y) ,

+

a PDE with respect to the forward (i.e. future) variables (t, y); (G ε )∗ is the formal adjoint operator of G ε . Moreover, the densities are the fundamental solutions of the backward and the forward equation. In the time-homogeneous case, i.e. if b and σ are independent of t as in (2.1), one then gets for the transition densities pε (s, x; t, y) = pε (0, x; t − s, y) =: pεt−s (x, y)

(s < t) ,

from the above PDEs the Kolmogorov backward equation d d ∂ pεt (x, y) ε ∂ 2 pεt (x, y) ∂ pεt (x, y) = bi (x) + aij (x) ∂t ∂xi 2 i,j=1 ∂xi ∂xj i=1

≡ G ε (x) pεt (x, y)

62


and the equivalent Kolmogorov forward equation (Fokker-Planck equation) d d ε ∂ pεt (x, y) ∂ ∂2 = − bi (y) pεt (x, y) + aij (y) pεt (x, y) ∂t ∂y 2 ∂y ∂y i i j i=1 i,j=1 ε ∗ ε ≡ G (y) pt (x, y) .

Further on, consider the time-homogeneous case (2.1). A stationary probability distribution for X ε is a probability measure ρε on Rd which is invariant with respect to the Markov transition probabilities (Ptε )t , Ptε (x, . ) := Px { Xtε ∈ . } ≡ P{ Xtε,x ∈ . } in the sense that

ρε ( . ) = Rd

Ptε (x, . ) ρε (dx)

or equivalently f (x) ρε (dx) = Rd

(t ≥ 0, x ∈ Rd ) ,

Rd

Ttε f (x) ρε (dx)

(t ≥ 0) ;

(f ∈ C c (Rd , R) ) ,

where Ttε f (x)

:= Rd

f (y) Ptε (x, dy) .

The stationary measure ρε is the “equilibrium (limiting) distribution as t → ∞”, in the sense that for any x ∈ Rd t→∞ Ttε f (x) −−−−→ f (y) ρε (dy) (f ∈ C b (Rd , R) ) Rd

and t→∞

Ptε (x, B) −−−−→ ρε (B)

(2.4)

for all B ∈ B(Rd ) such that ρε (∂B) = 0. If a stationary probability distribution exists, assumption (E) implies its uniqueness and the existence of a Lebesgue density pε (x) > 0 of ρε , the reason being that (E) entails strong ellipticity in the Lie-algebraic sense of Ichihara and Kunita [Ic-Ku 74, p.250]. The densities pεt then converge to the “equilibrium distribution” in the sense that for all x, y ∈ Rd t→∞

pεt (x, y) −−−−→ pε (y) ; pε , being the Lebesgue density of ρε , uniquely solves the respective autonomous differential equations


63

0 = G ε (x) pε (x) (Kolmogorov backward equation) and 0 = (G ε (x))∗ pε (x) (Kolmogorov forward equation, Fokker-Planck equation) pε (x) dx = 1. For any f ∈ C b (Rd , R) and ε ∈ (0, ε0 ), the function

such

that

uεf (t, x) := Ex f (Xtε ) ≡ E [ f (Xtε,x ) ] is the unique solution of the Cauchy problem for (2.1): d d ∂uε ε ∂uε ∂ 2 uε = G ε uε ≡ bi + aij ∂t ∂xi 2 i,j=1 ∂xi ∂xj i=1

(t > 0),

(2.5)

uε (0, . ) = f. This function uεf converges to the “equilibrium distribution as t → ∞” as well, in the sense that for any x ∈ Rd t→∞ uεf (t, x) −−−−→ f (y) ρε (dy) . (2.6) Rd

Concluding these remarks, we note that a general, sufficient condition for the existence of a stationary probability distribution ρε is (E) together with (B) There exists a bounded domain S ⊂ Rd with smooth boundary such that for all compact sets K ⊂ Rd , sup Ex τSε,x < ∞ ,

x∈K

where τSε,x denotes the hitting time of S for the process X ε,x given by (2.1). (for a proof see Khasminskii [Kh 80, Sec.IV.4] and [Kh 60]). Remark 2.2.1 (Explicit formula for the stationary density in the gradient case). Suppose that the SDE under consideration is (2.2), i.e.: σ = idRd and b = −∇U for some suitable potential function U ∈ C ∞ (Rd , R) (U being in C 2 (Rd , R) would suffice here). Note that in dimension d = 1 any drift is the gradient x of a potential function, U (x) := − 0 b(s)ds. Then the above Fokker-Planck equation takes the form

64


d ε ∂ ∂U ε + pε = 0 , (G ) p ≡ p ∂xi ∂xi 2 i=1 ε ∗

ε

where denotes the Laplace operator. A direct calculation of the respective derivatives shows that x → e−2 U(x)/ε fulfills the above PDE ; hence, a stationary probability measure for (2.2) exists if and only if Nε := e− 2U(x)/ε dx < ∞ , Rd

in which case pε (x) = Nε−1 e− 2U(x)/ε

(x ∈ Rd ) .

Jacquot [Ja 92, p.347] shows that a sufficient condition for the finiteness of the normalization constant Nε is the following: (U)

U → ∞ and |∇U | → ∞ as |x| → ∞ .

Note, however, that nevertheless (U) does not imply that U has positive curvature for all sufficiently large x, as can be seen for example from U (x) := x2 2 − 2 cos x, where x ∈ R. Example 2.2.2 (Density pε for the potential function U 1 ). Consider for example (2.2) with the potential function U1 , 3 4 2 3 x1 − x31 − 3 x21 + x1 x2 + x42 , 2 3 2

U1 (x) :=

as in (1.34) of example 1.5.3, where c := 1. The potential is sketched in the figure 2.1, where the different depths of the wells are emphasized by drawing level sets, too. The density pε of the stationary measure has the following qualitative shape as depicted in the contour plot of figure 2.2: The peaks are located at 10 8 6 4 2 0

−2

−1 −0.5 x_2

0 0.5 1

−2

−1.5

−1

−0.5

Fig. 2.1 The potential function U1 (x1 , x2 ) =

0.5

0

1

1.5

2

x_1

3 2

x41 −

2 3

x31 − 3 x21 + x1 x2 +

3 2

x42


65

2.5 2 1.5 1 0.5 0

x_2

−1 −0.5

0 0.5 1

−2

−1.5

−1

−0.5

0 x_1

0.5

1

1.5

2

Fig. 2.2 Sketch of the density pε ∼ e−2U1 /ε

the (local) minima of the potential and the mass concentrates at the global minimum in the small noise limit ε → 0. This feature is characteristic for all potential SDEs (2.2): The local minima of U correspond to the local maxima of the density of the invariant distribution; these are the preferential states of the process X ε as t → ∞. Similarly, in the non-gradient case the system X ε from (2.1) accumulates near the points K1 , . . . , Kl ; a corresponding statement for X ε on time scales T (ε) will appear later (see theorems 2.5.5 and 2.5.6). For the Ornstein-Uhlenbeck process even the time-dependent transition densities can be obtained explicitly: Example 2.2.3 (Ornstein-Uhlenbeck process, one-well potential function). Consider the special case of a linear drift b(x) = −βx for a constant β > 0 in dimension d = 1 and choose σ := 1. Then the SDE (2.2) (or (2.1) respectively) becomes √ dXtε = −U (Xtε ) dt + ε dWt , X0ε = x0 , x

where U (x) := − 0 b(y)dy = β2 x2 denotes a quadratic one-well potential. Then the transition probability densities pεt (x, . ) are given by

1 (x − x0 αt )2 , exp − pεt (x, x0 ) = 2 σt2 2π σt2 where

αt := e−β t

and

σt2 :=

ε (1 − e−2βt ) . 2β

This follows from a direct verification of the Kolmogorov forward equation (Fokker-Planck equation). In other words, Xtε is normally distributed where the mean is given by the trajectory of the deterministic system, Xt0 = x0 e−βt , and the variance is σt2 .

66


Remark 2.2.4 (SDE of gradient type whose potential exhibits singularities). Consider the above SDE of gradient type (2.2) √ dXtε,x = −∇U (Xtε,x ) dt + ε dWt ; in the previous situation the drift b ≡ −∇U was supposed to be an element of C ∞ (Rd , Rd ) (or at least C 1 (Rd , Rd )); in particular, U is defined on the whole of Rd and the stationary density is calculated as in 2.2.1. For some applications it is however necessary to diminish the state space Rd to an open set D. In the upcoming course of this exposition we will be ε concerned with the exit times τD of X ε from open sets D ⊂ Rd . However, ε one could also force the system X not to leave D at all. For this purpose fix a potential function U ∈ C(Rd , R ∪ {+∞}) such that for D :=

x ∈ Rd : U (x) < ∞

the restriction of U to D is C 1 , U D ∈ C 1 (D, R) . Further assume that

|∇U (x)|2 e−4 U (x)/ε dx < ∞

D

and Nε := where the convention is used. Then

Rd

e− 2U(x)/ε dx < ∞ ,

e− 2U( . )/ε := 0 on D pε (x) = Nε−1 e− 2U(x)/ε

is the Lebesgue density of a probability measure ρε and there exists a process (Xtε )t≥0 with initial distribution P ◦ (X0ε )−1 = ρε which is a weak2 solution of the gradient SDE (2.2), √ dXtε = −∇U (Xtε ) dt + ε dWt , up to a terminal time T∞ for which Pρε {T∞ < ∞} = 0 ; here Pρε denotes the law of (Xtε )t≥0 starting with the stationary distribution ρε and T∞ being terminal means that it is the limit of an increasing sequence of stopping times.3 2 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.42]. See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.16].


67

This result is due to Meyer and Zheng [My-Zh 85] and has been cited in its above form by Kunz [Kz 02, p.16f.,27]. We finish this section by quoting a general result which does not contain an assertion on the stationary measure. However, it suitably concludes the above considerations on PDEs corresponding to uniformly non-degenerate SDEs (2.3) and will be used later in sketching the proof of theorem 3.1.6. A reference for this theorem is Friedman [Fri 75, p.146f.] among others. Theorem 2.2.5. Consider the SDE dXtx = b (t, Xtx ) dt + σ (t, Xtx ) dWt Xsx

(t ≥ s ≥ 0) ,

=x ∈R , d

where the coefficients are functions b ∈ C ∞ (R+ × Rd, Rd ) and σ ∈ C ∞ (R+ × Rd , Rd×d ); assume that any σ(t, . ) satisfies the ellipticity condition (E) for some universal constant c > 0. The partial differential operator corresponding to this SDE is G(s, x) :=

d i=1

bi (s, x)

d ∂ 1 ∂2 + aij (s, x) . ∂xi 2 i,j=1 ∂xi ∂xj

Fix some bounded, open domain D ⊂ Rd with smooth boundary ∂D, some time horizon T > 0 and let θD := min τD (s, x) , T , where τD (s, x) := inf{t ≥ 0 : Xtx ∈ / D} denotes the first exit time4 of X•x from D. Furthermore, let ψ ∈ C(∂Q, R) be some continuous function5 defined on the boundary of Q := (0, T ) × D. Then the boundary value problem

∂ + G(s, x) v = 0 on Q ∂s v = ψ on ({T } × D) ∪ ([0, T ] × ∂D) is uniquely solved by v(s, x) := Es,x ψ θD , XθD . 4

Since D is open, τ ε,x is a stopping time with respect to the underlying standard filtration (Ft )t≥0 ; see e.g. Hackenbroch and Thalmaier [Hb-Th 94, 3.12.(ii)]. 5 This function ψ(s, x) is not to be confused with the stochastic process ψtε (ω) as defined in equation (1.4). Since these two objects will not be considered simultaneously, there is no ambiguity.

68


2.3 The large deviations principle This section is intended to sketch the fundamental principles for the SDE (2.1) from which the exit time law shall be deduced in the next section. Standard references underlying the following exposition are the monographs by Freidlin and Wentzell [Fr-We 98] and Dembo and Zeitouni [De-Zt 98]. At the beginning of this chapter the action functional and the quasipotential have already been mentioned (see p.55f.) in order to set up the assumptions 2.1.1. This section clarifies that the action functional indeed provides the rate function of a large deviation principle; in the next section the importance of the quasipotential in exit time considerations will be accounted for, thus illustrating its interpretation as cost function. The general setup for a large deviation principle is that of a family {µε }ε>0 of probability measures on a space E which we assume for simplicity to be a separable metric space equipped with its completed Borel-σ-Algebra B(E); the goal is to characterize the behavior of {µε }ε>0 on B(E) as ε → 0 in terms of a rate function I on an exponential scale. Definition 2.3.1 (Large Deviation Principle (LDP)). Let E be a separable metric space with its (completed) Borel-σ-algebra B(E). 1) A function I : E → [0, ∞] is a rate function, if it is lower semicontinuous, i.e. if the level sets {x ∈ E : I(x) ≤ α} are closed; since E is metric, an equivalent condition is that for all x ∈ E, lim inf I(xn ) ≥ I(x) . xn →x

A rate function I is good, if the level sets {x ∈ E : I(x) ≤ α} are compact. 2) A family {µε }ε>0 of probability measures on B(E) satisfies the large deviation principle (LDP) with rate function I, if − inf◦ I ≤ lim inf ε log µε (Ψ) ≤ lim sup ε log µε (Ψ) ≤ − inf I Ψ

ε→0

ε→0

(2.7)

Ψ

for all Ψ ∈ B(E) , where Ψ◦ and Ψ denote the topological interior and closure of the set Ψ, respectively. A large deviation principle is “pushed forward” by continuous mappings; this is the content of the following proposition. For the straightforward verification we refer to Freidlin and Wentzell [Fr-We 98, p.81] and Dembo and Zeitouni [De-Zt 98, p.127]. The latter reference also contains more general versions of the contraction principle such as the “almost continuous case”, where the function F is measurable and a suitable limit of continuous functions; see [De-Zt 98, p.133].

2.3 The large deviations principle

69

Proposition 2.3.2 (Contraction principle - continuous case). Let F : E 1 → E 2 be a continuous function between separable metric spaces E 1 , E 2 and let I (1) : E 1 → [0, ∞] be a good rate function. Then I (2) : E 2 → [0, ∞] ,

I (2) (x) :=

inf

F−1 ({x})

I (1) ( . )

(where inf ∅ ≡ ∞) is a good rate function on E 2 . If, in addition, I (1) governs a LDP for a family of probability measures {νε }ε>0 on E 1 , then I (2) governs a LDP for {µε }ε>0 := { νε ◦ F−1 }ε>0 on E 2 . Proposition 2.3.2 is used in proving the following theorem 2.3.4. Beforehand, the corresponding notation is fixed: Notation 2.3.3. Let T > 0 be an arbitrary time horizon. Then the following function spaces over the time interval [0, T ] are defined: Cx := Cx ([0, T ], Rd) := {f : [0, T ] → Rd continuous, f0 = x}

(x ∈ Rd )

and H1 := H1 ([0, T ], R ) := d

·

gs ds : g ∈ L ([0, T ], R ) 2

d

,

0

the absolutely continuous functions starting in 0 with square integrable derivative. Furthermore, let pr[0,T ] denote the restriction map pr[0,T ] : (Rd )[0,∞) → (Rd )[0,T ] ,

pr[0,T ] (f ) := f [0,T ] .

Theorem 2.3.4 (LDP for strong solutions of SDE). Let X ε,x be the solution of (2.1), √ dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , X0ε,x = x , where b ∈ C ∞ (Rd , Rd ) and σ ∈ C ∞ (Rd , Rd×d ) are assumed to be bounded together with their first derivatives. Let µε denote the law of X ε,x on Cx ([0, T ], Rd ) for a fixed time T > 0 , µε := P ◦

pr[0,T ] ◦ X•ε,x

−1

.

Then {µε }ε>0 satisfies the LDP with good rate function I(f ) ≡ I[0,T ],x (f ) :=

{ g∈H1 : ft = x +

t 0

inf b(fs ) ds +

t 0

σ(fs ) g˙ s ds }

1 2

T 2

| g˙ t | dt . 0

70


If a ≡ σσ ∗ is strictly positive definite, as in assumption (E), then this x as defined on p.55, i.e. rate function is the action functional I0T ⎧ T 2 ⎨1 a(fs )−1/2 [f˙s − b(fs )] ds I(f ) = 2 0 ⎩ ∞

, f − x ∈ H1 , f −x∈ / H1 .

Sketch of Proof. 1) √ Consider the case that b( . ) = 0, σ( . ) = idRd and Schilder’s x = 0 first, i.e. Xtε,x = εWt . In this case the assertion is due to √ theorem (Dembo and Zeitouni [De-Zt 98, p.185f.]): The law of εWt on C0 ([0, T ], Rd ), √ νε := P ◦ (pr[0,T ] ◦ ( ε W• ))−1 , satisfies the LDP with good rate function T 1 |g˙ s |2 ds BM I[0,T ],0 (g) := 2 0 ∞

, g ∈ H1 ,g ∈ / H1 .

2) Now consider a general drift b and initial condition x, but again let σ( . ) = idRd , √ dXtε,x = b(Xtε,x ) dt + ε dWt , X0ε,x = x . For any g ∈ C0 , the integral equation t b(fs )ds + gt ft = x +

(t ∈ [0, T ])

0

admits a unique continuous solution f ; this gives rise to a well-defined mapping F : C0 → Cx , F(g) := f . The map F is itself continuous by the Lipschitz continuity of b and Gronwall’s Lemma. Thus the contraction principle 2.3.2 applies and yields an LDP, as √ pr[0,T ] ◦ X•ε,x = F pr[0,T ] ◦ ε W• by the SDE for X ε,x which implies that µε := νε ◦ F−1 is the law of X ε,x on Cx . Due to the contraction principle 2.3.2 and Schilder’s theorem the corresponding rate function is I(f ) = =

inf

{g∈C0 : f =F(g)}

BM I[0,T ],0 (g)

inf

{ g∈H1 : f (t) = x +

t 0

b(fs ) ds + gt }

1 2

T 2

| g˙ t | dt ; 0

2.3 The large deviations principle

71

in the last equation it has been used that g ∈ H1 if and only if f = F(g) ∈ H1 + x ; in this case one obtains that g˙ t = f˙t − b(ft ) which yields together with the last equation that 2 1 T ˙ I(f ) = ft − b(ft ) dt ; 2 0 otherwise, if f = F(g) ∈ / H1 + x, then g ∈ / H1 and I(f ) = ∞ due to Schilder’s theorem. 3) Finally consider a general diffusion coefficient σ; in this case the map F defined analogously as above as F(g) := f via t t b(fs ) ds + σ(fs )g˙ s ds (t ∈ [0, T ]) ft = x + 0

0

is not necessarily continuous. Therefore, one approximates X ε,x by X ε,m,x ,

√ ε,m,x ε,m,x ε,m,x dt + ε σ X mt dWt , = b X mt X0ε,m,x = x dXt m

m

and uses the “almost continuous version” of the contraction principle which had already mentioned before; see Dembo and Zeitouni [De-Zt 98, p.214f., 133]. Further details concerning the above results are contained in the theorems 4.1.1 and 5.3.2. by Freidlin and Wentzell [Fr-We 98], in Wentzell and Freidlin [We-Fr 70, §1] as well as in Dembo and Zeitouni [De-Zt 98, Sec.5.6]. Remark 2.3.5. Consider the situation of the above theorem and assume that a is strictly positive definite. Then the rate function (action functional)

x I(f ) = I0T (f ) ≡

⎧ 1 ⎪ ⎪ ⎨

2

⎪ ⎪ ⎩

∞

T 0

0 12 −1/2 ˙ fs − b(fs ) ds , f absolutely continuous a(fs )

and f0 = x, , otherwise

vanishes if and only if f is a solution path of the ODE x˙ = b(x) on the time interval [0, T ] ; the action functional hence weighs the deviation of a path f from being a deterministic solution in the L2 -norm. The following continuity consequence of the LDP 2.3.4 will be used in the proof of the exit time law 2.4.6 (more precisely, in the lemmas 2.4.8, 2.4.9 and 2.4.10); it states that the above LDP also holds true uniformly with respect to the initial condition. For a proof see Dembo and Zeitouni [De-Zt 98, Cor.5.6.15]. A similar assertion will appear later in the case of degenerate SDEs; see corollary 3.2.1.

72


Theorem 2.3.6 (Uniform asymptotics). Consider the situation of the above theorem 2.3.4 and fix some compact set K ⊂ Rd . Then it follows for all open sets G ⊂ C([0, T ], Rd ), lim inf ε log inf P {X•ε,y ∈ G} ≥ − sup inf I[0,T ],y (f ) , ε→0

y∈K

y∈K f ∈G

and for all closed sets F ⊂ C([0, T ], Rd ), lim sup ε log sup P {X•ε,y ∈ F } ≤ − ε→0

y∈K

inf

y∈K, f ∈F

I[0,T ],y (f ) .

2.4 Exit probabilities for non-degenerate systems In this section we are interested in the noise-induced exit (exit time, exit location) from a neighborhood of an equilibrium point of the corresponding deterministic system. Throughout this section we consider the SDE (2.1) √ X0ε,x = x dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , under the assumptions 2.1.1. Again, the law of (Xtε,x )t≥0 is denoted by Px and Ex is the corresponding expected value. Notation 2.4.1 (First exit time). Let D be a bounded, open domain in Rd with smooth boundary ∂D. Then the random variable ε,x τ ε ≡ τ ε,x ≡ τD := inf{t ≥ 0 : Xtε,x ∈ / D}

denotes the first exit time of X•ε,x from D. Since D is open, τ ε,x is a stopping time6 with respect to the underlying standard filtration (Ft )t≥0 . Remark 2.4.2. The first exit time τ ε and the first exit location can be characterized for any ε in terms of solutions of PDEs involving the generator G ε of X ε : 1) f1 (t, x) := Px {τ ε ≤ t} is the unique solution of ε ∂f (t, x) , t > 0 , x ∈ D G f (t, x) = ∂t ⎧ ⎨ 0 , t = 0, x ∈ D f (t, x) = ⎩ 1 , t > 0 , x ∈ ∂D ; let Q := (0, ∞) × D; then f1 is continuous at all points of Q \ {(0, x) : x ∈ ∂D}. 6

See e.g.: Hackenbroch and Thalmaier [Hb-Th 94, 3.12.(ii)].

2.4 Exit probabilities for non-degenerate systems

73

2) f2 (x) := Ex τ ε is the unique solution of G ε f = −1 , on D f = 0 , on ∂D . 3) For any g ∈ C(∂D, R), f3 (x) := Ex (g(Xτεε )) is the unique solution of G ε f = 0 , on D f = g , on ∂D ; f2 and f3 are continuous on D. These differential equations are similar in spirit to the ones investigated in section 2.2 ; they are well known: 1) can be found e.g. in Friedman [Fri 76, p.347] and Freidlin and Wentzell [Fr-We 98, p.107], 2) and 3) are cited from Dembo and Zeitouni [De-Zt 98, p.222]; also see Hackenbroch and Thalmaier [Hb-Th 94, Sec.6.6]. The above PDEs are difficult to solve, especially in higher dimensions, and will not be used in the sequel. Instead, the asymptotic behavior of τ ε as ε → 0 is investigated by means of the LDP for X ε . Theorem 2.4.3 (Consequence of the LDP 2.3.4 for the first exit time). Let D be a bounded, open domain in Rd with smooth boundary ∂D and first-exit time / D} , τ ε,x ≡ inf{t ≥ 0 : Xtε,x ∈ where X ε,x is the solution of the SDE (2.1) under assumption (E), starting in x ∈ D. Furthermore, let I be the rate function (action functional) for X ε,x on Cx ([0, T ], Rd); see 2.3.4. Then it follows for t ∈ [0, T ] that ⎧ ⎫ ⎨ ⎬ /D lim ε log Px {τ ε ≤ t} = − inf I[0,T ],x (f ) : f ∈ Cx ([0, T ], Rd), ∃ f (s) ∈ ε→0 ⎩ ⎭ s∈[0,t] ≡ − inf V (x, y; s) : s ∈ [0, t], y ∈ /D , where V (x, y; s) := inf { I[0,s],x (f ) : f ∈ Cx ([0, s], Rd ) , f (s) = y} . Proof. For simplicity, the proof shall be given only for the case that σ is a constant (invertible) matrix; in doing so we follow Freidlin and Wentzell (theorem 4.1.2 and example 3.3.5 in [Fr-We 98]) who consider σ = idRd . For the general case the reader is referred to Wentzell and Freidlin [We-Fr 70, Th.2.1] and Friedman [Fri 76, Th.14.4.1].

74


By theorem 2.3.4 the LDP holds for the law µε of X ε,x on Cx with rate function I ≡ I[0,T ],x , so (2.7) applies and it is thus left to verify: inf◦ I[0,T ],x ≤ inf I[0,T ],x , Ψ

where

Ψ :=

Ψ

f ∈ Cx ([0, T ], Rd) : ∃

f (s) ∈ /D

.

s∈[0,t]

As Ψ is open in Cx , Ψ is closed, Ψ = Ψ . Furthermore, inf I[0,T ],x < ∞ , Ψ

because for some (even for any) fixed y ∈ ∂D, f ∈ Ψ, where f (u) := x + u t (y − x), and I[0,T ],x (f ) < ∞ due to the assumption (E). Suppose that the above claim were false, i.e. that inf◦ I[0,T ],x > inf I[0,T ],x . Ψ

Ψ

Hence, there exists ϕ ∈ Ψ \ Ψ◦ such that inf Ψ◦ I > I(ϕ). In particular, ϕ is absolutely continuous, since I(ϕ) < ∞. As ϕ ∈ Ψ, there exists s ≤ t such that ϕ(s) ∈ /D.

..................... ...... ......... ..... ...... .... ..... ... .... . . ... .. . . ... ... ... . ... .. . ... .. . ... .. . ... .. . ... ... .... ... ... ... δ ... .. ... .. ... ... .... ... ... .... .. .... ... .... .. .. . .... ... .... ... .... .... .... ... .. ... .... .. .... .... .. ... .. .... .... . ... ..... .... .. .... ... ... .... ..... ... .... .. ... ... ... .... ..... ... ... ... .. .. ... .. ... ... .. .. ... .. .... ... .. .. .. .. .. .. ... .. . . .. .. .. ... . . . .... ... .... ... ... .. ... ..... . . .. . ... . . . ... ... ... .... ... ...... ..... ... ....... ........ ... ................................... . . . . . . . . ... . . .......................................................... .... ... . ... ... ... ... ... ... ... ... .... . . . . ..... ..... ...... ..... ........ ........................

D

ϕ(t)

q

ϕ (t)

q

ϕ(s) q

q xδ

q

x

Fig. 2.3 Sketch of ϕ and ϕδ within the domains D and D∗

D∗


75

Now for any δ > 0 fix some xδ ∈ D∩B(ϕ(s), δ) , where B(ϕ(s), δ) denotes the open ball with center ϕ(s) and radius δ . The function ϕδ (r) := ϕ(r) +

r δ x − ϕ(s) s

(r ∈ [0, T ])

is also absolutely continuous and belongs to Ψ◦ . Hence, the proof is comδ→0 pleted, if it is shown that I(ϕδ ) −−−→ I(ϕ) , in contradiction to the choice of ϕ as inf Ψ◦ I > I(ϕ) . For this purpose define the auxiliary functions 1 Aδr := a− 2 ϕ˙ δ (r) − b(ϕδ (r)) and

1 Br := a− 2 ϕ˙ (r) − b( ϕ(r))

(r ∈ [0, T ]) (r ∈ [0, T ]) ,

where a ≡ σσ ∗ , in order to get I(ϕδ ) − I(ϕ) 2 2 1 1 T − 1 δ = a 2 ϕ˙ (r) − b(ϕδ (r)) − a− 2 ϕ˙ (r) − b( ϕ(r)) dr 2 0 T 2 1 δ 2 Ar dr − Br ≡ 2 0 T 1 δ2 δ 2 2 δ δ |Ar | + |Br | − Ar , Br − Br , Ar dr Ar , Br − |Br | + = 2 0 T 1 δ (Ar − Br ), (Aδr − Br ) (Aδr − Br ), Br + dr = 2 0 T 2 δ 1 T δ Ar − Br dr Ar − Br , Br dr + = 2 0 0 2 δ 1 δ A − B L2 ([0,T ],Rd ) . ≡ (A − B), B L2 ([0,T ],Rd ) + 2 Therefore the claim has been reduced to showing that δ→0

Aδ − B −−−→ 0 in L2 ([0, T ], Rd ) ; a sufficient condition for this assertion is that δ→0

Aδr − Br −−−→ 0 uniformly with respect to r ∈ [0, T ] . For this purpose fix an open domain D∗ ⊂ Rd such that D∗ ⊃ D ∪ {ϕ(r) : r ∈ [0, T ]} ∪ {ϕδ (r) : r ∈ [0, T ], δ ∈ (0, 1]} ; δ→0

since ϕδ −−−→ ϕ uniformly on [0, T ], this domain D∗ can be chosen to be bounded. Since b has been assumed to be C ∞ in (2.1), there is a (local)

76


Lipschitz constant C < ∞ such that | b(w1 ) − b(w2 ) | ≤ C |w1 − w2 |

(w1 , w2 ∈ D∗ ) .

Therefore, using that ϕ˙ δ (r) = ϕ(r) ˙ + 1s (xδ − ϕ(s)) , it follows altogether that 0 1 δ Ar − Br ≡ a− 12 ϕ˙ δ (r) − b(ϕδ (r)) − a− 12 ϕ˙ (r) − b( ϕ(r)) 1 1 1 1 = a− 2 (xδ − ϕ(s)) + a− 2 ϕ(r) ˙ − a− 2 ϕ(r) ˙ s − 12 δ − 12 − a b(ϕ (r)) + a b( ϕ(r)) 1 1 − 12 δ ≤ a x − ϕ(s) + a− 2 b(ϕδ (r)) − b(ϕ(r)) s

11 δ −2 δ ≤ a x − ϕ(s) + C ϕ (r) − ϕ(r) s δ→0

−−−→ 0 , uniformly with respect to r ∈ [0, T ] .

The above theorem yields information about the distribution of exit times of non-degenerate stochastic systems. The corresponding assertion concerning degenerate systems by Hernández-Lerma will appear later in 3.1.7. Theorem 2.4.3 furthermore motivates the following definition which had been anticipated at the beginning of this chapter (see p.56) for formulating assumption (V) in 2.1.1; the corresponding cost function in the context of degenerate systems will then appear in 3.1.2. Definition 2.4.4 (Quasipotential). Let I be the good rate function for the solution X ε of (2.1) as provided by theorem 2.3.4 . Then V : Rd × Rd × R>0 −→ R+ , V (x, y; s) := inf I[0,s],x (f ) : f ∈ Cx ([0, s], Rd ) , f (s) = y denotes the cost of forcing the system X ε to connect x and y in time s (in the sense of theorem 2.4.3) . The function V : Rd × Rd −→ R+ , V (x, y) := inf V (x, y; s) s>0

is the quasipotential of y with respect to x ; it is considered as the cost of forcing the system X ε to connect x and y eventually (see theorem 2.4.6). For a point O ∈ Rd ,


77

V : Rd −→ R+ , V (y) := V (O, y) denotes the quasipotential of the system X ε (with respect to O) . The meaning of “quasipotential” is clarified in the following proposition which is cited from Freidlin and Wentzell [Fr-We 98, p.118f.]: Proposition 2.4.5. Let D be a bounded open domain in Rd , suppose that σ = idRd and let the drift b ∈ C(Rd , Rd ) derive from a potential U with an orthogonal component L on D and let b have a unique equilibrium in D, i.e. suppose that b(x) = −(∇U )(x) + L(x) (x ∈ D) , where U ∈ C 1 (D, R) and L ∈ C(Rd , Rd ) are functions such that (∇U )(x), L(x) = 0 (x ∈ D) and for some O ∈ D, one can state that U (O) = 0 as well as and (∇U )(x) = 0

U (x) > 0

(x ∈ D \ {O}) .

Then for all x ∈ D for which U (x) ≤ min U , the quasipotential V with respect ∂D

to O is given by:

V (x) ≡ V (O, x) = 2 U (x) .

If in addition U ∈ C 2 , L ∈ C 1 and x ∈ D is some point, then the rate function I(−∞,T ],O , defined analogously to I[0,T ],O , has a unique extremal ϕ on the set d f ∈ C (−∞, T ], R : lim f (s) = O , f (T ) = x ; s→−∞

furthermore, this extremal ϕ is the solution of the ODE ϕ(s) ˙ = + ∇U (ϕ(s)) + L(ϕ(s)) (s ∈ (−∞, T ]) , ϕ(T ) = x . In the general case, σ( . ) ≡ idRd , the above proposition remains true, if D is endowed with the Riemannian metric ds2 :=

d

a(x)−1

ij

dxi dxj

i,j=1

and , as well as ∇ now denote the scalar product and the Riemannian gradient with respect to this metric, respectively; see Freidlin and Wentzell [We-Fr 72, Th.1].

78


Further properties of the quasipotential V are the following (see Freidlin and Wentzell [Fr-We 98, Ch.4]): V (O, . ) is Lipschitz continuous, but not necessarily differentiable ([Fr-We 98, p.108,119]). The function V (x, y, s) satisfies a Hamilton-Jacobi equation ([Fr-We 98, p.107]). In case that V (O, . ) is continuously differentiable, a corresponding Jacobi equation follows; from this Jacobi variational equation a decomposition for b as in proposition 2.4.5 can be deduced ([Fr-We 98, p.119]). Furthermore, in the situation of proposition 2.4.5 one obtains that V (O, x) > 0 if and only if x = O (see Day and Darden [Day-Dar 85, Cor.2]). Next the fundamental exit law for non-degenerate systems will be discussed. It concerns the noise induced first exit of X ε,x , given by (2.1), √ dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , X0ε,x = x ∈ D from a bounded, open domain D ⊂ Rd with smooth boundary ∂D. For this purpose, it is not necessary to impose the set of requirements 2.1.1 to the full extent. Instead, the following assumptions are underlying: (A1) There exists a unique stable equilibrium7 point O ∈ D of the deterministic system dXt0,x = b(Xt0,x) dt ,

X00,x = x ,

(2.8)

to which D is attracted.8 (A2) All trajectories of the deterministic system starting in ∂D converge to O (as t → ∞). (A3) V := inf V (O, . ) < ∞ . ∂D

(A4) There exist K, ρ0 > 0 such that for all ρ ≤ ρ0 and all x, y ∈ Rd for which |x − z| + |y − z| ≤ ρ

for some z ∈ ∂D ∪ {O} ,

there is a function u ≡ uρ;x,y ∈ L2 ([0, Tρ ], Rd ) such that u∞ < K and k(Tρ ) = y , where

t

b(k(s)) ds +

k(t) := x + 0

t

σ(k(s)) u(s) ds , 0 ρ→0

and where Tρ ≥ 0 is a time such that Tρ −−−→ 0. 7 O is an (asymptotically) stable equilibrium point of the deterministic dynamical system, if for any neighborhood B1 of O there exists another neighborhood B2 ⊂ B1 such that all trajectories of the deterministic system starting in B2 converge to O (as t → ∞) without leaving B1 ; of course, b(O) = 0 . 8 D is attracted to O, if all trajectories of the deterministic system X 0,x starting in D converge to (the equilibrium position) O (as t → ∞) without leaving D .


79

In the situation described by the assumptions 2.1.1 the above requirements (A1)-(A4) are fulfilled, if for some i ∈ {1, . . . , l}, the domain D satisfies that O := Ki ∈ D and D ⊂ Di , where t→∞

Di := {x ∈ Rd : Xt0,x −−−−→ Ki } denotes the domain of attraction of Ki under the deterministic motion X 0 . The fact that necessarily D ⊂ Di , is due to (A1) and (A2) which exclude any other equilibrium (i.e. any other element in {K1 , . . . , Kl } \ {Ki}) from being in D. (A4) is implied by (E), see Dembo and Zeitouni [De-Zt 98, p.224]. In general, (A2) prevents that b(x), N (x) = 0, ∀x ∈ ∂D, where N (x) is the outer normal to ∂D at x ; in this situation ∂D is a characteristic boundary; for studies on this case see Day [Day 90] and the references therein. Dembo and Zeitouni [De-Zt 98, Cor.5.7.16] investigate the situation when (A2) is skipped, but when the boundary is not necessarily characteristic. The above assumptions (A1)–(A4) are taken from Dembo and Zeitouni [De-Zt 98, p.221ff.]; this reference is also underlying to the subsequent discussion. Since this section is intended to provide the argumentation in outlines, some of the proofs will only be sketched and the reader is referred to Dembo and Zeitouni [De-Zt 98, Sec.5.7] for details. These results are due to Freidlin and Wentzell; see [We-Fr 70, §3] and [Fr-We 98, §§4.2,4.4]. Theorem 2.4.6. Let the assumptions (A1)-(A4) be satisfied and let τ ε,x ≡ inf{t ≥ 0 : Xtε,x ∈ / D} , denote the first exit time of X ε,x , given by (2.1), from a bounded, open domain D ⊂ Rd with smooth boundary ∂D. Then it follows 1) for the first exit time: For all x ∈ D and δ > 0, lim Px e(V −δ)/ε < τ ε < e(V +δ)/ε = 1 , ε→0

and for all x ∈ D,

lim ε log Ex τ ε = V ;

ε→0

2) for the first exit position: If N ⊂ ∂D is a closed set for which inf V (O, . ) N

> V , then for any x ∈ D lim Px {Xτεε ∈ N } = 0 ;

ε→0

thus, if V (O, . ) has a unique minimum z ∗ on ∂D, then for any x ∈ D and δ > 0, lim Px Xτεε − z ∗ < δ = 1 . ε→0

80


The proof of Theorem 2.4.6 relies on the following lemmas. Here, Bρ (O) and Sρ (O) will denote the closed ball and the sphere around O with radius ρ, respectively; furthermore, the radii of all balls and spheres appearing are chosen so small such that the balls and spheres are contained in D. Lemma 2.4.7 (Continuity of V given (A4)). Assume condition (A4). Then there exits for any δ > 0 a sufficiently small ρ > 0 such that sup

inf V (x, y, t) < δ

(2.9)

x,y∈Bρ (O) t∈[0,1]

as well as

sup x,y∈Rd : inf (|x−z|+|y−z|)≤ρ

inf V (x, y, t) < δ . t∈[0,1]

(2.10)

z∈∂D

Proof. Given x, y ∈ Rd for which |x−z|+|y −z| ≤ ρ for some z ∈ ∂D∪{O}, let k, u and K, Tρ be the functions and constants made available by (A4). Due to theorem 2.3.4, 1 t I[0,t],x (k) = inf | g(s) ˙ |2 ds . { g∈H1 : k(s) = x + 0s b(k(r)) dr + 0s σ(k(r)) g(r) ˙ dr } 2 0 Hence, (A4) implies that V (x, y; Tρ ) ≡ inf I[0,Tρ ],x (f ) : f ∈ Cx ([0, Tρ ], Rd ) , f (Tρ ) = y ≤ I[0,Tρ ],x (k) 1 Tρ 2 | u(s) | ds ≤ 2 0 ≤ K 2 Tρ /2 , which can become arbitrarily small for an appropriate choice of ρ, again due to (A4). Next, five lemmas are formulated from which 2.4.6 then can be proved. Here, lemma 2.4.7 is needed for 2.4.8 and 2.4.10. The LDP for X ε will be used in terms of theorem 2.3.6 in the proofs of the first three of these lemmas. In doing so, the boundedness conditions on b und σ in theorem 2.3.6 (see 2.3.4) are tacitly assumed to be satisfied. This is no restriction, since the system is only examined until its first exit time τ ε from D which only9 depends on the values of b and σ on D.

9

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.22].


81

Lemma 2.4.8 (Uniform lower bound on the exit probability for starts near O). Assume the set of conditions (A). For any η > 0 there is then a ρ0 > 0 such that for all ρ ∈ (0, ρ0 ], there exists T0 < ∞ for which lim inf ε log inf Px {τ ε ≤ T0 } > − V + η . ε→0

x∈Bρ (O)

Proof. Given η > 0, apply lemma 2.4.7 for δ := η6 , to get ρ0 > 0 such that (2.9) and (2.10) hold for ρ0 — and hence also for all ρ ∈ (0, ρ0 ]. Fix such a ρ and an arbitrary x ∈ Bρ (O): (2.9) provides a path ψ x and tx ∈ [0, 1] such that ψ x (tx ) = O ,

ψ x (0) = x ,

η ; 3

I[0,tx ],x (ψ x ) ≤ δ
0 and z ∈ ∂D such that ψ 0 (0) = O ,

I[0,t0 ],O (ψ 0 ) ≤ V +

ψ 0 (t0 ) = z ,

η ; 6

for this choice of z, (2.10) yields a ψ z , tz ∈ [0, 1] and y ∈ / D for which dist(y, ∂D) = ρ and ψ z (0) = z ,

I[0,tz ],z (ψ z ) ≤ δ ≡

ψ z (tz ) = y ,

η ; 6

finally, let ψ y := X 0,y denote the solution curve of the deterministic system (2.8) started in y and considered until time ty := 2 − (tx + tz ), ψ y (0) = y ,

I[0,2−(tx +tz )],y (ψ y ) = 0 .

Juxtaposing ψ x , ψ 0 , ψ z and ψ y results in a path φx which is defined on [0, T0 ], where T0 := tx + t0 + tz + ty ≡ t0 + 2 and for which I[0,T0 ],x (φx ) ≤ I(ψ x ) + I(ψ 0 ) + I(ψ z ) + I(ψ y ) < V + η . Using these functions φx , x ∈ Bρ (O), the set : ρ Ψ := ψ ∈ C([0, T0 ], Rd ) : ψ − φx ∞ < 2 x∈Bρ (O)

is open and { X ε,x ∈ Ψ } ⊂ { τ ε,x ≤ T0 }. Therefore it follows from theorem 2.3.6 that lim inf ε log ε→0

inf x∈Bρ (O)

Px {τ ε ≤ T0 } ≥ lim inf ε log ε→0

≥−

sup

inf x∈Bρ (O)

P {X•ε,x ∈ Ψ}

inf I[0,T0 ],x (φ)

x∈Bρ (O) φ∈Ψ

82


≥−

sup x∈Bρ (O)

I[0,T0 ],x (φx )

> −(V + η) . Lemma 2.4.9 (X ε cannot stay in D arbitrarily long without approaching O). Assume the set of conditions (A). Then we have for any ρ > 0, lim lim sup ε log sup Px σρε > t = −∞ , t→∞

where of O,

σρε

ε→0

x∈D

denotes the first hitting time of either ∂D or a small neighborhood σρε,x := inf{t ≥ 0 : Xtε,x ∈ Bρ (O) ∪ ∂D} .

Sketch of Proof. For all x ∈ Bρ (O), σρε,x = 0 ; thus, only initial values x ∈ D \ Bρ (O) are of relevance. The set of functions which do not leave the closure of the latter set, Ψt := ψ ∈ C([0, t], Rd ) : ψ(s) ∈ D \ Bρ (O) for all s ∈ [0, t] (t > 0) is closed and { σ ε,x > t } ⊂ { X ε,x ∈ Ψt } for x ∈ D \ Bρ (O). Hence, theorem 2.3.6 yields lim sup ε log ε→0

sup x∈D\Bρ (O)

Px {σ ε > t} ≤ lim sup ε log ε→0

sup

Px {X•ε ∈ Ψt }

x∈D\Bρ (O)

≤ − inf I[0,t],φ(0) (φ) , φ∈Ψt

which reduces the claim of the lemma to proving that the right hand side diverges (t → ∞). Via (A2), a Gronwall argument (b is C ∞ , hence Lipschitz on D) and the compactness of D \ Bρ (O) one can get T < ∞ such that for all x ∈ D \ Bρ (O), the solution φx (t) := Xt0,x of the deterministic system (2.8) is contained in the ball B2ρ/3 (O) for all t ≥ T . In order to obtain a contradiction suppose the divergence inf φ∈Ψt t→∞ I[0,t],φ(0) (φ) − −−− → ∞ were wrong; so imagine that there exists an M < ∞ such that for all n ∈ N, there is some ψ n ∈ ΨnT for which I[0,nT ],ψn (0) (ψ n ) ≤ M ; merely considering times nT is no restriction, since I is additive. Dissecting ψ n into n pieces and using the additivity of I again, one obtains φn ∈ ΨT n→∞ such that I[0,T ],φn (0) (φn ) ≤ M/n −−−−→ 0 . As the rate function I is good, ΨT ∩ {I ≤ 1} is compact, providing a limit point φ∗ ∈ ΨT of (φn )n . I being lower semicontinuous, I[0,T ],φ∗ (0) (φ∗ ) = 0 follows and φ∗ is necessarily a solution curve of the deterministic system (2.8). Due to the definition of T this implies that φ∗ (T ) ∈ B2ρ/3 (O), contradicting the fact that φ∗ (T ) ∈ / Bρ (O)◦ , as an element of ΨT .


83

Lemma 2.4.10 (Bound on the probability of leaving D before further approaching O). Assume the set of conditions (A). Then for all closed sets N ⊂ ∂D, lim sup lim sup ε log sup Py Xσερε ∈ N ≤ − inf V (O, . ) . ρ→0

ε→0

y∈S2ρ (O)

N

Sketch of Proof. Define VN,δ := min[inf N V (O, . ) − δ , 1δ ] for δ > 0. Due to the previous lemma 2.4.9 there exists T < ∞ such that lim sup ε log sup Py σρε > T < −VN,δ . ε→0

y∈S2ρ (O)

Now, one applies theorem 2.3.6 to the closed set Φ := φ ∈ C([0, T ], Rd ) : φ(t) ∈ N for some t ∈ [0, T ] to see that −VN,δ also bounds the exponential growth rate of supy∈S2ρ (O) Py {X•ε ∈ Φ} from above (as ε → 0), where ρ ≡ ρ(δ) derives from (2.9). The same bound on the exponential rate holds true for Py Xσερε ∈ N ≤ Py {X•ε ∈ Φ} + Py σρε > T . Finally, take δ → 0. The final two lemmas are not based on the large deviations principle. Remarkably, the assertion of the next lemma is not uniform with respect to the initial point; in contrast, the other lemmas contain uniformity information. This is why theorem 2.4.6 does not hold uniformly on D (but only uniformly on compact subsets of D). Lemma 2.4.11 (The probability of approaching O without leaving D is large). Assume the set of conditions (A). Then it follows for any ρ > 0 and x ∈ D that lim Px Xσερε ∈ Bρ (O) = 1 . ε→0

Sketch of Proof. Since σρε,x = 0 for x ∈ Bρ (O), fix x ∈ D\Bρ (O). Again, there is T < ∞ such that Xt0,x ∈ Bρ/2 (O) for t ≥ T . Now it holds for ∆ := min dist {φx (t) : t ∈ [0, T ]}, ∂D , ρ > 0 that Xσερε ∈ ∂D ⊂ X•ε − X•0 [0,T ] > ∆/2 and the probability of the latter event can be estimated from above by means of a Gronwall argument and the Burkholder-Davis-Gundy maximal inequality.10 This upper bound converges to 0 as ε → 0. By the definition of σρε this is the converse of the claim. See Dembo and Zeitouni [De-Zt 98, p.234f.] for details.

10

See e.g. Dembo and Zeitouni [De-Zt 98, E.3] or Hackenbroch and Thalmaier [Hb-Th 94, 4.63].

84


Lemma 2.4.12 (Upper bound on the distance of X ε from its starting point). Assume the set of conditions (A). For all ρ, c > 0 there exists T ≡ T (ρ, c) < ∞ such that ; lim sup ε log sup P ε→0

x∈D

sup |Xtε,x − x| ≥ ρ

< −c .

t∈[0,T ]

Sketch of Proof. Also in this case a Gronwall argument is applied to |Xtε,x − x| . By the time change theorem11 of martingale theory the upper bound, hence obtained, can be further modified such that the statement is seen to hold true. See Dembo and Zeitouni [De-Zt 98, p.235f.] for details. Proof of Theorem 2.4.6: 1) Let x ∈ D and δ > 0 be fixed. First, the bounds in probability for τ ε,x , ε→0 Px τ ε ≥ e(V +δ)/ε −−−→ 0 and Px

τ ε ≤ e(V −δ)/ε

ε→0

−−−→ 0 ,

will be proved; afterwards, the assertion on Ex τ ε will then follow from these arguments. (a) τ ε,x < e(V +δ)/ε Fix η > 0. Lemma 2.4.8 yields ρ, ε0 > 0 and T0 < ∞ such that inf x∈Bρ (O)

η

Px {τ ε ≤ T0 } > e−(V + 2 )/ε

(ε < ε0 ) .

With this choice of ρ lemma 2.4.9 provides some T1 < ∞ such that η 1 sup Px σρε > T1 < e− 4 ε (ε < ε0 ) . x∈D

Furthermore, choose ε0 sufficiently small such that eη/(2 ε) − eη/(4 ε) ≥ 1

(ε < ε0 ) .

Setting T := T0 + T1 , the definitions of σρε and τ ε , the strong Markov property and the previous string of estimates imply that for all ε < ε0 :

11

See e.g. Dembo and Zeitouni [De-Zt 98, E.2] or Hackenbroch and Thalmaier [Hb-Th 94, 5.24].


qε

85

ε,X ε ε := inf Px {τ ε ≤ T } ≥ inf Px σρε ≤ T1 , τ σρ ≤ T0 x∈D x∈D ≥ inf Px σρε ≤ T1 · inf Px {τ ε ≤ T0 } x∈D x∈Bρ (O) η − V > 1 − e−η/(4 ε) e ( + 2 )/ε η

≥ e−η/(2 ε) e−(V + 2 )/ε

=

e−(V +η)/ε .

An iteration of the strong Markov property12 implies that sup Px {τ ε > kT } ≤ (1 − q ε )k

(k ∈ N) ;

x∈D

more precisely, for k = 1 this is just the definition of q ε ; for k > 1, ε

Px {τ ε > kT } = Px {τ ε > (k − 1)T , τ ε,X(k−1)T > T } 5

6 ε,x F(k−1)T H ◦ X(k−1)T +• , = E 1{τ ε,x >(k−1)T } · 1 − E where H is defined on the path space by H := 1{τ ≤T } for τ (f ) := inf{t ≥ 0 : ft ∈ / D}, i.e. τ ε,x ≡ τ ◦X•ε,x . With (T H)(z) := E(H ◦X•ε,z ) the strong Markov property13 implies: ε,x EF(k−1)T H ◦ X(k−1)T +• ε,x ≡ E 1{τ ≤T } ◦ X•ε,z ; = (T H) ◦ X(k−1)T ε,x z=X(k−1)T

plugging this into the previous equation one gets : 5

6 ε ε,z Px {τ > kT } ≤ E 1{τ ε,x >(k−1)T } · 1 − inf E 1{τ ≤T } ◦ X• z∈D

≡ 1 − q ε Px {τ ε > (k − 1)T } and thus by the induction assumption (IA): sup Px {τ ε > kT } ≤ x∈D

IA k 1 − q ε sup Px {τ ε > (k − 1)T } ≤ 1 − q ε . x∈D

This induction result yields together with the previously obtained bound on q ε :

12

Such an iterative application of the strong Markov property will also appear in the last chapter which is the reason for us to calculate details explicitly here. 13 See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.32 & 6.41].

86


sup Ex τ ε ≤ sup T x∈D

x∈D

Px {τ ε > kT } ≤ T

k∈N0

(1 − q ε )k

k∈N0

T = ε ≤ T e(V +η)/ε , q

(†)

the upper bound on the mean exit time. For η := inequality implies:

δ 2

, Chebyshev’s

sup Px {τ ε ≥ e(V +δ)/ε } ≤ e−(V +δ)/ε sup Ex τ ε ≤ T e−δ/(2ε) −−−→ 0 . ε→0

x∈D

x∈D

(b) τ ε,x > e(V −δ)/ε Fix ρ > 0 (not necessarily as above) such that S2ρ (O) ⊂ D and define θ0 ε,x τm ε,x θm+1

:= :=

0 , ε,x inf { t ≥ θm : Xtε,x ∈ Bρ (O) ∪ ∂D } (m ∈ N0 ) , ε,x ε,x ε,x inf { t ≥ τm : Xt ∈ S2ρ (O) } , Xτm ∈ Bρ (O) (m ∈ N0 ) . := ∞ , Xτε,x ∈ ∂D m

......................................................................................... ................................ ................ ....................... ............ .................... .......... ................... ......... ............ ........ . . . . . ...... .... . ..... . . ..... . . .... ... .... . .. .... . ... . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .......... ... 2ρ ....... . . . . .. . . . . . . ... . . . ....... . ..... . . . . . . .. . . . . . . ......... ............... ...... ... .. . . . ... ...... . . . . . . . ...... ..... .. ...... ........ .. . .... .... ... ... ........ ... . . . . . . . . . . . . ... ... ε,x .... ... ... .... .. . . . ... ... .. . .................. .. . . . . .. 0 . . . . . . . . . . ... ............................... ................. .................. ...... .. .. . . . . ... .......... .. .................. . .. .. . . . . . . . . ... . ... .. ....... .. . . . ... .... . .. . . .. ... ...... .... ... . . . .. ... ... .. .. ... . .. .... .... .. .. . . .... ... ... ... .... .. . .. .... ... ... ... ... ... ... .. .. ... .. ... .. ... ... .. ..... .. ... ... .............. .. ... .. ... ... .. .. .. .. .... ... ... . . . . . .... . ... ... ... .... . ... ... ... .... ........ ε,x ... .. ... .. ....... ... .... ... ... .. .. ..... . ... .. ... .... ....... .................... 1 ................ ... ... ..... . ... . . . . ........... . . . . . . . . . . . . ... ... . .. .. ... ..... ......... ... ... ... ... ε,x ..... ... ................ ... ..... ... ... ... ............... .. .... .... ... ... ..... .. ... ...............1 ..... ... . . ... . . . . . . . . . . . . . .......... ...... ... ... ... .. . ...... .. ... .. ........ .. ....... .. .. ....... ..... ... ......... .... ... .. .................. .......................... ... ... ... .......... ... .. ... . ... . . . . . . . . ............... ... ε,x .... .. ... ..... ... ..... ........ 2 ... ..... ........ ............... ... ........................................... ......... . . . . . . . . ................ . . . . . . ............. ........... . ............ ........................... ............................. ................. ................. ..... ....... .......... .................... ......................... ............... ....................... ........................................................................................................................

S

xq

τ

(0)

r

O

s

τr

rθ

r

D

θ

r

τ2ε,x

= τ ε,x

X•ε,x

ε,x ε,x Fig. 2.4 Sketch of the stopping times τm and θm+1 .

ε,x From these definitions, it follows that on {τ ε,x = τm } for m ≥ 1, X•ε,x hits Bρ (O) before it hits ∂D. Thus for all m ≥ 1, ε } ≤ sup Py Xσερε ∈ ∂D . Px {τ ε = τm y∈S2ρ (O)


87

By applying lemma 2.4.10 with N := ∂D, ρ and ε0 > 0 can be chosen such that δ sup Py Xσερε ∈ ∂D < e(−V + 2 )/ε y∈S2ρ (O)

for all ε < ε0 . Consequently, ρ > 0 has been fixed such that for all m ∈ N and ε < ε0 , ε } < e−(V − 2 )/ε . sup Px {τ ε = τm δ

(††)

x∈D

ε,x ε,x −τm−1 ≤ T2 (where From the above definitions it also follows that if θm T2 < ∞ is some parameter to be specified in a moment), then X•ε,x must cover the distance ρ between Bρ (O) and S2ρ (O) in time T2 , i.e. ε ε,y ε sup |Xt − y| ≥ ρ (m ∈ N). Px θm − τm−1 ≤ T2 ≤ sup P y∈D

t∈[0,T2 ]

Furthermore, using lemma 2.4.12 with the above choice of ρ and c := V + δ/2, one can choose T2 := T (ρ, c) < ∞ and modify ε0 > 0 such that for all ε < ε0 , δ ε,y sup P sup |Xt − y| ≥ ρ < e−(V − 2 )/ε . y∈D

t∈[0,T2 ]

Consequently, T2 < ∞ has been fixed such that for all m ∈ N and ε < ε0 : δ ε ε sup Px θm − τm−1 ≤ T2 < e−(V − 2 )/ε . (†††) x∈D

Assume for the moment that τ ε,x ≤ k T2 for some k ∈ N : By definition ε,x there is a random variable m ∈ N0 such that τ ε,x = τm ; here, either ε,x m ≤ k or m > k, but in the latter case X• has already performed m > k hits of Bρ (O), (see figure 2.4); if the lengths of these excursion ε,x ] were ≥ T2 for all n = 0, . . . , m − 1 (≥ k), then time intervals [τnε,x , τn+1 k T2 < m T2 ≤ τ0ε,x +

m−1

ε,x τn+1 − τnε,x = τ ε,x ≤ k T2 ,

n=0

a contradiction. Therefore it follows for all k ∈ N0 that { τ ε,x ≤ k T2 } ⊂

k :

ε,x { τ ε,x = τm } ∪

m=0

k : m=1

ε,x ε,x θm − τm−1 ≤ T2

88


and thus Px { τ ε ≤ k T 2 } ≤

k

ε Px { τ ε = τm } +

m=0

k

ε ε P x θm − τm−1 ≤ T2

m=1

≤ Px { τ = ε

τ0ε

} + 2ke

−(V − δ2 )/ε

for all ε < ε0 , where in the last estimate, (††) and (†††) have been used. Now plug in = < k := T2−1 e(V −δ)/ε + 1 ( . denoting the integer part) to get: Px τ ε ≤ e(V −δ)/ε ≤ Px { τ ε ≤ k T2 } ≤ Px { τ ε = τ0ε } + 2 k e−(V − 2 )/ε δ

≤ Px { Xσερε ∈ / Bρ (O) } + 4 T2−1 e−δ/(2ε)

ε→0

−−−→ 0

by lemma 2.4.11. Altogether, the first claim concerning the first exit time is proven. lim ε log Ex τ ε = V

(c)

ε→0

The upper bound, lim sup ε log Ex τ ε ≤ V , has ε→0

already been verified in (†). For the lower bound note that Chebyshev’s inequality states that Px {τ ε > e(V −δ)/ε } ≤ e−(V −δ)/ε Ex τ ε and thus, as Px {τ ε > e(V −δ)/ε } → 1, it follows that for any δ > 0, ε→0 ε log Ex τ ε ≥ V − δ + ε log Px τ ε > e(V −δ)/ε −−−→ V − δ + 0 . 2) lim Px {Xτεε ∈ N } = 0 ε→0

Here, N ⊂ ∂D denotes a fixed closed set such

that VN := inf N V (O, . ) > V ; if inf N V (O, . ) = ∞, then some VN ∈ (V , ∞) is fixed instead. Again, for any ρ > 0 (to be specified later), Px {Xτεε ∈ N } ≤ Px Xσερε ∈ / Bρ (O) + sup Py {Xτεε ∈ N } . y∈Bρ (O)

By lemma 2.4.11 the first summand on the right tends to 0 as ε → 0 for any choice of ρ. Thus it remains to be verified that also


89 ε→0

Py {Xτεε ∈ N } −−−→ 0 .

sup y∈Bρ (O)

Fix η ∈ 0, VN3−V and choose ρ, ε0 > 0 according to lemma 2.4.10 such that sup z∈S2ρ (O)

Pz {Xσερε ∈ N } ≤ e−(VN −η)/ε

(‡)

ε,x and the strong Markov property it for all ε < ε0 . From the definition of τm follows that for all T3 ∈ R+ and κ ∈ N0 , ;

sup |Xtε,z − z| ≥ ρ

sup Pz {τκε ≤ κ T3 } ≤ κ sup P

z∈D

z∈D

;

t∈[0,T3 ]

on the other hand, by applying lemma 2.4.12 with the above choice of ρ and c := VN − η, one can choose T3 ≡ T3 (ρ, VN , η) < ∞ such that the latter term is further estimated as ; ≤ e−(VN −η)/ε

sup |Xtε,z − z| ≥ ρ

sup P z∈D

t∈[0,T3 ]

for all ε < ε0 . Consequently, there is T3 < ∞ such that for all ε < ε0 and κ ∈ N0 : (‡‡) sup Pz {τκε ≤ κ T3 } ≤ κ e−(VN −η)/ε . z∈D

ε,Xθε,y ε,y ε ε,y ε,y and τ ε,y ≡ θ ε,y + σ m , the Since14 τ ε,y > τm−1 ∈ Fτm−1 ⊂ Fθ m m m ρ strong Markov property implies here that for any y ∈ D : Py

ε ε ∈ N Xτεm ∩ {τ ε > τm−1 } =

=

dP EFθm 1 X ε,y ∈N ε,y

{

ε,y τ ε,y >τm−1

}

ε,y {τ ε,y >τm−1 }

ε

≤ Py {τ >

ε τm

dP P Xσε,z ε ∈ N ρ

ε τm−1 }

·

sup z∈S2ρ (O)

Pz

z=Xθε,y ε (.)

∈N . m

Xσερε

Now it follows for all y ∈ Bρ (O), κ ∈ N and ε < ε0 : Py {Xτεε ∈ N } ≤

κ

ε ε ε ∈ N , τ Py Xτεm > τm−1

m=1

≤κ

sup z∈S2ρ (O)

+ Py {τ ε > τκε }

Pz Xσερε ∈ N + Py {τ ε > κ T3 } + Py {τκε ≤ κ T3 }

≤ 2κ e−(VN −η)/ε + Py {τ ε > κ T3 } ; 14

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.93f.].

90


the first inequality is a general decomposition of the set {Xτεε ∈ N } in which it has been used that τ ε,y > τ0ε,y = 0 for all y ∈ Bρ (O) by definition; the second step follows from the previous application of the strong Markov property and the last inequality is due to (‡) and (‡‡). Further reducing ε0 (if needed) such that (†) holds true for all ε < ε0 , using Chebyshev’s inequality and plugging in the integer part of e(V +2η)/ε , < = κ := e(V +2η)/ε , and using that V − VN + 3η < 0 due to the choice of η, the previous estimate implies: lim sup

sup

ε→0

y∈Bρ (O)

Py {Xτεε ∈ N }

T (V +η)/ε ≤ lim sup 2κ e−(VN −η)/ε + e κ T3 ε→0 - . T ≤ lim sup 2e V −VN +3η /ε + ε→0 T eη/ε − e−(V +η)/ε

=

0.

3

As has been observed above, this concludes the proof of the claim, lim Px {Xτεε ∈ N } = 0. ε→0 lim Px Xτεε − z ∗ < δ = 1 follows hereby with N := {z ∈ ∂D : ε→0

|z − z ∗ | ≥ δ}.

Remark 2.4.13 (the corresponding Stratonovich SDE). Instead of proposing the Itˆ o-SDE (2.1) as above, it is also intriguing to consider the corresponding Stratonovich-SDE, √ X0ε,x = x , dXtε,x = b (Xtε,x ) dt + ε σ (Xtε,x ) ◦ dWt , where “◦” denotes the stochastic integral in the sense of Stratonovich.15 This equation is equivalent to the Itˆ o-SDE 0 1 √ ε dXtε,x = b (Xtε,x ) + σ (Xtε,x ) Dσ (Xtε,x ) dt + ε σ (Xtε,x ) dWt , 2 where the coordinates of σ(x) Dσ(x) are given by

σ(x) Dσ(x)

k

:=

d i,j=1

σij (x)

∂σkj (x) ∂xi

for k = 1, . . . , d . Hence, one also needs to cope with an ε-dependent drift 15

See e.g. Arnold [Ar 98, Ch.2] or Hackenbroch and Thalmaier [Hb-Th 94, Ch.5].

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

bε (x) := b (x) +

91

ε σ (x) Dσ (x) 2

in the Itˆ o equation. Large deviation results on such equations can also be obtained: Freidlin and Wentzell (see [Fr-We 98, p.154f.] and [We-Fr 70, p.7ff.]) prove that the action functional is the same as for (2.1), if the coordinates of bε converge to those of b uniformly in x as ε → 0. Therefore it would be conceptually equivalent to investigate the above Stratonovich-SDE instead of (2.1), as Freidlin [Fr 00] does indeed; also see Carmona and Freidlin [Cm-Fr 03] who assume the coefficients to be (globally) bounded. In general, one then needs to assure that ε ε→0 Dσ (Xtε,x ) σ (Xtε,x ) −−−→ 0 2 uniformly with respect to x. It seems as if this assumption should be added to Freidlin’s [Fr 00] premises. However, the rationale of our work is to restrict the assumptions on b and σ to a minimum in order not to overburden the technical apparatus for installing locality into the treatment of linear differential systems: Since in most applications (see section 2.6), where σ = idRd , the Itˆ o-SDE and the Stratonovich-SDE coincide, it seems reasonable to stick to the Itô-SDE.

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior This subsection describes the concept of metastability, made precise by the convergence of the system to the sublimiting distributions on the respective time scales; the exposition here follows Freidlin [Fr 77], [Fr 00]; also see Freidlin and Wentzell [Fr-We 98, Ch. 6] as well as Carmona and Freidlin [Cm-Fr 03]. It is required throughout the section that the standing assumptions 2.1.1 are satisfied. For describing the behavior of X ε a hierarchy of “cycles” in the state space, together with their rotation and exit rate as well as their main state and the entrance point to the next cycle is needed. More precisely, the cycles will be defined as certain subsets of L := {1, . . . , l} , equipped with a cyclic order which expresses the order of transition between the domains of (Ki )i∈L . In this construction the constants Vij := V (Ki , Kj )

(i, j ∈ L)

will play a central role. Due to (K) the entries of this matrix (Vij )i,j∈L are zero on the diagonal and strictly positive otherwise. In case that the sets Ki

92


do not consist of one point only, but are compact subsets of Rd , this definition of Vij would read Vij := V (x, y) for some x ∈ Ki and y ∈ Kj

(i, j ∈ L) ;

this definition then does not depend on the choices of x and y due to assumption (K3) in remark 2.1.2. Hence, the matrix (Vij )i,j∈L is well defined also in this case. One more definition is needed (Freidlin and Wentzell [Fr-We 98, p.177]) now. Definition 2.5.1 (arrows, W-graphs on L). Let L := {1, . . . , l} be as above (or an arbitrary finite set whose elements are labeled as 1, . . . , l). (a) An arrow (α → β) is an ordered tuple (α, β) ∈ L2 where α and β are called the initial point and the endpoint of the arrow, respectively. (b) A set g of arrows is called a graph on L. (c) Let a subset W ⊂ L be fixed; then a set g of arrows is called a W -graph on L, if 1) (α → β) ∈ g implies that α ∈ L \ W and α = β ; 2) for any α ∈ L\W there exists exactly one element of g whose initial point is α; 3) for any α ∈ L\W there exists a sequence in g leading from α into W (i.e. a sequence α → α1 → α2 → · · · → αm−1 → αm of arrows contained in g such that αm ∈ W ) . Given the initial two assumptions 1) and 2), 3) is equivalent to the following one: 4) There are no closed loops in g (i.e. for any sequence α0 → α1 → α2 → · · · → αm−1 → αm of arrows contained in g, it follows that α0 = αm ) . Let GW (L) := { g : g W -graph on L} denote the set of all W -graphs on L; if W consists of one element i only, one defines Gi (L) := G{i} (L) , the set of i-graphs on L. In other words, Gi (L) is the set of all sets g of arrows (α → β) such that α ∈ L \ {i}, to each initial point α exactly one element of g is attached and by forming sequences α0 → α1 → α2 → · · · → αm−1 → αm of arrows in g no closed paths are possible: α0 = αm .


93

Example 2.5.2 (Simple cases of i-graphs). Let L ≡ {1, . . . , l}. Then one obtains for l = 1: G1 (L) = ∅ , l = 2: G1 (L) = {(2 → 1)} and G2 (L) = {(1 → 2)} , and l = 3: G1 (L) = {(2 → 1), (3 → 1)} , {(2 → 1), (3 → 2)} , {(3 → 1), (2 → 3)} . 1

r oS S 7 S Sr r 3 2

1

1

r

3 r

SS o S Sr 2

r

r 3

7 r2

Fig. 2.5 Visualization of the 1-graphs on {1, 2, 3}

The Hierarchy of cycles The hierarchy shall describe the transitions of X ε,x between the different domains of attraction which belong to K1 , . . . , Kl . Successively the k-cycles will be defined. The cycles of any rank k are subsets of L equipped with a cyclic order. Furthermore, to any cycle there will be assigned the main state, the rotation rate and the exit rate. The notation and terminology coined here will be illustrated in example 2.6.2 for the instructive situation of a two-well potential function. k = 0: A 0-cycle is an element of L, i.e. the set of all 0-cycles is defined as C (0) := L. M : C (0) → L , M (i) := i defines the main state of a 0-cycle; the rotation rate R and the stationary distribution rate mi vanish on 0-cycles by definition, R(i) := 0

and

mi (i) := 0

(i ∈ C (0) ≡ L) ,

respectively. The exit rate E(i) of a 0-cycle i is given by the mapping ⎧ ⎨ min ViJ , l ≡ |L| ≥ 2 , (2.11) E(i) := J∈L\{i} E : C (0) → (0, ∞] , ⎩ ∞ , l =1. The meaning of the terminus “exit rate” (which will be also defined for cycles of higher order) will be explained in theorem 2.5.3; it states that E yields the precise exponential rates for the exit time of the respective domains, analogously to the exit time law 2.4.6.

94


Now if l ≥ 2 (i.e. if the situation is non-trivial for what follows) and if i ∈ L is fixed, one can choose J ≡ J(i) ∈ L \ {i} such that this minimum (2.11) is attained; by (G) it is actually postulated that this J is unique; thus there is a well-defined mapping J : C (0) → C (0) , implicitly given by Vi,J(i) = E(i) ≡ min ViJ . J∈L\{i}

This definition of J is also expressed by saying that J(i) follows after i. k = 1: Applying the mapping J successively yields a sequence {J r (i)}r∈N0 in C (0) for each i ∈ C (0) . As C (0) ≡ L is a finite set, this sequence returns to itself at step r(i) := min{ r : J n (i) = J r (i) for some n < r }

(i ∈ C (0) ) .

At the step r(i) the sequence becomes periodic. Therefore it decomposes into 0-cycles i, J(i), . . . , J n(i)−1 (i) and the closed loop {J n(i) (i), J n(i)+1 (i), . . . , J r(i) (i) = J n(i) (i)}. If the cyclic order of the loop (which is simply a subset of L) shall be emphasized, it is written as succession of arrows, J n(i) (i) → J n(i)+1 (i) → · · · → J r(i) (i) = J n(i) (i) ; this loop does not contain i, J(i), . . . , J n(i)−1 (i) by definition. We conclude that each sequence {J r (i)}r∈N0 is characterized by the objects i, J(i), . . . , J n(i)−1 (i) , J n(i) (i) → J n(i)+1 (i) → · · · → J r(i) (i) = J n(i) (i) which are called the 1-cycles generated by i. Let C (1) denote the set of all 1-cycles generated by some i ∈ C (0) ; its elements are called 1-cycles. Now M : C (1) → L , implicitly given by the requirement VM(C),J(M(C)) = max Vi,J(i) i∈C

(2.12)

defines the main state of a 1-cycle C ; again, the uniqueness of such an element M (C) and hence the well-definedness of the main state mapping M is postulated in assumption 2.1.1(G). The rotation rate R of 1-cycles is given by R : C (1) → R+ ,

R(C) := max Vi,J(i) , i∈C

(2.13)


95

the stationary distribution rate mC for a 1-cycle C is the map16 m C : C → R+ ,

mC (i) := R(C) − Vi,J(i)

and the exit rate E(C) of a 1-cycle C is defined by ⎧ ⎨ min (mC (i) + Vij ) , C (1) ≥ 2, (1) / E : C → (0, ∞], E(C) := i∈C,j ∈C ⎩ ∞ , C (1) = 1.

(2.14)

If C (1) consists of one element only, this is equal to L, considered as a point if (1) set; in this case the recursive definition of cycles stops. Otherwise,(1) C ≥ 2, there is a cyclic order on these 1-cycles: Fix an arbitrary C ∈ C and define i∗ , j ∗ ∈ L by the requirement mC (i∗ ) + Vi∗ j ∗ =

min (mC (i) + Vij ) ≡ E(C) .

i∈C,j ∈C /

By (G) we assume that i∗ and j ∗ are unique, since they are uniquely determined by (2.14). Now let J(C) ∈ C (1) be the (unique) 1-cycle which contains j ∗ . This procedure provides a well-defined mapping J : C (1) → C (1) and one says that the 1-cycle J(C) follows after the 1-cycle C. k > 1: Assume that the set C (k) of k-cycles has already been defined and that the main state map M , the rotation rate R, the stationary distribution rate m, the exit rate E and the “follow”-map J have been extended to C (k) . Iterating J yields a sequence {J r (C)}r∈N0 in C (k) for each C ∈ C (k) . As C (k) is finite, this sequence returns to itself at step r(C) := min{ r : J n (C) = J r (C) for some n < r }

(C ∈ C (k) ) .

Therefore the sequence decomposes into C, J(C), . . . , J n(C)−1 (C) , J n(C) (C) → J n(C)+1 (C) → · · · → J r(C) (C) = J n(C) (C) which are called the (k + 1)-cycles generated by C. C (k+1) denotes the set of (k + 1)-cycles, i.e. of all (k + 1)-cycles generated by some C ∈ C (k) . For a

16

Freidlin [Fr 00, p.338] defines the stationary distribution rate mC for a 1-cycle C with the opposite sign, i.e. mC (i) := Vi,J(i) − R(C); however, this would result in a negative rate; here, we use the corrected definition of mC (equivalently of E(C)) for 1-cycles from Carmona and Freidlin [Cm-Fr 03, (2.2)].

96


given C ∈ C (k+1) the requirement Vαβ = min min min g∈GM (C)

i∈C g∈Gi (C)

(α→β)∈g

Vαβ

(2.15)

(α→β)∈g

uniquely determines M ≡ M (C) ∈ C(⊂ L) thus extending the main state map M : C (k+1) → L on (k + 1)-cycles; recall that Gi (C) denotes the set of i-Graphs on C as in definition 2.5.1. The rotation rate R of (k + 1)-cycles is given by R : C (k+1) → R+ ,

R(C) := max{ E(C ) : C ∈ C (k) , C ⊂ C } ;

(2.16)

the stationary distribution rate mC for a (k + 1)-cycle C is the map mC (i) := min Vαβ − min m C : C → R+ , g∈Gi (C)

(α→β)∈g

g∈GM (C) (C)

Vαβ

(α→β)∈g

and the exit rate E(C) of a (k + 1)-cycle C is defined by ⎧ ⎨ min (mC (i) + Vij ) , C (k+1) ≥ 2 , (k+1) / → (0, ∞] , E(C) := i∈C,j ∈C E :C ⎩ ∞ , C (k+1) = 1 . (2.17) If C (k+1) consists of one element only, this is equal to L as a point set and the recursive definition of cycles stops. Otherwise, if C (k+1) ≥ 2, there is a cyclic order on the (k + 1)-cycles again: Fix an arbitrary C ∈ C (k+1) and define i∗ , j ∗ ∈ L by the requirement mC (i∗ ) + Vi∗ j ∗ =

min (mC (i) + Vij ) ≡ E(C) ,

i∈C,j ∈C /

where mC (i) denotes the newly defined stationary distribution rate for (k+1)cycles. By (G) we assume that i∗ and j ∗ are unique, since they are uniquely determined by (2.17). Now let J(C) ∈ C (k+1) be the unique (k + 1)-cycle which contains j ∗ ; i∗ is called the exit point of C and j ∗ is the entrance point of J(C). This procedure now provides a well-defined map J : C (k+1) → C (k+1) and we say that the (k + 1)-cycle J(C) follows after the (k + 1)-cycle C. Example 2.6.2 shall give an illustration for the above definitions. There, a two well potential function is considered and the previous construction of the hierarchy of cycles is seen to terminate at k = 1. For a discussion of the three well situation which corresponds to a maximal degree k = 2 see Freidlin [Fr 00, p.344].


97

Next, an explanation of the above terminology “exit rates” shall be cited. For this purpose let t→∞

Di := {x ∈ Rd : Xt0,x −−−−→ Ki }

(i ∈ L)

denote the domain of attraction of Ki under the deterministic motion X 0 as before which is an open subset of Rd . Furthermore, define the open set .◦ : Di D(C) := i∈C

for cycles C. The set D(C) is connected, since by taking the closures Di also the lower dimensional submanifolds are taken into account which are attracted to one of the points Kl+1 , . . . , Kl and which thus separate the domains D1 , . . . , Dl ; see remark 2.1.2 and assumption 2.1.1 (K1). Hence, > investigating the open set D(C) := i∈C Di instead17 would result in considering a disconnected set. Its topological components are the sets Di , separated by those lower dimensional submanifolds which are attracted to one of the points Kl+1 , . . . , Kl . Therefore, the corresponding exit time from D(C) coincides with the exit time from the domain Di containing the initial condition x. However, this is not the time of interest in the following discussion; rather one is concerned with the exit time from the whole cycle C in the sense that X ε,x exits the domain “spanned” by the domains Di for which i ∈ C and which also includes the separatrices between them. Theorem 2.5.3 (The exit rates). Let τCε,x := inf{ t > 0 : Xtε,x ∈ / D(C) } denote the first exit time of X ε,x from D(C). Then for any x ∈ D(C), lim ε log Ex τCε = E(C)

ε→0

and for all δ > 0, lim Px e(E(C)−δ)/ε < τCε < e(E(C)+δ)/ε = 1 .

ε→0

This generalization of theorem 2.4.6 is taken from Freidlin [Fr 00, p.339], Carmona and Freidlin [Cm-Fr 03, p.61f.] and Freidlin and Wentzell [Fr-We 98, Th.6.6.2], respectively. Being technically very involved the proof of this theorem is beyond the scope of this book: It is necessary to consider balls around the points Ki for which stopping times are then defined analogously as in the proof of theorem 2.4.6 above; figure 2.4 then depicts the case for l = 1. For 17

As Freidlin [Fr 00] and Carmona and Freidlin [Cm-Fr 03] do

98


the arguments of the proof see Freidlin and Wentzell [Fr-We 98, Ch.6]; also see Freidlin [Fr 77]. Note that the exit rates E( . ) considered here coincide with the rates C( . ) as used in the formulation of theorem 6.6.2 by Freidlin and Wentzell [Fr-We 98]: More precisely, let π be some cycle and define Vαβ , (2.18) C(π) := A(π) − min min i∈π g∈Gi (π)

where A(π) :=

min

g∈GL\π (L)

(α→β)∈g

Vαβ

(2.19)

(α→β)∈g

(see p.199 and p.201 in Freidlin and Wentzell [Fr-We 98]). Now GL\π (L) consists of those graphs whose arrows take their courses in π and whose last endpoint is in L \ π; this is the same as considering graphs in π terminating in i ∈ π (elements of Gi (π) in other words) and proceeding to some j ∈ / π subsequently. Hence, unwinding the definitions of graphs and main states yields that for any cycle π of order k > 1, Vαβ − min min Vαβ C(π) ≡ min g∈GL\π (L)

⎛

= min ⎝ min i∈π,j ∈π /

g∈Gi (π)

⎛ = min ⎝ min i∈π,j ∈π /

i∈π g∈Gi (π)

(α→β)∈g

g∈Gi (π)

(α→β)∈g

⎞

Vαβ + Vij ⎠ −

(α→β)∈g

(α→β)∈g

Vαβ −

min

min

g∈GM (π) (π)

g∈GM (π) (π)

(α→β)∈g

Vαβ ⎞

Vαβ + Vij ⎠

(α→β)∈g

≡ min (mπ (i) + Vij ) i∈π,j ∈π /

≡ E(π) . Therefore the above theorem coincides with Theorem 6.6.2 by Freidlin and Wentzell [Fr-We 98]. Furthermore, note that the compactness assumption on the state space which is underlying § 6.6 by Freidlin and Wentzell [Fr-We 98] is replaced here by assumption (V), i.e. by considering the compact set {y : V (0, y) ≤ V˜ } for some large V˜ instead which X ε leaves only with an exponentially small probability; see Freidlin [Fr 77].

The support of the sublimiting distribution Using the hierarchy of cycles we shall now define the metastable state Kµ(x,ζ) for an initial value x and a time scale eζ/ε . Again the presentation follows


99

Freidlin and Wentzell ([Fr 77], [Fr 00] and [Fr-We 98, § 6.6]). An illustration will be provided in example 2.6.2 . Fix an initial value l : Di . x ∈ i=1

This set has full Lebesgue measure by assumption 2.1.1 (K). Choose i(x) ∈ L such that x ∈ Di(x) and let C (k) (x) denote the (unique) element of C (k) such that i(x) ∈ C (k) (x), i(x) ≡ C (0) (x) ⊂ C (1) (x) ⊂ · · · ⊂ C (κ−1) (x) ⊂ C (κ) = L , where κ < ∞ denotes the maximal rank in this sequence. This yields the finite sequences Ek (x) := E C (k) (x) and Rk (x) := R C (k) (x) (k = 0, 1, . . . , κ) of the corresponding exit and rotation rates, respectively. For these rates it follows that 0 ≡ R0 (x) < E0 (x) ≡ Vi(x),J(i(x)) ≤ R1 (x) ≤ E1 (x) ≤ · · · ≤ Rk (x) ≤ Ek (x) ≤ · · · ≤ Rκ (x) ≤ Eκ (x) ≡ ∞ Now fix ζ ∈ R>0 \

κ :

{Rk (x), Ek (x)}

(2.20)

k=0

and define

m∗ ≡ m∗ (x) ∈ {−1, 0, 1, . . . , κ − 1}

by Em∗ (x) < ζ < Em∗ +1 (x) , where we put E−1 (x) := 0 for definiteness. There are two cases to distinguish: ∗ ζ > Rm∗ +1 (x): In this case define µ(x, ζ) := M C (m +1) (x) . ∗

ζ < Rm∗ +1 (x): Since Rm∗ +1 (x) ≡ max{E(C ) : C ∈ C (m ) , C ⊂ C (m there is a ∗ ∗ Cˆ (m ) ∈ C (m )

such that

∗ ∗ Cˆ (m ) ⊂ C (m +1) (x)

∗

+1)

(x)},

∗

and E(Cˆ (m ) ) > ζ ;

∗ we also assume that Cˆ (m ) is the first m∗ -cycle satisfying this statement ∗ ∗ which follows after C (m ) (x) in C (m +1) (x). Again there are two cases to distinguish: ∗

∗

ζ > R(Cˆ (m ) ): In this case define µ(x, ζ) := M (Cˆ (m ) ) .

100


∗ ζ < R(Cˆ (m ) ): By the same argument as above there is a ∗ ∗ Cˆ (m −1) ∈ C (m −1)

∗ ∗ Cˆ (m −1) ⊂ Cˆ (m ) ,

such that

E(Cˆ (m

∗

−1)

)>ζ

∗

and Cˆ (m −1) is the first (m∗ − 1)-cycle which follows after the (m∗ − 1)∗ cycle containing the entrance point of Cˆ (m ) . Again there are two cases to distinguish: ∗

∗

ζ > R(Cˆ (m −1) ): In this case define µ(x, ζ) := M (Cˆ (m −1) ) . ∗ ζ < R(Cˆ (m −1) ): We proceed successively as in the previous step till we find ∗ ∗ Cˆ (m −n) ∈ C (m −n)

such that

E(Cˆ (m

∗

−n)

) > ζ > R(Cˆ (m

∗

−n)

).

Since R( . ) = 0 on 0-cycles, this condition is fulfilled after a finite number ∗ n ≤ m∗ of further steps; then define µ(x, ζ) := M (Cˆ (m −n) ). By the previous distinction of the possible cases one gets a well defined index function - l . . κ : : µ : Di × R>0 \ {Rk (x), Ek (x)} −→ L . (2.21) i=1

k=0 l >

Definition 2.5.4 (Metastable states). Fix x ∈

Di and ζ > 0, where ζ

i=1

is not contained in a finite exceptional set depending on x (see (2.20)). Let T (ε) be a time scale which is logarithmically equivalent to eζ/ε , abbreviated as T (ε) ! eζ/ε in the sequel, i.e. let T be a function T : (0, ε0 ) → R>0 , where ε0 > 0, such that 0 < lim ε log T (ε) = ζ. ε→0

Then the metastable state for the initial value x and the time scale T (ε) is Kµ(x,ζ) , where µ(x, ζ) ∈ L denotes the index function defined in (2.21).


101

The following theorem clarifies the behavior of X ε on the time scales T (ε). Theorem 2.5.5 (Sublimiting distribution). Let the assumptions 2.1.1 be satisfied and consider the solution X ε of SDE (2.1). Fix an initial value >l x ∈ i=1 Di and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Then for all open sets B ⊂ Rd , / B, 0 , Kµ(x,ζ) ∈ ε→0 ε ε PT (ε) (x, B) ≡ Px XT (ε) ∈ B −−−→ 1 , Kµ(x,ζ) ∈ B . In particular, XTε,x (ε) converges in probability to the respective metastable state, P XTε,x − −−→ Kµ(x,ζ) . (ε) − ε→0

Furthermore, for all functions f ∈ C b (Rd , R), ε→0 uεf T (ε), x −−−→ f Kµ(x,ζ) , where uεf (t, x) ≡ Ex f (Xtε ) denotes the solution of the Cauchy problem (2.5) for (2.1). The first and the third claim are a reformulation of Freidlin’s [Fr 77] theorem. The statement on the convergence in probability follows from the first part simply by taking Bη := B Kµ(x,ζ) , η for arbitrary η > 0 and hence ε→0 − K | > η ≡ PTε (ε) (x, Bη ) −−−→ 0 . P | XTε,x µ(x,ζ) (ε) Note that in citing Freidlin’s [Fr 77] theorem the notation from (2.18) and (2.19) is used again: C(π) as defined in (2.18) equals Freidlin’s [Fr 77] “c(π)” and A(π) as given by (2.19) coincides with Freidlin’s [Fr 77] “ (π)” and “A(π)”. Furthermore, note that Freidlin [Fr 77] formally assumes the coefficient functions bi and aij of the SDE (of the generator G ε ) to be globally bounded. This assumption, however, is not used in the course of the paper and can be done away; see Freidlin [Fr 00]. For additional discussions see Freidlin and Wentzell [Fr-We 98, p.202f.]; further arguments and proofs can also be found in the work by Li and Qian [Li-Qi 99] and [Li-Qi 98]. This theorem in particular shows that the transition probabilities have different limits for different choices of the time scale and these limits do depend on the initial value x. Since the resultant convergence in probability implies weak convergence, one gets that the distribution of XTε,x (ε) converges weakly to the Dirac measure at the respective metastable state,

a

−1 w −− −−→ δKµ(x,ζ) . P ◦ XTε,x (ε) ε→0

102


This fact is expressed by calling Kµ(x,ζ) the support of the sublimiting distributions for x and the time scale T (ε) ! eζ/ε , since in this sense δKµ(x,ζ) is the sublimiting distribution, in contrast to the limiting distribution ρε (section 2.2) for which it was obtained in equations (2.4) and (2.6) that t→∞

Ptε (x, B) −−−−→ ρε (B) and t→∞

uεf (t, x) −−−−→

f (y) ρε (dy) Rd

independently of x. Since the above hierarchy of cycles and the corresponding main states, rotation rates and exit rates — which all describe the behavior of X ε — do not depend on chance, the long-time evolution of this system has an intrinsic deterministic component; this is called the quasi-deterministic approximation of X ε (Freidlin [Fr 00]). More precisely, Freidlin’s [Fr 00] theorem 1 on the quasi-deterministic behavior of the solution X ε of the SDE (2.2) states the following, where L denotes the Lebesgue measure on R : Theorem 2.5.6 (Quasi-deterministic behavior). Let the assumptions l > 2.1.1 be satisfied. Fix x ∈ Di and a time scale T (ε) ! eζ/ε , where ζ > 0 i=1

is not contained in a finite exceptional set depending on x (see (2.20)). Then for any c, Γ > 0 P L t ∈ [0, Γ] : Xtε,x > c − K −−−−→ 0 . µ(x,ζ) T (ε) ε→0

Sketch of Proof. According to the definition of µ(x, ζ) in the distinction of cases preceding (2.21), there exist m∗ ≡ m∗ (x) ∈ N0 ∪ {−1} , a number s ∈ {0, . . . , m∗ + 1} of steps and a cycle ∗ ∗ Cˆ (m +1−s) ∈ C (m +1−s)

and

such that

E(Cˆ (m

∗

+1−s)

) > ζ > R(Cˆ (m

∗

+1−s)

)

∗ µ(x, ζ) = M (Cˆ (m +1−s) ) .

Applying theorem 2.5.5 to the time scale T(ε) := t0 T (ε) ! eζ/ε , for some t0 > 0, yields that for η > 0, lim Px XtεT (ε) ∈ / B(Kµ(x,ζ) , η) for all t ∈ [0, t0 ] = 0 . ε→0

The exit rate theorem 2.5.3 directly implies ∗ δ ∈ (0, E(Cˆ (m +1−s) ) − ζ) and all y ∈ B(Kµ(x,ζ) , η), lim Py

ε→0

ˆ (m∗ +1−s) )−δ)/ε

Γ T (ε) < e(E(C

< τCεˆ (m∗ +1−s)

that = 1 ,

for

all


since E(Cˆ (m

∗

+1−s)

lim Px

ε→0

103

) > ζ ; in other words,

∗ XtεT (ε) ∈ D(Cˆ (m +1−s) ) for all t ∈ [t0 , Γ]

= 1.

By the definition of the rotation rate, it follows that E(C ) ≤ ∗ ∗ R(Cˆ (m +1−s) ) < ζ for all cycles C Cˆ (m +1−s) . The exit rate theorem ∗ 2.5.3 therefore implies that for all δ ∈ (0, ζ − R(Cˆ (m +1−s) )) , z ∈ D(C ) and t1 > 0, lim Pz τCε < e(E(C )+δ)/ε < t1 T (ε) = 1 ε→0

and hence, lim Pz

ε→0

XtεT (ε) ∈ D(J(C )) for some t ∈ (0, t1 )

= 1.

Now fix η > c; then the closed ball Bc (Kµ(x,ζ) ) is contained in the open ball B(Kµ(x,ζ) , η). Piecing together the previous time scales t0 T (ε) > ∗ exp(R(Cˆ (m +1−s) )/ε) by using the Markov property, it can be seen that the system X ε,x mostly stays near Kµ(x,ζ) during the interval [0, ΓT (ε)]. The ∗ system also visits cycles C Cˆ (m +1−s) ; however, the lengths of these excursions is of exponential order smaller than ζ. Following this sketch of proof the claim can be deduced; see Freidlin [Fr 00, p.340]. Freidlin [Fr 00, Cor.1] also notes the following corollary. It provides the decisive connection between the LDP and the quasi-deterministic approximation defined above. Corollary 2.5.7 (Boundedness in probability I). Let the assumptions l > Di and a time scale T (ε) ! eζ/ε , where ζ > 0 2.1.1 be satisfied. Fix x ∈ i=1

is not contained in a finite exceptional set depending on x (see (2.20)). Then for the set Fx,ζ := { y ∈ Rd : V (x, y) ≤ ζ } , it follows that for any Γ > 0 ε ∈ / F for some t ∈ [0, Γ] = 0 . lim Px XtT x,ζ (ε) ε→0

Furthermore, one obtains that for any p > 0 ε P X . T (ε) − Kµ(x,ζ) p −− −−→ 0 . ε→0 L ([0,Γ],L)

Sketch of Proof. The set Fx,ζ is compact (hence bounded) due to assumption (V). By fixing a continuation of b and σ outside the open set Fδx,ζ := {y ∈ Rd : dist(y, Fx,ζ ) < δ} for some δ, the large deviation principle

104


2.3.4 can be applied. This approach provides the asymptotic behavior for the exit time from Fδx,ζ in terms of the rate function and, hence also, in terms of the quasipotential as in theorem 2.4.3, since the first exit time from Fδx,ζ does not depend on the specific continuation of b and σ on Fδx,ζ . This proves the first statement. The second claim can be proven from the first statement and theorem 2.5.6 by a similar argument as will be performed in theorem 2.5.10 below. The subsequent conclusion has also been drawn by Freidlin [Fr 00, Cor.2]: Corollary 2.5.8 (Boundedness in probability II). Let the assumptions l > Di and a time scale T (ε) ! eζ/ε , where ζ > 2.1.1 be satisfied. Fix x ∈ i=1

0 is not contained in a finite exceptional set depending on x (see (2.20)). Furthermore, suppose that ζ < Vµ(x,ζ),J(µ(x,ζ)) . Then for the set Hx,ζ := { y ∈ Rd : V (Kµ(x,ζ) , y) ≤ ζ } , it follows that for any Γ > 0 and c ∈ (0, Γ] ε lim Px XtT = 1 . (ε) ∈ Hx,ζ for all t ∈ [c, Γ] ε→0

Sketch of Proof. First note that also the set Hx,ζ is bounded due to assumption (V). As in the sketch of proof for theorem 2.5.6 µ(x, ζ) is the main state of the cycle corresponding to ζ. Hence, any neighborhood of Kµ(x,ζ) is hit before c T (ε) with probability converging to 1. However, as in corollary 2.5.7 above, the bound on the quasipotential on the respective set Hx,ζ (or Fx,ζ ) provides the time scale on which this set cannot be left. More precisely, the exit time law 2.4.6 for D := Hδx,ζ := {y ∈ Rd : V (Kµ(x,ζ) , y) < ζ + δ}, for small δ > 0, and O := Kµ(x,ζ) implies, since V := inf ∂Hδx,ζ V (Kµ(x,ζ) , . ) = ζ + δ , that for all y ∈ D and δ > 0, lim Py eζ/ε < τHε δ = 1 ; ε→0

x,ζ

in other words, lim Px

ε→0

XtεT (ε) ∈ Hδx,ζ for all t ∈ [c, Γ]

= 1.

The assumption that the claim is false can be led to a contradiction this way.


105

Remark 2.5.9. As has already been remarked before, the property of Ki consisting of one single point might be relaxed: If Ki were a compact set, such that also (K3) and (K4) are satisfied, then the assertions would hold true, if suitably modified. For instance, theorem 2.5.6 would state the following: For any c, Γ > 0 ε,x P L t ∈ [0, Γ] : dist XtT > c , K −−−−→ 0 , µ(x,ζ) (ε) ε→0

where dist(x, B) ≡ inf y∈B |x − y|. Also see Freidlin [Fr 77, Remark 1] and [Fr 00]. Now we prove a consequence of the metastability which is not contained in the workings by Freidlin and Wentzell. This theorem is of central importance for the investigations of local Lyapunov exponents to come. Its final assertion can be regarded as a “sublimiting ergodic theorem”. Theorem 2.5.10 (Consequences of the metastability of X ε,x ). Let the l > assumptions 2.1.1 be satisfied. Fix x ∈ Di and a time scale T (ε) ! eζ/ε , i=1

where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Moreover, consider some continuous function f ∈ C(Rd , K). Then for all open sets B ⊂ K , / B, 0 , f (Kµ(x,ζ) ) ∈ ε→0 ε Px f XT (ε) ∈ B −−−→ 1 , f (Kµ(x,ζ) ) ∈ B . In particular, f (XTε,x (ε) ) converges in probability, P f XTε,x −− −−→ f (Kµ(x,ζ) ) . (ε) ε→0 Since the resultant convergence in probability implies weak convergence, one gets that the distribution of f (XTε,x (ε) ) converges weakly to the Dirac measure at the corresponding point, −1 w ) P ◦ f (XTε,x −− −−→ δf (Kµ(x,ζ) ) . (ε) ε→0 Furthermore, 1 T (ε)

0

T (ε)

f Xuε,x du =

0

1

P f Xtε,x − −−→ f (Kµ(x,ζ) ) . T (ε) dt − ε→0

Proof. The first statement follows from theorem 2.5.5 together with the continuity of the function f . The second and the third statements follow immediately from the first one. It remains to verify the fourth assertion:

106


For this purpose we fix an arbitrary η > 0; then ; 1 T (ε) ε,x P f Xu ( . ) du − f Kµ(x,ζ) > η T (ε) 0 1 ε,x = P > η f Xt·T ( . ) dt − f K µ(x,ζ) (ε) 0 1 ε,x ≤P f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 1 ε,x ε = P τR ( . ) ≤ 1 , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 1 ε,x + P τRε ( . ) > 1 , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0

≡ Aε + B ε , where

Aε := P

1

τRε ( . ) ≤ 1 , 0

ε,x f X dt > η ( . ) − f K µ(x,ζ) t·T (ε)

and B := P ε

τRε ( . )

> 1, 0

and where τRε := inf

1

ε,x , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η

ε,x ≥ R t > 0 : Xt·T ( . ) − K µ(x,ζ) (ε)

(ε, R > 0)

ε,x denotes the first exit time of Xt·T (ε) ( . ) from the open ball B Kµ(x,ζ) , R with center Kµ(x,ζ) and radius R. ε→0

ε→0

It needs to be shown that Aε −−−→ 0 and B ε −−−→ 0 which is done in the sequel: First, we have that ε→0

Aε ≤ P { τRε ≤ 1 } −−−→ 0 , if R is sufficiently large; namely, choose R sufficiently large such that Fx,ζ ⊂ B(Kµ(x,ζ) , R) and use that X ε is bounded in probability (corollary 2.5.7 with Γ = 1). For estimating B ε fix ω ∈ {τRε > 1} and c > 0 in order to get:

ε,x f Xt·T (ε) (ω) − f Kµ(x,ζ) dt 0 ε,x = f Xt·T (ε) (ω) − f Kµ(x,ζ) dt 1

ε,x (ω)−Kµ(x,ζ) > c t∈[0,1] : Xt·T (ε)


+

ε,x (ω)−Kµ(x,ζ) ≤ c t∈[0,1] : Xt·T (ε)

107

ε,x f Xt·T (ε) (ω) − f Kµ(x,ζ) dt

ε,x ≤ maxB(Kµ(x,ζ) ,R) f − f Kµ(x,ζ) · L t ∈ [0, 1] : |Xt·T (ε) (ω) − Kµ(x,ζ) | > c

+ maxB(Kµ(x,ζ) ,c) f − f Kµ(x,ζ)

;

now if c is chosen sufficiently small such that the latter summand is bounded by η/2, η maxB(Kµ(x,ζ) ,c) f − f Kµ(x,ζ) < 2 (f is continuous by assumption) and if one defines m := maxB(Kµ(x,ζ) ,R) f − f Kµ(x,ζ) , then it follows from the previously obtained estimate that 1 ε,x ε ε ≡ P τR ( . ) > 1 , B f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 ≤ P ω : τRε (ω) > 1 ,

≤

η m · L t ∈ [0, 1] : > η − Kµ(x,ζ) | > c + 2 η ε,x P L{ t ∈ [0, 1] : | Xt·T (ε) ( . ) − Kµ(x,ζ) | > c } > 2 m ε,x | Xt·T (ε) (ω)

ε→0

−−−→ 0 by the stochastic convergence stated in theorem 2.5.6 with Γ = 1.

Remark 2.5.11 (Further concepts of metastability). The concept of metastability used here is the one as described by Freidlin [Fr 00]. Other authors also address related phenomena and offer different routes to metastability. We only mention a few of them. Cassandro et al. [Cs-Ga-Ol-Va 84] choose the “pathwise approach”: Here, the Curie-Weiss model (an Ising spin system) and the contact process (taking its values in the power set of Z), two Markov processes with discrete state spaces, are investigated; time averages are taken along each path and these averages are then shown to converge to measure valued jump processes. The same group, Galves et al. [Ga-Ol-Va 87], also applies this rationale to the gradient SDE (2.2) for a potential function with two wells. Surveys on this approach are given by Vares [Va 96] and Olivieri and Vares [Ol-Va 05]. Bovier et al. [Bv-Ec-Gd-Kn 04] and [Bv-Gd-Kn 05] develop a potential theoretic approach: By calculating capacities metastable exit times can be deduced. These authors are then able to obtain precise prefactors for the exit times, too.

108


Huisinga et al. [Hui-Mey-Sch 04] study metastability using the “V -uniform ergodicity” of the process; this is a spectral theoretic assumption from which these authors deduce that the generator admits a spectral gap and the eigenfunctions provide a decomposition of the state space into “almost-absorbing subsets”. Remark 2.5.12 (Simulated annealing). A stochastic process which is closely related to the gradient SDE (2.2) is the “simulated annealing process”, i.e. the solution process X of the SDE (t ≥ 0) , dXt = − ∇U (Xt ) dt + ε(t) dWt where ε(t) → 0 is a time-varying noise-intensity. For the time dependent “temperature” ε(t) one can then consider cooling schedules ε(t) := εc (t) ≡

c log(2 + t)

(t ≥ 0)

for different choices of c > 0. Hwang and Sheu [Hw-Sh 90, Th.3.3] prove that, under appropriate conditions, there is some cooling schedule such that X minimizes U ; more precisely, there is a constant d∗ > 0 such that for all c > d∗ , t→∞ P0,x {Xtc ∈ S} −−−− → 1 uniformly with respect to initial conditions x in compact sets, where d S := z ∈ R : U (z) = min U ( . ) Rd

and where X c denotes the solution of the above simulated annealing SDE for ε(t) = εc (t). There are many more interesting features of the simulated annealing process and interesting connections to the spectral theory of the generator of (2.2) which are beyond the scope of this book. Instead, the reader is referred to Hwang and Sheu [Hw-Sh 90], Royer [Roy 89], Chiang et al. [Cg-Hw-Sh 87] and Geman and Hwang [Gem-Hw 86] among others.

2.6 Sample systems The first example, taken from Freidlin and Wentzell [Fr-We 98, 4.3.2], lies at the core of our investigations. On the one hand it is an illustrative example for the Freidlin-Wentzell theory; on the other hand it alludes to the manner in which the local Lyapunov exponents shall generalize the Lyapunov exponents of linear ODEs with constant coefficients which are given by the real parts of the eigenvalues of the coefficient matrix; see 1.5.2.

2.6 Sample systems

109

Example 2.6.1 (Multi-dimensional Ornstein-Uhlenbeck process). Consider the SDE (2.1) with linear drift b(x) := Ax and σ = idRd , √ dXtε,x = AXtε,x dt + ε dWt , X0ε,x = x ∈ Rd , where A ∈ Rd×d is a constant matrix. The origin K1 := 0 is an equilibrium of b and assuming that A is normal and A + A∗ is negative definite (i.e. its eigenvalues, which are real by symmetry, are strictly negative), then one gets for X 0 = eAt (the propagator) that ∗

| Xt0,x |2 = eAt x , eAt x = eA t eAt x , x ∗

= e(A+A

)t

x,x

A+A∗ t

≤ |x| e 2

t→∞

−−−−→ 0 ,

i.e. K1 attracts Rd , D1 = Rd . Now the vector field b(x) ≡ Ax can be decomposed as b = −∇U + L , where U (x) := −

1 (A + A∗ )x, x 4

and

L(x) :=

1 (A − A∗ )x . 2

More precisely, let (ek )dk=1 denote the canonical basis of Rd ; then 1 ∇x (A + A∗ )x, x 4 1 ∂ = ek (aij + aji ) xj xi 4 ∂xk i,j

−(∇U )(x) ≡ +

k

1 = ek (aij + aji ) (δkj xi + δki xj ) 4 i,j,k

=

1 4

i,j

ej (aij + aji )xi +

1 1 ei (aij + aji ) xj = (A + A∗ )x . 4 i,j 2

The normality of A implies that this decomposition of b is orthogonal: For all x ∈ Rd ,

1 1 (A + A∗ )x , (A − A∗ )x −(∇U )(x) , L(x) = 2 2 1 Ax, Ax + 0 − A∗ x, A∗ x = 4 1 ∗ A Ax, x − AA∗ x, x = 0 . = 4

110


Proposition 2.4.5 thus implies for the quasipotential that V (K1 , x) = 2 U (x) for all x ∈ Rd . Theorem 2.5.6 yields that for all initial values x ∈ Rd , all time scales T (ε) ! eζ/ε for some parameter ζ > 0 and all c, Γ > 0, the following holds true: P > c L t ∈ [0, Γ] : Xtε,x −−−−→ 0 . T (ε) ε→0

Example 2.6.2 (Two well potential). Consider the gradient SDE (2.2) with a two well potential function U as sketched in the following figure 2.6, where v < V ; for a numerical example see (1.34). U

v V

K1

K3

K2

Fig. 2.6 A two well potential function U : Rd → R (cf. figure 1)

Here, the quasi-potential is calculated by means of proposition 2.4.5 as V (K1 , K3 ) = 2 (U (K3 ) − U (K1 )) ≡ 2v , V (K2 , K3 ) = 2 (U (K3 ) − U (K2 )) ≡ 2V and hence also V (K1 , K2 ) = 2v and V (K2 , K1 ) = 2V , since following the deterministic path X 0 amounts to a vanishing rate function. The assumption 2.1.1 (K) is satisfied with l = 2 and the stable points K1 , K2 ; the saddle point K3 (l = 3) is sorted out, since it violates the stability requirement (K2). In other words, we are concerned with . 0 2v (Vij )i,j∈L ≡ V (Ki , Kj ) i,j∈L = , 2V 0 where L := {1, 2}. Furthermore, the respective domain of attraction of Ki is Di , i ∈ L, where D1 denotes the shallow well (around K1 ) and where D2 is the deep well (around K2 ), respectively, both excluding K3 . The quasideterministic approximation is given by the following data here: The set of

2.6 Sample systems

111

0-cycles is C (0) = L ≡ {1, 2} . On the 0-cycles the main state, the rotation rate and the stationary distribution rate are trivial, i.e. M (i) = i, R(i) = 0 and mi (i) = 0, respectively, where i ∈ L. The exit rates are E(1) = V (K1 , K2 ) = 2v

and

E(2) = V (K2 , K1 ) = 2V .

The “follow-mapping” J is given by J(1) = 2

and

J(2) = 1 .

The possible 1-cycles are (1 → 2 → 1) and (2 → 1 → 2) which both describe the same cyclic order; hence, C (1) = (1 → 2 → 1) and the main state is

M (1 → 2 → 1) = 2

by (2.12). The rotation rate as defined in (2.13) is R (1 → 2 → 1) ≡ max Vi,J(i) = V21 = 2V ; i∈{1,2}

furthermore, the stationary distribution rate m(1→2→1) for the 1-cycle (1 → 2 → 1) is given by m(1→2→1) (1) ≡ R (1 → 2 → 1) − V12 = 2V − 2v = 2(V − v) and m(1→2→1) (2) ≡ R (1 → 2 → 1) − V21 = 2V − 2V = 0 . The exit rate of the 1-cycle (1 → 2 → 1) is E (1 → 2 → 1) = ∞ due to (2.14), since C (1) = 1. The latter fact also means that C (1) is equal to L, considered as a point set. Hence, the recursive definition of cycles stops at k = 1 for the two well potential function. Next, we use the above findings to determine the support of the sublimiting distribution, i.e. we calculate the metastable states corresponding to initial values and time scales as given by definition 2.5.4 and the distinction of cases preceding it. Firstly, fix x ∈ D1 . The sequence of cycles belonging to x is then 1 = i(x) ≡ C (0) (x) ⊂ (1 → 2 → 1) ≡ C (1) (x) = L

112


and the corresponding exit and rotation rates are E0 (x) ≡ E(1) = 2v , E1 (x) ≡ E (1 → 2 → 1) = ∞ and R0 (x) ≡ 0 ,

R1 (x) ≡ R (1 → 2 → 1) = 2V ,

respectively, which we collect as 0 ≡ R0 (x) < 2v = E0 (x) ≤ 2V = R1 (x) < ∞ = E1 (x) . Due to (2.20) we fix ζ ∈ R>0 \ {2v, 2V } and then we have −1 , if ζ < 2v , ∗ ∗ m ≡ m (x) = 0 , if ζ > 2v . By virtue of the distinction of cases leading to (2.21) we immediately get that µ(x, ζ) = 1, if m∗ = −1, i.e. if ζ < 2v, since ζ > 0 = R0 (x) . Otherwise, if m∗ = 0, i.e. if ζ > 2v, then µ(x, ζ) = 2, if also ζ > 2V = Rm∗ +1 (x) ; if m∗ = 0 and if, moreover, ζ ∈ (2v, 2V ), then we also get µ(x, ζ) = 2, since for ∗ ∗ ∗ Cˆ (m ) := 2 we have E(Cˆ (m ) ) = 2V > ζ and ζ > R(Cˆ (m ) ) ≡ 0 then. All in all, we get for x ∈ D1 and ζ ∈ R>0 \ {2v, 2V } that µ(x, ζ) = 1, if ζ < 2v and µ(x, ζ) = 2 otherwise. Secondly, fix x ∈ D2 . The sequence of cycles corresponding to x is then 2 = i(x) ≡ C (0) (x) ⊂ (1 → 2 → 1) ≡ C (1) (x) = L and the respective exit and rotation rates are E0 (x) ≡ E(2) = 2V , and R0 (x) ≡ 0 ,

E1 (x) ≡ E (1 → 2 → 1) = ∞ R1 (x) ≡ R (1 → 2 → 1) = 2V ,

which we again collect as 0 ≡ R0 (x) < 2V = E0 (x) ≤ 2V = R1 (x) < ∞ = E1 (x) . On the basis (2.20) we fix ζ ∈ R>0 \ {2V } and then we have −1 , if ζ < 2V , m∗ ≡ m∗ (x) = 0 , if ζ > 2V ; both cases directly yield that µ(x, ζ) = 2. Hence, we can summarize our above findings as follows: Fix some ζ ∈ R>0 \ {2V }, then

2.6 Sample systems

113

⎧ ⎪ ⎨2 µ(x, ζ) = 2 ⎪ ⎩ 1

, x ∈ D2 , x ∈ D1 and ζ > 2v , x ∈ D1 and ζ < 2v .

Now fix an initial value x ∈ Rd \ {K3 } and a time scale T (ε) ! eζ/ε for such a parameter ζ ∈ R>0 \ {2V }. Then it follows from theorem 2.5.6 : 1) If x ∈ D1 and ζ < 2v , then for any c, Γ > 0, P > c − K L t ∈ [0, Γ] : Xtε,x −− −−→ 0 ; 1 T (ε) ε→0 2) in all other cases (x ∈ D1 and ζ > 2v ; x ∈ D2 ) for any c, Γ > 0 , L

P > c t ∈ [0, Γ] : Xtε,x − K −− −−→ 0 . 2 T (ε) ε→0

In the remainder of this section several real-world situations will be discussed which can be modeled by the SDEs (2.2) and (2.1). In doing so, the conceptual scope of these equations from an applications’ point of view shall be demonstrated. As in equation (2.2) the diffusion coefficient of the SDE (2.1) will mostly be taken as σ( . ) = idRd . From an applications’ point of view this is not a severe restriction, but avoids technicalities; see e.g. 2.4.13.

Economic time series modeling Example 2.6.3 (Vasicek model for the interest rates with several wells). The classical Vasicek model [Vc 77] describes the short term interest rate18 as solution of the SDE √ dXtε,x = −β(Xtε − R) dt + ε dWt , X0ε,x = x , where β, R, x > 0 . This equation fits into the framework of the SDEs (2.1) 2 x and (2.2) via b(x) := −β(x − R), d = 1 and U (x) := − 0 b(y)dy = β ( x2 − Rx) . U is a quadratic one-well potential and after a translation by R, X ε coincides with the Ornstein-Uhlenbeck process as described in example 2.2.3. Therefore, the transition densities are known and t √ Xtε,x = R + (x − R)e−βt + ε e−β(t−u) dWu 0

18

For an overview of interest rate models see e.g. Gibson et al. [Gs-Lh-Pr-Ta 99] and [Gs-Lh-Ta 01].

114


is normally distributed, where the mean is given by the trajectory of the deterministic system Xt0,x . A characteristic feature of this process is the mean-reversion with respect to the one stable equilibrium point K1 := R. Hence, the assumptions 2.1.1 hold with l(= l ) = 1. Therefore, the previous large deviation considerations can be readily applied to the Vasicek interest rate model; see Callen et al. [Ca-Gv-Xu 00]. Using non-parametric tests, however, A¨ıt-Sahalia [A¨ı 96] rejects such linear drift specifications for the short term interest rate. In order to better capture the underlying economic effects, regime switching models seem necessary (see A¨ıt-Sahalia [A¨ı 96, p.397]). In the literature such spot rate shifts have mostly been modeled by replacing the above mean parameter R by a Markov chain (Rt )t with finite state space (see Landen [Lan 00]): E.g. if (Rt )t can attain two possible values R(1) < R(2) , this amounts to a low-mean regime and an (exceptional) high-mean regime. However, the resulting drift b(t, x) := − β (1) x − R(1) 1{Rt =R(1) } − β (2) x − R(2) 1{Rt =R(2) } , β (1) , β (2) > 0, is discontinuous as time evolves. A¨ıt-Sahalia, [A¨ı 96, p.397f.] and [A¨ı 99], suggests to take a “two-regime potential drift” instead, i.e. to consider the gradient SDE (2.2) (in dimension d = 1) with a potential function U as in figure 2.6 of example 2.6.2. Here, the well at R(1) is supposed to be deeper than the well of the exceptional level R(2) . The Freidlin-Wentzell theory underlying the two-regime potential model (2.2) then allows to calculate the mean exit times from the regimes; see A¨ıt-Sahalia [A¨ı 96]). There is, however, one severe drawback of this model: As in the one-well case (the original Vasicek model) the interest rates may become negative. One possibility to overcome this disadvantage is to define the potential as ∞ on R≤0 ; see remark 2.2.4 above and the underlying references Meyer and Zheng [My-Zh 85] and Kunz [Kz 02]. Example 2.6.4 (Prices of energy commodities). Borovkova et al. [Br-Dh-Re-Tu 03] and Anderluh and Borovkova [Ad-Br 04] use the gradient SDE (2.2) to model prices of energy commodities (heating oil, gasoline) and agricultural commodities (coffee, cocoa, soybean). Focusing on oil prices these authors estimate the potential function U and fit the model to the time series of daily closing prices of Brent North Sea oil from 1991 to 1998. As Borovkova et al. [Br-Dh-Re-Tu 03] note, the oil price was generally known to have several “preferred regions” at 14, 18 and 23 dollars per barrel during this period of time and most trading occurs there: The price “clusters” at these levels and deviates from them relatively briefly. As in example 2.6.3 above this additive noise model can admit negative prices; again, see remark 2.2.4 above.

2.6 Sample systems

115

More examples for the application of “metastability” to economic (or more generally: socio-political) systems can be found in the literature: Weidlich [Wd 71] studies polarization phenomena in society (e.g. formation of opinion) akin to Ising spin models; in the continuous limit case this leads to Fokker-Planck equations on compact intervals. Bouchaud and Cont [Bd-Ct 98] use the gradient SDE (2.2) with a cubic potential to model the motion of the instantaneous return ut of a stock Xt . The exit of ut from the one well is then interpreted as “crash” of the stock. However, Bouchaud and Cont [Bd-Ct 98] deduce that in this situation the stock price Xt diverges to −∞. A more realistic model of crashes on financial markets therefore has to investigate, more precisely, to which regime Xt moves in case of a crash. It seems reasonable to use a gradient ansatz (2.2) in which the potential function is time-dependent and mostly attains the onewell shape (“regular regime”). In exceptional situations the potential might feature a second well; the regime of the latter well then represents the (lower) price level to which the stock might crash. The depth of the regular well then determines the crash probability and expected crash time by the exit time law. Such an approach to the investigation of crashes is supported by oneperiod trading models; see Gennotte and Leland [Ge-Ll 90]. This is work in progress. Haag et al. [Hg-Wd-Mn 85] consider a macroeconomic potential function to study the structural change of an economy. This also leads to a timedependent potential function. However, Haag et al. [Hg-Wd-Mn 85] do not add white noise, but a (deterministic) periodic forcing to model business cycle effects (e.g. Kondratieff cycles, . . . ). Here, one might alternatively investigate a time-periodic potential function upon which additive white noise is imposed. In this situation it would then be interesting to investigate the phenomenon of “stochastic resonance”; see Pavlyukevich [Pv 02] and Freidlin [Fr 00].

Models for climate systems Example 2.6.5 (Thermohaline circulation). The Thermohaline circulation (THC) describes the density driven circulation of ocean water: The respective components are the temperature (“thermo”), as cold water is denser than warm water, and the salinity (“haline”), as saltwater is denser than freshwater. The present state of the THC in the Atlantic Ocean is that warm surface water from the equator area flows to the north, cools and sinks down near Greenland and Iceland; this cold deep water is then transported back to the south. An approximation of this phenomenon can be described by the use of box models: The classical examples by Stommel [Sto 61] and Cessi [Ce 94] use

116


two boxes for modeling the equatorial and polar basins of the North Atlantic Ocean, respectively19 : 6

evaporation: F/2

6

precipitation: −F/2

? mass exchange Q ≥ 0:

box “1” (low-latitude warm water): temperature T1 (t) [◦C] salinity S1 (t) [psu]20 kg density ρ1 (t) [m 3 ] depth H volume V

[m] [m3 ]

Q/2

Q/2

?

box “2” (high-latitude cold water):

-

temperature T2 (t) [◦C] salinity S2 (t) [psu] kg density ρ2 (t) [m 3] depth H volume V

[m] [m3 ]

Fig. 2.7 Schematic depiction of the thermohaline circulation

The mass exchange function Q [s−1 ] is supposed to depend only on the (non-dimensional) difference of the respective densities, ρ :=

ρ1 − ρ2 := αS ( S1 − S2 ) − αT ( T1 − T2 ) , ρ0

where ρ0 denotes a reference density and αS [psu−1 ] and αT [(◦C)−1 ] are constants. In the presence of a freshwater flux F [m s−1 ], which stems from evaporation and precipitation and influences the salinities, and assuming a relaxational forcing for the temperatures, the time-dependent quantities T1 , T2 , S1 and S2 are then governed by the differential equations

1 1 θ d T1 (t) = − − Q ρ(t) T1 (t) − T2 (t) T1 (t) − dt tr 2 2

1 1 θ d T2 (t) = − − Q ρ(t) T2 (t) − T1 (t) T2 (t) + dt tr 2 2 F 1 d S1 (t) = S0 − Q ρ(t) S1 (t) − S2 (t) dt 2H 2 F 1 d S2 (t) = − S0 − Q ρ(t) S2 (t) − S1 (t) ; dt 2H 2 here, the following constants are used: tr [s] is the relaxation time for the temperatures towards ± θ2 [◦C] (if mass exchange were absent) and S0 [psu] is a reference salinity. 19 20

Square brackets are used here to indicate physical units (dimensions). “psu” denotes Practical Salinity Units, a scale for salinity.

2.6 Sample systems

117

From the above system of ODEs one gets that the salinity and temperature differences, and S := S1 − S2 , T := T1 − T2 are governed by the differential equations 1 d T (t) = − T (t) − θ − T (t) Q αS S(t) − αT T (t) dt tr F d S(t) = S0 − S(t) Q αS S(t) − αT T (t) . dt H Now the exchange function is specified as Q(ρ) :=

1 q 2 ρ , + td V

where td [s] is the “diffusive” time scale, q [m3 s−1 ] is proportional to the pressure difference (Poiseuille’s law) and V [m3 ] denotes the volume of one of the boxes. Note that Stommel [Sto 61], who pioneered the THC research by box models, used the exchange function QSto (ρ) :=

1 q |ρ| , + td V

which we will not take into account for convenience, since it exhibits a plane at which it is not differentiable. Reasonable choices for the constants in use are for example tr = 25 days, θ = 20◦C, F = 2.3 m year−1 , H = 4.5 · 103 m, S0 = 35 psu, αS = 0.75 · 10−3 psu−1 , td = 219 years ,

αT = 0.17 · 10−3 (◦C)−1 ,

q = 1.0 · 1012 m3 s−1 ,

ρ0 = 1029 kg m−3 ,

V = 1.1 · 1016 m3 ,

according to Cessi [Ce 94]. Introducing the dimensionless variables x1 :=

T1 − T2 T ≡ θ θ

and scaling time by td ,

and

x2 :=

αS (S1 − S2 ) αS S ≡ αT θ αT θ

tnew := td · told ,

the above ODEs are rewritten as 0 2 1 d x1 (t) = − α x1 (t) − 1 − x1 (t) 1 + ν 2 x1 (t) − x2 (t) (2.22) dt 1 0 2 d x2 (t) = p − x2 (t) 1 + ν 2 x1 (t) − x2 (t) , dt

118


where α :=

td , tr

ν 2 :=

q td (αT θ)2 V

and

p :=

αS S0 td F . αT θ H

Using the above sample values for the parameters one would obtain the numerical approximations α = 3.20 · 103 ,

ν 2 = 7.3

and

p = 0.9 .

Note that the above ODE (2.22) can also be considered as deterministic slow-fast system, in which x1 represents the fast variable and x2 is the slow variable; see Berglund and Gentz [Bg-Gen 06]. Now we consider two ways of introducing stochastic dynamics in this system: a) Let b denote the two-dimensional drift vector of this ODE system (2.22), i.e.

2 2 1 + ν x ( x − x ) − α (x1 − 1) − 1 1 2 . b(x1 , x2 ) := p − x2 1 + ν 2 ( x1 − x2 )2 This drift fits into the setting imposed on (2.1); adding additive random √ perturbations of intensity ε one gets the following SDE for the temperature and salinity differences: √ dXtε = b (Xtε ) dt + ε dWt . For appropriate choices of the parameters this THC-drift b has three equilibria, two of which are stable and one is not; the former states are then the metastable states K1 and K2 of the THC. Note that the above design of the THC-drift is due to Cessi [Ce 94]. If one uses Stommel’s choice [Sto 61] QSto (ρ) =

1 q |ρ| + td V

of the exchange function, the resulting drift after introducing dimension-less variables is

− α (x1 − 1) − x1 [ 1 + ν | x1 − x2 | ] bSto (x1 , x2 ) := p − x2 [ 1 + ν | x1 − x2 | ] which exhibits the same qualitative features, but also non-smoothness. b) Another route to a stochastic equation of motion for the THC is to consider the freshwater flux as driving stochastic process, i.e. to add white noise to p , which will be described next: For this purpose we first replace (2.22) by a one-dimensional equation, which can be justified since for a large α the first summand in the x1 -equation forces the temperature difference x1 to stay near 1. Hence, by substituting x1 = 1 into (2.22) one gets the

2.6 Sample systems

119

one-dimensional equation for the salinity difference x2 : 0 2 1 d x2 (t) = p − x2 (t) 1 + ν 2 1 − x2 (t) dt = − U x2 (t) , where integrating the one-dimensional drift leads to the potential function ν2 4 2 ν2 3 (1 + ν 2 ) 2 x − x + x − px. 4 3 2 √ Adding small additive random perturbations of intensity ε of the freshwater flux, as announced, this leads to the SDE for the salinity difference, √ dXtε = − U (Xtε ) dt + ε dWt , U (x) :=

where W is a Brownian motion in R. This SDE again fits into the setting of (2.1); depending on p and ν 2 , the potential U has several local minima which are then the metastable states of the salinity difference of the THC. For further details we refer to Cessi [Ce 94]. More references on the THC are given by Imkeller and Monahan [Im-Mo 02]. Example 2.6.6 (El Ni˜ no-Southern Oscillation). El Ni˜ no is an anomaly of the sea surface temperature in the tropical Pacific Ocean. This short term phenomenon recurs on time scales of several (6–10) years and has important consequences such as reduced fishing in the eastern Pacific Ocean, low grain yields in south Asia and Australia and high crop yields in the North American prairies (see Hansen et al. [Hn-Hd-Jo 98] and the references therein). Furthermore, the hurricane activity in the western Atlantic Ocean is reduced during the season following the El Ni˜ no event and returns to normal only in the second summer following an El Ni˜ no event (see Gray [Gy 84]). a) Wang et al. [Wg-Bc-Fg 99] describe the El Ni˜ no system by the SDE (2.1) in R2 with the drift function . a1 x1 + a2 x2 + a3 x1 (x1 − c x2 ) − 2 x31 b(x) := b (2x2 − x1 ) − 2 x32 and σ( . ) := idR2 . Here, the (nondimensional) variables x1 and x2 represent the anomalous sea surface temperature and the thermocline depth, respectively; a1 , a2 , a3 , b, c ∈ R are constants. Depending on the choice of the parameters this drift exhibits either a stable regime, a limit cycle regime or several equilibria. In the latter case the system can be adjusted such that there are three unstable fixed points and two stable equilibria of b; one of the latter points (the “warm” state) represents the El Ni˜ no situation.

120


For details such as typical values of the parameters see Wang et al. [Wg-Bc-Fg 99]. b) The multi-dimensional Ornstein-Uhlenbeck process of example 2.6.1 can also be used for describing the El Ni˜ no-Southern Oscillation, as has been carried out by Penland and Sardeshmukh [Pl-Sa 95b] and [Pl 96]. More precisely, consider the SDE (2.1) with linear drift b(x) := Ax , √ dXtε = AXtε,x dt + ε σ dWt , X0ε,x = x ∈ Rd , where A, σ ∈ Rd×d are constant matrices. Here, σ is additionally invertible, A again attracts D1 = Rd towards the origin K1 := 0 and the dimension is d := 15. If the eigenvalues (characteristic roots) are non-real, there are components of X 0 which show oscillatory behavior. In the randomly perturbed case ε > 0, X ε exhibits random periodicity. One can then calculate the respective rotation numbers as in Arnold [Ar 98, Sec.6.5] for the stationary situation (i.e. for fixed ε > 0). This rotational behavior also implies random periodicity for the sea surface temperature which can be generated through X ε . For more references to the literature on stochastic (or deterministic) models of the El Ni˜ no phenomenon see Imkeller and Monahan [Im-Mo 02]. Example 2.6.7. A potential function with n wells is for example also given by

n 1 1 |x − zi |2 U (x) = |x|2 − , ki exp − 2 2 i=1 ci where x ∈ Rd ; ki > 0, ci > 0 and zi ∈ Rd are the parameters representing the depth, width and position of the i-th well. Teng et al. [Tg-Mo-Fy 04] use this potential as a model for the extratropical northern hemisphere atmosphere. The gradient SDE (2.2) is also used by Nicolis and Nicolis [Nc-Ni 81] and Sutera [Su 81] as a general model for climate transitions.

Further examples Finally, we briefly list some more examples for the interested reader. Again, the goal is not to compile a complete list, but to hint at the various possibilities of application. Example 2.6.8 (Saksaul tree population). Freidlin and Svetlosanov [Fr-Sv 76] consider (2.2) with the potential function U (x) := −

a a l x + 3 arctan( κ · x ) + x2 , 2 κ κ 2

2.6 Sample systems

121

where a, κ, l are real parameters such that 0 < 2lκ a < 1. This potential function has a local minimum at K1 := 0, a global minimum at ⎛ ⎞ 3

2 a ⎝ 2lκ ⎠ K2 := 1+ 1− 2lκ2 a and a saddle at

⎛ ⎞ 3

2 a ⎝ 2lκ ⎠ K3 := 1− 1− . 2lκ2 a

Choose an initial condition x ∈ D2 , the domain of attraction of K2 . Then the process X ε,x represents the population of the Saksaul, a certain tree in the Gobi Desert which can best resist the drought there. Freidlin and Svetlosanov [Fr-Sv 76] then calculate the mean exit time from D2 by the exit time law. Such an exit and hence an approach of K1 = 0 means the extinction of the tree population. The parameters a, l and κ parametrize the sprout rate, the extinction rate and the influence of neighboring trees, respectively. Example 2.6.9 (Evolution). Newman et al [Nm-Co-Ki 85] use (2.2) to describe Neo-darwinian evolution: X ε depicts the population mean of some genetically determined character; the function −U is then interpreted as the mean fitness of the population. Natural selection pushes the population √ mean X ε towards higher values of this fitness landscape. The parameter ε measures the magnitude of the random genetic variations relative to that of natural selection. Example 2.6.10 (The Stochastic Disk Dynamo Model for the polarity of the earth’s magnetic field). The stochastic disk dynamo is given by the SDE (2.1) in R2 for the drift function

− ν1 x1 + x1 x2 , b(x1 , x2 ) := − ν2 x2 + 1 − x21 where ν1 , ν2 > 0 are constants such that ν1 ν2 < 1. This drift has two stable equilibria at √ K1,2 := ± 1 − ν1 ν2 , ν1 and an unstable fixed point at K3 := (0, 1/ν2 ). Ito and Mikami, [Ito-Mik 96] and [Ito 88], use this system as a model for the earth’s magnetic field: The stable equilibria K1 and K2 then correspond to the two respective polarities.

122


The case of a compact state space Instead of considering a diffusion on Rd one could also consider a diffusion X ε on a d-dimensional compact manifold. In this case (V) is not necessary. Furthermore, a density satisfying the Fokker-Planck equation is always normalizable in the compact case. The following SDE provides an example of a system on the circle for which the stationary density can be calculated as in remark 2.2.1 due to the onedimensionality of its state space: Example 2.6.11 (Noisy north-south-flow). The noisy north-south-flow is the solution X ε := αε for the SDE √ d αεt = − cos αεt dt + ε dWt on the compact state space S 1 ≡ R/2πZ " [0, 2π). The state space being one-dimensional, the system can be thought of as the random perturbation of the north-south-flow d αt = − U (αt ) dt

with the potential

U (α) := sin α

on S 1 ; this deterministic dynamical system has two equilibria: αN := π2 which is unstable and αS := 3π 2 which is stable. Calling αN and αS the north pole and the south pole of the “planet” S 1 , respectively, one gets an explanation of the name of this system. The assumptions (S) and (E) are clearly met; (K) holds true with l = 1, π K1 := αS ≡ 3π 2 ; additionally, l = 2 and K2 := αN ≡ 2 . Furthermore, also (G) is satisfied. The invariant measure of (Xtε )t≥0 ≡ (αεt )t≥0 can be also calculated explicitly: As in remark 2.2.1 on stationary measures, the invariant measure has a density pε (α) with respect to the Lebesgue measure on the circle, which needs to solve the Fokker-Planck-equation d ε d2 ε − cos(α) pε (α) = 0 p (α) − 2 2 dα dα for α ∈ (0, 2π), fulfill the normalization constraint

2π

pε (α) dα = 1

0

and additionally the continuity (periodicity) requirement pε (0) = pε (2π) . The solution to this problem is given by Khasminskii, [Kh 67] and [Kh 80, Sec.VI.8], as

2.6 Sample systems

1 p (α) = cε ε ε

where

123

5 1 +

W(2π) − 1 2π 0

W(s)ds

6

α

W(u)du

W(α)−1 ,

0

2 W(α) := exp + U (α) ε

and cε is the normalization constant; this can be seen by a direct calculation. Furthermore, the one peak of the stationary density pε is located at the south pole αS and in the small noise limit ε → 0 the invariant measure converges weakly to δαS . The latter property can be either directly verified from the above formula or be deduced from corresponding results by Freidlin and Wentzell, [We-Fr 69] and [Fr-We 98, Th.6.4.2], since the state space is compact. This noisy north-south-flow system has been considered by Carverhill [Cv 85b, p.290ff.]; note that Carverhill [Cv 85b, p.290] defines the noise free north-south-flow as the stereographic projection of the flow ηt (x) := xe−t in R to the unit circle; it can be calculated that this is equivalent to the above potential function U (α) := sin α on S 1 . A rotated version (a noisy westeast-flow so to say) has been investigated and simulated by Carverhill et al. [Cv-Cl-Ew 86, p.54ff.]. Carverhill [Cv 85b] calculates the (one) Lyapunov exponent λε for this system αεt , obtains a Furstenberg-Khasminskii-type formula for it and notes that in the small noise limit ε → 0, λε converges to −U (αS ) = sin(− π2 ) = −1. A more general investigation concerning the invariant measures of diffusion processes on a circle with small diffusion is given by Nevel’son [Nev 64]. The impetus for us mentioning this system on the circle here is twofold: On the one hand it illustrates how the results from the Freidlin-Wentzell theory also hold in the setting of compact state spaces. On the other hand — what is more — elliptic SDEs of the above type typically arise in the study of the angle (and hence of the Lyapunov exponent) of linear, two-dimensional stochastic systems with white noise. The crucial property when studying the SDE (1) is, however, that its angle satisfies a differential equation with real noise; see (1.6). Therefore, when considering the angle system of (1), the Freidlin-Wentzell results only hold for the noise process X ε , but not for the angle process; in particular, when examining (1) for (local) Lyapunov exponents, the Freidlin-Wentzell results cannot be applied to the angle process. For this reason we study exit probabilities of degenerate stochastic systems in the following chapter.

Chapter 3

Exit probabilities for degenerate systems

This chapter is devoted to the investigation of the exit probabilities for degenerate stochastic differential systems depending on small parameters. In doing so, the exposition follows Hern´ andez-Lerma [HL 80], [HL 81] and Fleming [Fl 80], [Fl 78], [Fl 77]. The object underlying all our considerations is the stochastic system (1). For obtaining information about the exponential growth rates of Z ε , the formulas (1.5) and (1.8) advise to investigate the movement of the direction ψ ε or the angle αε (n = 2) of Z ε , respectively; focusing on the two-dimensional case here this means, more precisely, that one needs assertions concerning the behavior of the system ¯ (X ε , αε ) dt , dαεt = h t t √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ; see (1.6). The results of the previous chapter imply that X ε mostly lingers near Kµ(x,ζ) . However, if A(Kµ(x,ζ) ) ∈ R2×2 has two different eigenvalues, αε still can oscillate at two equilibrium points, A1 Kµ(x,ζ) and A2 Kµ(x,ζ) (see definition 1.2.4) — the former being attracting and the latter being repelling. Therefore, the results of this chapter will be applied (in section 4.4) to quantify the exit of αε from a neighborhood of A2 (Kµ(x,ζ) ); see theorem 4.4.6. As far as we are aware the only results available for exit probabilities of such degenerate systems are proven by Hern´ andez-Lerma and Fleming in the papers cited above, Dupuis and Kushner [Du-Ks 89], Barles and Blanc [Bar-Bln 97] and Freidlin [Fr 69]: Dupuis and Kushner [Du-Ks 89] assume that b ≡ b(x, u) also depends on the values of a control function u, and consider the problem of determining u as to minimize the exit probability of (X ε , αε ), in the above notation, over a given time interval as ε → 0. Barles


125

126

3 Exit probabilities for degenerate systems

and Blanc [Bar-Bln 97] consider the SDE (2.1) in which σ might degenerate and investigate the exit times from domains converging to a subset of ∂D, where D is a domain which the deterministic system X 0 uniformly exits. Freidlin [Fr 69] considers the associated Dirichlet problem for the SDE (2.1) under the condition that a ≡ σσ ∗ is just non-negative definite and constructs “generalized solutions”, i.e. solutions which converge to 0 towards the boundary of the domain. Note that the above stochastic equation for αε is a real noise system, i.e. the differential equation is not driven by white noise, but is a random differential equation due to its dependence on the diffusion X ε . Real noise systems and associated large deviation properties also have been investigated by Freidlin [Fr 73b], Nguyen Viet Phu [NgVP 74a] and [NgVP 74b] and Grin [Gn 75]; also see Freidlin and Wentzell [Fr-We 98, p.132ff.]. However, these authors consider systems of the type αεt = b(αεt , εXt ) , where b is a differentiable drift function, X is a Gaussian stochastic process and ε becomes small. Since this is a different setting, the corresponding results cannot be applied to the angle system αε of (1) which had been mentioned before. In the sequel, the findings by Hern´ andez-Lerma [HL 80] and [HL 81] shall be presented. Since these results are well documented (also see Fleming [Fl 80], [Fl 78] and [Fl 77]), we allow ourselves to sketch the arguments only and refer to these papers for more details.

3.1 Exit probabilities for degenerate systems depending on a small parameter The system under consideration is the stochastic process (Xtε , Ytε )t≥0 in Rd × Rm defined by the SDE dYtε = F (t, Xtε , Ytε ) dt , √ dXtε = ˆb (t, Xtε ) dt + ε σ ˆ (t, Xtε ) dWt ,

(3.1)

where the coefficients are smooth functions F : R+ × Rd × Rm → Rm , ˆb : R+ × Rd → Rd and σ ˆ : R+ × Rd → Rd×d ; ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which again denotes a Brownian motion in Rd on a standard filtered probability space (Ω, F , P, (Ft )t≥0 ) ; the initial condition is (Xsε , Ysε ) = (x, y) ∈ Rd × Rm

3.1 Exit probabilities for degenerate systems depending on a small parameter

127

for some initial time s ≥ 0 and the solution process is denoted by (Xtε,x , Ytε,y )t≥s . Furthermore, Ps,x,y denotes the law of this process conditioned on the above initial value. In the sequel, we will work with a fixed bounded, open domain D ⊂ Rm and a time horizon T > 0. Furthermore, set a ˆ := σ ˆσ ˆ∗ , where ∗ denotes the transpose of the respective matrix as usual. Note that the notation in use here differs from the one used by HernándezLerma [HL 80] and [HL 81] : The x- and the y-variable are interchanged in order to reserve “x” for the direction in which X ε (as defined in chapter 2) is acting. Notation 3.1.1 (First exit time). Let D be a bounded, open domain in Rm with smooth boundary ∂D. Then the random variable T ε ≡ T ε (s, x, y) ≡ TDε (s, x, y) := inf { t ≥ s : Ytε,y ∈ / D} denotes the first exit time of (Ytε,y )t≥s from D. Again, T ε is a stopping time with respect to the underlying standard filtration (Ft )t≥0 . Notation 3.1.2 (Value function). Let u ∈ C 1 (R+ , Rd ) be a (“control”) function and let y 0 be the path defined by dy 0 (t) = F (t, u(t), y 0 (t)) dt under the initial condition (u(s), y 0 (s)) = (x, y) ∈ Rd × D . Then

ϑ ≡ ϑD (s; x, y; u) := inf t ≥ s : y 0 (t) ∈ /D

denotes the corresponding exit time of y 0 from D and U ≡ UD (s; T ; x, y) := u ∈ C 1 (R+ , Rd ) : u(s) = x and ϑD (s; x, y; u) ≤ T is the (possibly empty) set of control functions for which an exit happens before T > 0. Using this notation, define the value function 0 (s; T ; x, y) I 0 ≡ I 0 (s, x, y) ≡ ID ⎧ ϑD (s;x,y;u) ⎪ ⎨ inf K t, u(t), u(t) ˙ dt := u∈UD (s;T ;x,y) s ⎪ ⎩ ∞

, U = ∅, , U = ∅,

where K : R + × Rd × Rd → R + ,

K(s, x, q) :=

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) . a 2

128


Here, a ˆ(s, x)−1/2 again denotes the (unique) d × d-matrix whose square is σ (s, x) σ ˆ (s, x)∗ ]−1 . Also a ˆ(s, x)−1/2 the positive definite matrix a ˆ(s, x)−1 ≡ [ˆ is a symmetric matrix; hence one can rewrite K as K(s, x, q) =

0 1 1 q − ˆb(s, x) , a ˆ(s, x)−1 q − ˆb(s, x) . 2

Assumption 3.1.3 (on the coefficients of the SDE (3.1)). The data of the SDE (3.1) are supposed to satisfy: (a) F , ˆb and σ ˆ are continuous on their respective domains of definition and σ ˆ takes its values in the invertible matrices; furthermore, F ∈ C ∞ (R>0 × ˆ ∈ C ∞ (R>0 × Rd , Rd×d ) and Rd × Rm , Rm ) , ˆb ∈ C ∞ (R>0 × Rd , Rd ) , σ −1 ∞ d d×d σ ˆ ∈ C (R>0 × R , R ) are bounded with bounded first derivatives (on the respective spaces). (b) The matrix a ˆ≡σ ˆσ ˆ ∗ is strictly positive definite, i.e. there is a constant c1 > 0 such that 2 c1 |x2 |2 ≤ a ˆ(t, x1 )x2 , x2 ≤ c−1 1 |x2 |

(t ∈ R+ ; x1 , x2 ∈ Rd ) .

(c) Let Lˆε (s, x, y) :=

d

d m ∂2 ∂ ˆbi (s, x) ∂ + ε a îj (s, x) + Fi (s, x, y) ∂x 2 ∂x ∂x ∂y i i j i i=1 i,j=1 i=1

denote the generator of (X ε , Y ε ) from (3.1); then the corresponding backward operator ∂ + Lˆε (s, x, y) ∂s is supposed to be hypoelliptic in Q0 := R>0 × Rd × Rm , i.e. for any distribution v ≡ v(s, x, y) we have that v is a C ∞ function in ∂ every open subset of Q0 in which ( ∂s + Lˆε )v is a C ∞ function (H¨ ormander [H¨ o 67, p.147]). (d) Let D be the fixed bounded, open domain in Rm with smooth boundary ∂D, let N (y) denote the outer normal to ∂D at y ∈ ∂D and define G0 := {(t, x, y) ∈ R>0 × Rd × ∂D : F (t, x, y), N (y) = 0} ; then it is assumed that Ps,x,y (t, Xtε , Ytε ) ∈ G0 for some t ∈ [s, T ] = 0


129

for all (s, x, y) in Q := (0, T ) × Rd × D , where T > 0 denotes a fixed time horizon. (e) For all s < T and (x, y) ∈ Rd × D , UD (s; T ; x, y) = ∅ .

Remark 3.1.4 (on the set of assumptions 3.1.3). on (c): Given the non-degeneracy assumption (b), uniform ellipticity, the hypoellipticity (c) of a parabolic differential operator with smooth data F, ˆb and a ˆ is implied by the following condition:

∂Fi (t, x, y) rank = m ∂xj 1≤i≤m , 1≤j≤d for all (t, x, y) ∈ Q0 ; see Hernández-Lerma [HL 81] and Hörmander [H¨ o 67, p.147-149]. For the arguments, which use the so-called “Levi method” of constructing a fundamental solution for the corresponding differential equation, we would like to refer to Ilin [Il 64], H¨ ormander [H¨ o 67, p.147– 149], Weber [Web 51] and Kolmogorov [Ko 34]. Ilin [Il 64] is using the stronger condition that m ≤ d and for some constant c2 > 0

∂Fi (t, x, y) 0 < c2 < det < c−1 2 ∂xj 1≤i,j≤m for all (t, x, y) ∈ R+ × Rd × Rm . The backward operator in (c) is ultraparabolic, i.e. its coefficient matrix of the second derivatives is non-negative definite and has rank less than d + m, where d + m + 1 is the number of “independent” variables (Ilin [Il 64]). Also see Ichihara and Kunita [Ic-Ku 74] for further results concerning hypoellipticity. Following Hern´ andez-Lerma [HL 81] and [HL 80] we consider the case that the coefficients F, ˆb and σ ˆ also depend on the time-parameter. Although the results of this section will be applied to the autonomous angle SDE (see theorem 4.4.6 in section 4.4)), we also present the more general case ∂ here. Note that calling ∂s + Lˆε (s, x, y) the backward operator to (3.1) is in complete agreement with the notation introduced in section 2.2 (see ∂ + Lˆε (s, x, y) = 0 is the backward parabolic p.61) according to which ∂s equation corresponding to (3.1) in the backward variables (s, x, y).

130


on (d): From the law of motion (3.1) for Y ε it follows that Y ε cannot cross ∂D at points y where F (x, y), N (y) < 0 ; more formally, further define G+ := {(t, x, y) ∈ R>0 × Rd × ∂D : F (t, x, y), N (y) > 0} in correspondence to G0 ; then it is a priori known that Ps,x,y (T ε , XTε ε , YTε ε ) ∈ G+ ∪ G0 = 1 for all (s, x, y) ∈ Q ; see Stroock and Varadhan [Str-Vdh 72, Sec.7] and Fleming [Fl 80]. Hence the assumption (d) asserts that exits via G0 do not occur either. Furthermore, the smoothness assumption on the boundary ∂D can be relaxed to assuming ∂D to be of class C 2 ; see Hernández-Lerma [HL 80] and [HL 81]. 0 on (e): This assumption guarantees that ID is a well defined function in [0, ∞).

Notation 3.1.5 (Exit and inclusion probabilities). Consider the SDE (3.1) under the above assumptions 3.1.3 and let T ε denote the first exit time of (Ytε,y )t≥s from the bounded, open domain D ⊂ Rm as defined in 3.1.1. Then Qε ≡ Qε (s, x, y) ≡ QεD (s; T ; x, y) := Ps,x,y {TDε ≤ T } denotes the exit probability corresponding to the fixed time horizon T > 0 and ε PD (s; T ; x, y) := 1 − QεD (s; T ; x, y) = Ps,x,y {TDε > T } denotes the inclusion probability. Theorem 3.1.6 (PDE for the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3. Then the exit probability Qε (s, x, y) is a solution of the boundary value problem

∂ ε ˆ + L (s, x, y) Qε (s, x, y) = 0 on Q ≡ (0, T ) × Rd × D (3.2) ∂s 1 on G+ T Qε (s, x, y) = 0 on {T } × Rd × D , + where all appearing partial derivatives are continuous functions and G+ T ⊂ G is defined as d G+ T := {(s, x, y) ∈ (0, T ] × R × ∂D : F (s, x, y), N (y) > 0} .


131

Furthermore, Qε is continuous on Q1 := Q ∪ ({T } × Rd × D) ∪ (s, x, y) ∈ G+ : s ∈ (0, T ) ≡ (0, T ] × Rd × D ∪ (s, x, y) ∈ (0, T ) × Rd × ∂D : F (s, x, y), N (y) > 0 .

Sketch of Proof. In addition to the process (Xt , Yt )t≥0 := (Xtε , Ytε )t≥0 from (3.1) also consider the non-degenerate system Xt , Ytδ t≥0 := ε ε,δ Xt , Yt t≥0 in Rd × Rm defined by the SDE √ ?t , dYtε,δ = F (t, Xtε , Ytε ) dt + δ dW √ dXtε = ˆb (t, Xtε ) dt + ε σ ˆ (t, Xtε ) dWt ,

(3.3)

?t )t≥0 is a Brownian motion in Rm independent of (Wt )t≥0 , where δ > 0, (W ?• W• , and the same initial condition is fulfilled, W ε ε,δ Xs , Ys = (x, y) ∈ Rd × D for the initial time s ≥ 0. The solution process is now denoted by Xtx , Ytδ,y ≡ Xtε,x , Ytε,δ,y . t≥s

t≥s

In particular, the X ε -component of this process coincides with the one from (3.1), but the new coupled system is non-degenerate in Rd × Rm . In addition to the first exit time T ε defined in 3.1.1 let T δ ≡ T ε,δ (s, x, y) ≡ TDε,δ (s, x, y) := inf t ≥ s : Ytε,δ,y ∈ /D denote the first exit time of (Ytε,δ,y )t≥s from D. Furthermore, for abbreviation set δ Y − Y := Y δ ( . ) − Y( . ) ≡ max Yuε,δ,y ( . ) − Yuε,y ( . ) , t [s,t] u∈[s,t]

where ( . ) indicates the dependence on chance and also set θδ := TDε,δ (s, x, y) ∧ T as well as θ := TDε (s, x, y) ∧ T , where ∧ denotes the minimum operation.

132


Then one obtains for all (x, y) ∈ Rd × D and 0 ≤ s < t ≤ T that δ→0 i) Y δ − Y t −−−→ 0 , δ→0

ii) θδ −−−→ θ and δ→0

iii) Yθδδ −−−→ Yθ almost surely. Here, i) is proven by use of a Gronwall argument and the Lipschitz continuity of F (assumption 3.1.3(a)); ii) can be deduced from assumption 3.1.3(d) and iii) then follows from i) and ii); see Hern´ andez-Lerma [HL 81, p.42f.] for details. Now for proving that Qε (s, x, y) is a smooth solution of the proposed PDE (3.2),

∂ + Lˆε (s, x, y) v = 0 on Q , ∂s it must be verified that Qε (s, x, y) is a distribution solution of this PDE, due to the hypoellipticity assumption 3.1.3(c). The backward parabolic equation corresponding to the SDE (3.3) is

∂ δ + Lˆε (s, x, y) + ∆y v = 0 , ∂s 2 where ∆y denotes the Laplace-operator with respect to the variable y ∈ Rm . Furthermore, let ψ ∈ C(∂Q, R) be some continuous function1 defined on the boundary of Q. Then the boundary value problem

δ ∂ ε ˆ + L (s, x, y) + ∆y v = 0 on Q ∂s 2 v = ψ on ∂ ∗ Q , where is solved by

∂ ∗ Q :=

{T } × Rd × D ∪ G+ T ,

. v δ (s, x, y) := Es,x,y ψ θδ , Xθδ , Yθδδ

This

theorem 2.2.5, since the noise coefficient matrix √ follows from δ Im √ 0 of (3.3), Im denoting the m-dimensional unit matrix, 0 εσ ˆ (t, . ) satisfies the ellipticity condition (E) for a time-independent bound due to 1

This function ψ(s, x) is not to be confused with the stochastic process ψtε (ω) as defined in equation (1.4). These two objects will not be considered simultaneously, so there is no ambiguity.


133

assumption 3.1.3(b). Note, however, that theorem 2.2.5 cannot be applied directly, since the domain Rd × D is not bounded; this can be completed by a truncation argument. Now specify ψ by functions ψk ∈ C(Q, R), k ∈ N, such that the following conditions are also fulfilled: 1 , if (s, x, y) ∈ G+ T , ψk (s, x, y) = 0 , if (s, x, y) ∈ {T } × Rd × D and dist(y, ∂D) > k1 , ψk (s, x, y) ∈ [0, 1] , if (s, x, y) ∈ {T } × Rd × D and dist(y, ∂D) ≤ and such that ψk ( . ) − ψj ( . ) −k,j→∞ −−−−→ 0

1 k

,

uniformly on all compact subsets of Q .

The solution of the above PDE with boundary condition ψk is denoted by vkδ , i.e. vkδ (s, x, y) := Es,x,y ψk θδ , Xθδ , Yθδδ . δ Due to the properties i), ii) and iii) of θ , Yθδδ noted above, it follows from the dominated convergence theorem and the continuity of ψk that δ→0 vkδ (s, x, y) −−−→ Es,x,y ψk θ, Xθ , Yθ =: Qεk (s, x, y) for any k ∈ N. Now one verifies that Qεk (s, x, y) is a distributional solution of the boundary value problem

∂ ε ˆ + L (s, x, y) v = 0 on Q ∂s v = ψk on ∂ ∗ Q . Then one proves that lim Qεk (s, x, y) = Qε (s, x, y)

k→∞

almost everywhere on Q, and this convergence then implies that also Qε (s, x, y) is a distributional solution of the proposed PDE (3.2),

∂ + Lˆε (s, x, y) v = 0 on Q . ∂s As has already been remarked before, the hypoellipticity assumption 3.1.3(c) then yields that Qε (s, x, y) is a smooth solution of this PDE. The second assertion of the theorem, i.e. the continuity of Qε on the set Q1 ≡ (0, T ] × Rd × D ∪ (s, x, y) ∈ (0, T ) × Rd × ∂D : F (s, x, y), N (y) > 0 ,

134


is verified by use of the continuity properties of the Feller semigroup of the process (X ε , Y ε ); see Hernández-Lerma [HL 81, p.44] for details. The following theorem calculates the asymptotics of the exit probability QεD (s; T ; x, y) of the degenerate system. Note the similarities of this assertion 0 and the corresponding value function ID (s; T ; x, y) with theorem 2.4.3 for non-degenerate systems and its corresponding rate function I[0,T ] as defined in theorem 2.3.4. Theorem 3.1.7 (Asymptotics of the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3. Then it follows for the exit probability QεD (s; T ; x, y) ≡ Ps,x,y {TDε ≤ T } that 0 lim ε log QεD (s; T ; x, y) = − ID (s; T ; x, y)

ε→0

0 (s; T ; x, y) denotes the value for any s < T and (x, y) ∈ Rd × D, where ID function as defined in 3.1.2,

0 (s; T ; x, y) ≡ I 0 ≡ ID

1 inf 2 u∈U

ϑ

0 1 2 ˆ(t, u(t))−1/2 u(t) ˙ − ˆb(t, u(t)) dt a

s

in which U ≡ UD (s; T ; x, y) ≡ u ∈ C 1 (R+ , Rd ) : u(s) = x and ϑD (s; x, y; u) ≤ T and

ϑ ≡ ϑD (s; x, y; u) ≡ inf t ≥ s : y 0 (t) ∈ /D

for the deterministic control system dy 0 (t) = F (t, u(t), y 0 (t)) dt

(t > s) ,

(3.4)

0

(u(s), y (s)) = (x, y) . Sketch of Proof. Define I ε as I ε ≡ I ε (s, x, y) := − ε log QεD (s; T ; x, y) for all s < T and (x, y) ∈ Rd × D. Then we have to prove that lim I ε = I 0 .

ε→0

1) The PDE for the exit probability Qε obtained in theorem 3.1.6 implies for I ε that

∂ + Lˆε (s, x, y) ∂s

I ε (s, x, y) =

d 1 ∂I ε (s, x, y) ∂I ε (s, x, y) a îj (s, x) 2 i,j=1 ∂xi ∂xj


135

on Q ≡ (0, T ) × Rd × D , and the boundary behavior of I ε is given by = 0 on G+ T , ε I (s, x, y) s→T −−−→ ∞ for (x, y) ∈ Rd × D . This PDE for I ε can be rewritten as d ε ∂2I ε ∂I ε + a îj (s, x) + F (s, x, y), ∇y I ε m + H s, x, ∇x I ε = 0 R ∂s 2 i,j=1 ∂xi ∂xj

on Q, where ∇x I ≡ ε

∂I ε ∂I ε ∂x1 , . . . , ∂xd d

and ∇y I ≡ ε

∂I ε ∂I ε ∂y1 , . . . , ∂ym

(3.5)

denote the

gradient with respect to x ∈ R and y ∈ R , respectively, and H is defined as H : R+ ×R ×R → R, d

d

m

d

d 1 ˆ H(s, x, p) := a îj (s, x)pi pj . bi (s, x) pi − 2 i,j=1 i=1

Note that the function K : R+ × Rd × Rd → R+ ,

K(s, x, q) ≡

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) a 2

defined in 3.1.2 is dual to H in the sense that K(s, x, q) = max H(s, x, p) − p, q p∈Rd

and

H(s, x, p) = min K(s, x, q) + p, q , q∈Rd

where . , . denotes the canonical scalar product in Rd ; see e.g. Freidlin and Wentzell [Fr-We 98, p.154].2

2 Freidlin and Wentzell [Fr-We 98, p.154,136f.] do not work with the duality defined above but with

H(s, x, p) := − H(s, x, −p) ≡

d

d ˆbi (s, x) pi + 1 a îj (s, x) pi pj 2 i,j=1 i=1

(in our notation) and its Legendre transformation L(s, x, q) := sup p∈Rd

p, q − H(s, x, p)

=

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) . a 2

The latter coincides with the function K used here, L(s, x, q) = K(s, x, q), as is immediately seen by a substitution p → −p in the maximization argument. For more details on duality and Legendre transformations see Rockafellar [Roc 97].

136


2) Setting formally ε = 0 in (3.5) one obtains the PDE of dynamic programming corresponding to the control problem which defines the function 0 (s; T ; x, y) ≡ I 0 (s, x, y) ≡ ID

ϑD (s;x,y;u)

inf u∈UD (s;T ;x,y)

K t, u(t), u(t) ˙ dt

s

in 3.1.2, that is ∂I 0 + F, ∇y I 0 m + H s, x, ∇x I 0 = 0 ; R ∂s

(3.6)

see Fleming and Rishel [Fl-Ris 75, p.80-84,104]. 3) Next, one introduces a new stochastic control problem whose dynamic programming PDE is (3.5) for each ε > 0: Consider dηtε = F (t, Ξεt , ηtε ) dt √ d Ξεt = vt dt + ε σ(t, Ξεt ) dWt (Ξεs , ηsε ) = (x, y) ∈ Rd × D , where s ≤ t ≤ T and where v is a nonanticipative ((Ft )t≥0 -adapted) stochastic process taking its values in Rd such that

T

| vt |2 dt < ∞

E s

and such that the resulting process (Ξεt , ηtε )t∈[s,T ] fulfills the assumption 3.1.3(d). A stochastic process v satisfying these properties is called admissible for the above control problem and the set of such admissible control processes is denoted by U ≡ U (s, x, y). Furthermore, fix some bounded Lipschitz continuous function Φ : R+ × Rm −→ R+ such that Φ(s, y) = 0 for all (s, x, y) ∈ G+ and define 5 J (s, x, y) := min E v∈U

6

ϑε

K(t, Ξεt , vt ) dt

ε

ε

+ Φ(ϑ

, ηϑε ε )

,

s

where ϑε is the minimum of T and the first exit time of η ε from D. Then one can prove that the minimum in the definition of J ε exists and that J ε (s, x, y) thus defined satisfies the PDE (3.5) on Q ≡ (0, T ) × Rd × D together with the boundary condition J ε (s, x, y) = Φ(s, y) for all (s, x, y) ∈ ∂ ∗ Q ≡ {T } × Rd × D ∪ G+ T .


137

The proof of this fact follows the same route as the one of theorem 3.1.6 and uses the so-called “verification theorem” of dynamic programming (see Fleming and Rishel [Fl-Ris 75, p.159]); since this argument goes beyond the framework of this book, we would like to refer to Hern´ andez-Lerma [HL 81, p.46f.]. 4) lim supε→0 I ε ≤ I 0 : Consider the system y 0 (t) from (3.4) for some control function u ∈ U such that ϑ < T and v := u˙ ∈ U . Then take the system (Ξεt , ηtε ) from step 3) for this (deterministic) control process v , fix T ∈ (ϑ, T ) and define T ε := T (η ε ) ∧ T , where T (η ε ) denotes the first exit time of η•ε from D; in particular, T ε ≤ T < T. Now it follows that 5 ε d T ε ∂2I ε ∂I ε ε + I (s, x, y) = −E a îj + F, ∇y I ε m R ∂t 2 ∂x ∂x i j s i,j=1 6

◦ t, Ξεt , ηtε dt + E I ε T ε , ΞεT ε , ηTε ε + vt , ∇x I ε Rd

5

T

ε

=E

H t, x, ∇x I ε − vt , ∇x I ε ◦ t, Ξεt , ηtε dt

s

5

0 1 +E I ε T ε , ΞεT ε , ηTε ε

Tε

K

≤E

6

t, Ξεt , vt

6 0 1 dt + E I ε T ε , ΞεT ε , ηTε ε ,

s

where the first equation is the Dynkin formula3 I ε (s, x, y) ≡ E[I ε ◦ Xs ] = E[I ε ◦ XT ε ] − E

Tε

(G I ε ) ◦ Xt dt s

for the process (Xt )t≥s := (t, Ξεt , ηtε )t≥s with generating operator G, the second equality is the PDE (3.5) for I ε found in step 1) and the estimate from above is immediate from the definition of K; note that “◦” denotes the composition of mappings here and is not to be confused with Stratonovich stochastic differentials. The second summand in the above bound can be seen to converge to 0 as ε → 0; the argument uses the fact that I ε = 0 on G+ T and an extension of theorem 1.1 by Freidlin and Wentzell [We-Fr 70]; see Hern´ andez-Lerma [HL 81, p.47f.] for details. 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.131].

138


The first summand can be seen to converge to ε → 0, since v = u˙ in the system of step 3). Hence, this implies that

ϑ

lim sup I ≤ ε

ε→0

ϑ s

K t, u(t), u(t) ˙ dt as

K t, u(t), u(t) ˙ dt

s

and thus lim supε→0 I ε ≤ I 0 by the definition of I 0 (see 3.1.2). 5) As a preparation for proving that also lim inf ε→0 I ε ≥ I 0 , one considers the deterministic analogue of the system defined in step 3), that is dηt0 = F (t, Ξ0t , ηt0 ) dt d Ξ0t = vt dt (Ξ0s , ηs0 ) = (x, y) ∈ Rd × D , where again s ≤ t ≤ T and where the set of admissible control functions U 0 ≡ U 0 (s, x, y) now consists of the continuous functions v ∈ C([s, T ], Rd ) such that T

| vt |2 dt < ∞ s

and such that possible exits of the generated trajectory (t, Ξ0t , ηt0 )t∈[s,T ] from R+ × Rd × D only occur through points in G+ . Let Φ be the function fixed in step 3) and define J (s, x, y) := min0 v∈U

ϑ0

K(t, Ξ0t , vt ) dt + Φ(ϑ0 , ηϑ0 0 ) ,

0

s

where (s, x, y) ∈ [0, T ) × Rd × D and where ϑ0 is the minimum of T and the first exit time of η 0 from D. Furthermore, define J 0 (s, x, y) = Φ(s, y) for (s, x, y) ∈ ∂ ∗ Q ≡ {T } × Rd × D ∪ G+ T which will play a role in the final step. Then we have lim inf J ε (s, x, y) ≥ J 0 (s, x, y) ε→0

for all (s, x, y) ∈ Q ≡ (0, T ) × Rd × D. The proof of this fact works with an “optimal feedback control” for the system defining J ε in step 3) and uses a stochastic version of the “principle of optimality in dynamic programming”; see Hernández-Lerma [HL 81, p.49f.] for details.


139

6) lim inf ε→0 I ε ≥ I 0 : Fix some Lipschitz continuous function Ψ : D → R+ such that > 0 on D , Ψ = 0 on ∂D . Furthermore, for M > 0 define 0 1 QεM (s, x, y) := Es,x,y exp − M Ψ YTε ε ∧T /ε , where T ε denotes the first exit time of Y ε from D as before; accordingly define ε (s, x, y) := − ε log QεM (s, x, y) . IM It follows immediately from the definitions that QεM (s, x, y) ≥ Es,x,y 1

Tε ≤T

≡ Qε (s, x, y)

ε and equivalently that IM (s, x, y) ≤ I ε (s, x, y). QεM (s, x, y) is seen to satisfy the same PDE (3.2) as Qε (s, x, y) together with the boundary condition 1 on G+ ε T QM (s, x, y) = exp [−M Ψ(y)/ε ] on {T } × Rd × D ; ε (s, x, y) satisfies the same PDE (3.5) as I ε (s, x, y) with the boundary IM condition 0 on G+ ε T IM (s, x, y) = M Ψ(y) on {T } × Rd × D ;

by remembering the boundary behavior of I ε , = 0 on G+ T , ε I (s, x, y) s→T −−−→ ∞ for (x, y) ∈ Rd × D , it becomes clear how I ε is being “truncated” near {T }×Rd ×D by considering ε IM . Now let Φ ∈ C(R+ × Rm , R+ ) be a Lipschitz function such that Φ(s, y) = 0 whenever (s, x, y) ∈ G+ and such that also 0 on G+ T Φ(s, y) = M Ψ(y) on {T } × Rd × D ; ε 0 and JM , the corresponding functions from steps 3) and 5) are denoted by JM respectively.

140


ε ε 0 In particular, the functions IM , JM and JM just defined are all equal to ∗ d Φ(s, y) whenever (s, x, y) ∈ ∂ Q ≡ ({T } × R × D) ∪ G+ T . By uniqueness it ε ε follows for the former two functions that IM = JM . Furthermore, comparing 0 the control problems in 3.1.2 and in step 5) defining I 0 and JM , respectively, 0 0 one gets that I ≤ lim inf JM . M→∞

Also using the estimate mentioned at the end of step 5) and the fact that ε ≤ I ε , it follows altogether that IM 0 I 0 (s, x, y) ≤ lim inf JM (s, x, y) M→∞

ε ≤ lim inf lim inf JM (s, x, y) M→∞

ε→0

ε = lim inf lim inf IM (s, x, y) M→∞

ε→0

≤ lim inf I ε (s, x, y) ε→0

for all (s, x, y) ∈ Q ≡ (0, T ) × Rd × D.

3.2 Uniform consequence for the exit probability Next we prove a consequence of theorem 3.1.7 which is not contained in the paper by Hern´ andez-Lerma [HL 81]: This corollary will be used later in the proof of theorem 4.4.6. Corollary 3.2.1 (Uniform asymptotics of the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3 and suppose that this SDE (3.1) is autonomous, i.e. F, ˆb and σ ˆ do not depend on t explicitly. 0 0 (x, y) := ID (0; T ; x, y). Furthermore, define ID Then it follows for the exit probability QεD (x, y) := QεD (0; T ; x, y) that 0 lim inf ε log min QεD (x, y) ≥ − max ID (x, y) ε→0

(x,y)∈G

(x,y)∈G

and 0 lim sup ε log max QεD (x, y) ≤ − min ID (x, y) ε→0

(x,y)∈G

(x,y)∈G

for all T > 0 and any compact set G ⊂ Rd × D. 0 Proof. First note that the functions under consideration, QεD and ID , are continuous with respect to (x, y); this follows from the respective PDEs in theorem 3.1.6 and in equation (3.6). More precisely, (s, x, y) → QεD (s, T, x, y) is continuous on (0, T] × Rd × D due to theorem 3.1.6 for any T > 0. Fixing

3.2 Uniform consequence for the exit probability

141

T := T + δ for given T and some δ > 0 this implies in particular, due to the time-homogeneity (i.e. the stationarity of the stochastic flow4 ), that (x, y) → QεD (x, y) = QεD (δ, T + δ, x, y) 0 is continuous on Rd × D. The continuity of (x, y) → ID (x, y) follows from 0 the fact that ID satisfies the PDE (3.6) of dynamic programming. This PDE asserts differentiability in the sense that there exists a tangent plane at each point (see Fleming and Rishel [Fl-Ris 75, p.83f.]) and hence continuity. 0 are continuous and the proposed Therefore the functions QεD and ID maxima and minima are attained indeed.

First estimate: For brevity define 0

I D,G :=

0 max ID (x, y)

(x,y)∈G

and fix some η > 0; it is to be shown that there is an ε0 > 0 such that for all ε < ε0 , 0 ε log min QεD (x, y) > − I D,G − η . (x,y)∈G

From theorem 3.1.7 it follows that for any (x, y) ∈ G, there exists ε(x,y) > 0 such that for all ε < ε(x,y) , 0 0 (x, y) − η ≥ − I D,G − η . ε log QεD (x, y) > − ID Due to the continuity of QεD it is hence possible to find a sufficiently small open ball B(x,y) and to choose ε(x,y) > 0 such that ε log

min (˜ x,˜ y)∈B(x,y)

0

QεD (˜ x, y˜) > − I D,G − η

for all ε < ε(x,y). Since G is compact, it can be covered by finitely many of k > such balls which are denoted by B(xi ,yi ) , i.e. G ⊂ B(xi ,yi ) . The bound i=1

for the parameter ε which corresponds to B(xi ,yi ) by virtue of the previous estimate is denoted by ε(xi ,yi ) , i ∈ {1, . . . , k}. Then one gets for all ε < ε0 := min ε(xi ,yi ) that

i=1,...,k

0

ε log min QεD (x, y) > − I D,G − η . (x,y)∈G

4

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.335].

142


Second estimate: The above compactness argument also works in this case. Define 0 (x, y) I 0D,G := min ID (x,y)∈G

and fix an η > 0. By theorem 3.1.7 there is for any (x, y) ∈ G an ε(x,y) > 0 such that 0 (x, y) + η ≤ − I 0D,G + η ε log QεD (x, y) < − ID for all ε < ε(x,y) . Due to the continuity of QεD and the compactness of G, it is possible to cover G by finitely many open balls B(x,y) on which ε log

max (˜ x,˜ y)∈B(x,y)

QεD (˜ x, y˜) < − I 0D,G + η .

Thus the above claim holds true, if ε0 is again chosen as the minimum of the corresponding (finitely many) parameters ε(x,y) .

Chapter 4

Local Lyapunov exponents

In this chapter the goal of obtaining the “local Lyapunov exponents” as sublimiting exponential growth rates is tackled. As already described, the system under consideration is the real-noise driven stochastic system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ,

(1)

where A ∈ C(Rd , Kn×n ) is a continuous matrix function (K = R or C), d ∈ N and n ∈ N are the dimensions of the state spaces of X ε and Z ε , respectively, ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which denotes a Brownian motion in Rd on a complete probability space (Ω, F , P) and X ε,x is a diffusion starting in x ∈ Rd , defined by the SDE (2.1) such that the assumptions 2.1.1 hold. For Z ε , solving the random vector differential equation dZtε = A Xtε,x (ω) Ztε dt, Z0ε = z ∈ Kn , we will use the equivalent notations Z ε : R+ × Ω × Rd × Kn −→ Kn , (t, ω, x, z) → Z ε (t, ω, x, z) ≡ Z ε (t, ω, x) z ≡ Ztε (ω, x) z ≡ Ztε (ω, x, z), as before, where Ztε (ω, x) ≡ Z ε (t, ω, x, . ) solves the random matrix differential equation dZtε = A Xtε,x (ω) Ztε dt , Z0ε = idKn . 1 The object of interest is the exponential growth rate T (ε) log | ZTε (ε) (ω, x, z) | on the time scale T (ε). Any limit as ε → 0 of this rate will be called local Lyapunov exponent of Z ε .


143

144

4 Local Lyapunov exponents

4.1 Local Lyapunov exponents: upper and lower bound The n eigenvalues (roots of the characteristic polynomial) of A(x) ∈ Kn×n are denoted by Λj (x) := Λj (A(x)) ∈ C

(j ∈ {1, . . . , n} , x ∈ Rd )

in the sequel. In particular, Λj (Ki ) ≡ Λj (A (Ki )) lists the n eigenvalues of the matrix A(Ki ) at the equilibrium positions K1 , . . . , Kl as given by assumption 2.1.1(K); this enumeration is chosen such that Re (Λ1 (Ki )) ≥ Re (Λ2 (Ki )) ≥ · · · ≥ Re (Λn (Ki )) for all i = 1, . . . , l and the distinct numbers in this list are denoted as λ1 (Ki ) > λ2 (Ki ) > · · · > λpi (Ki ) , where the quantity pi = p A (Ki ) also depends on the space point Ki . Note that if ε = 0, theorem 1.4.3 provides the Lyapunov exponent of the system Zt0 (x, z) as deterministic limit as one of the numbers λ1 (Ki ) , . . . , λpi (Ki ), where i ∈ L ≡ {1, . . . , l} is chosen such that x is in the domain of attraction of Ki according to assumption 2.1.1(K). Fixing the metastable state Kµ(x,ζ) for a specific time scale T (ε) ! eζ/ε and initial value x = X0ε,x of the noise process (see 2.5.4 for the definitions), the SDE for Z ε in (1) can be rewritten as dZtε = [ Aµ(x,ζ) + G(Xtε,x ) ] Ztε dt , where the fixed matrix Aµ(x,ζ) is defined as Aµ(x,ζ) := A(Kµ(x,ζ) ) and G( . ) := A( . ) − Aµ(x,ζ) is considered as stochastic perturbation of the deterministic coefficient matrix Aµ(x,ζ) . This perturbation is “small” on the chosen time scale. More precisely, we note the following consequences of section 2.5 on the metastability of X ε,x . Proposition 4.1.1 (Consequences of the metastability of X ε,x ). Consider and let X ε satisfy the assumptions 2.1.1. Fix >l the above situation ζ/ε x ∈ i=1 Di and T (ε) ! e , where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Furthermore, let C ∈ Kn×n be some invertible matrix. Then for all open sets B ⊂ R, ε −1 0, 0 ∈ / B, ε→0 ∈ B −−−→ C G XT (ε) C Px 1, 0 ∈ B .

4.1 Local Lyapunov exponents: upper and lower bound

145

−1 In particular, C G(XTε,x converges in probability to zero, (ε) ) C

P C G X ε,x C −1 −−− −→ 0. T (ε) ε→0 Since the resultant convergence in probability implies weak convergence, one −1 gets that the distribution of C G(XTε,x converges weakly to the Dirac (ε) ) C measure at zero, −1 w −1 P ◦ C G(XTε,x −− −−→ δ0 . (ε) ) C ε→0 Furthermore, T (ε) 1 −1 1 P C G Xuε,x C −1 du = C G X ε,x dt −−− −→ 0. t T (ε) C ε→0 T (ε) 0 0 (4.1) Proof. This is theorem 2.5.10 for the (continuous) function f := C G( . ) C −1 ≡ C A( . ) − Aµ(x,ζ) C −1 which vanishes at Kµ(x,ζ) , f (Kµ(x,ζ) ) ≡ C G(Kµ(x,ζ) ) C −1 = 0. Theorem 4.1.2 (Local Lyapunov exponents: upper and lower bound). Consider a mapping A ∈ C(Rd , Kn×n ) and let X ε satisfy the >l assumptions 2.1.1. Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Furthermore, fix z ∈ Kn \{0} and let Ztε (ω, x, z) be the corresponding solution process of (1). Then the exponential growth rate of Ztε (ω, x, z) on the time scale T (ε), 1 log ZTε (ε) ( . , x, z) , T (ε) is bounded from above by Re Λ1 (Kµ(x,ζ) ) and from below by Re Λn (Kµ(x,ζ) ) , where Kµ(x,ζ) denotes the metastable state corresponding to x and ζ as given by definition 2.5.4. More precisely, for any η > 0, 1 log ZTε (ε) ( . , x, z) P Re Λn (Kµ(x,ζ) ) − η ≤ T (ε) ε→0 ≤ Re Λ1 (Kµ(x,ζ) ) + η −−−→ 1. Proof. The SDE (1) under consideration for Z ε is rewritten as above as dZtε = [ Aµ(x,ζ) + G(Xtε,x ) ] Ztε dt ,

146

where and


Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) G( . ) ≡ A( . ) − Aµ(x,ζ)

which is “small” on the chosen time scale in the sense of proposition 4.1.1. In order to simplify the exposition, we also set (j = 1, . . . , n) Λj := Λj A Kµ(x,ζ) for the eigenvalues of Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) and choose an enumeration of these eigenvalues such that Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ), as before. We are going to view the SDE for Z ε as stochastic perturbation of d the linear ODE dt zt = Aµ(x,ζ) zt with constant coefficient matrix Aµ(x,ζ) ; x and ζ specify the metastable state Kµ(x,ζ) which remains unchanged during the whole treatment. This notation resembles the one used in the proof of the Hartman-Wintner-Perron theorem 1.4.3. It is the purpose of this theorem to investigate to which extent their arguments provide information in the stochastic case. In short, the Gronwall argument and proposition 4.1.1 will imply the upper and lower bounds: As in the proof of 1.4.3, we can assume that after a constant, invertible coordinate transformation, Aµ(x,ζ) is given in (complex) Jordan canonical form, ⎛ ⎞ Λ1 0 0 ⎜ a Λ 0 ⎟ 2 ⎜ ⎟ ⎜ 0 a Λ ⎟ 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Λ ⎜ ⎟, 4 Aµ(x,ζ) = ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ Λn−1 0 ⎠ a Λn where in this example Λ1 = Λ2 = Λ3 and Λn−1 = Λn ; again, the boxes depict lower Jordan blocks of the following qualitative shape: ⎞ ⎛ Λ0 ⎟ ⎜a Λ 0 ⎟ ⎜ ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟; J a := ⎜ . . . ⎟ ⎜ ⎟ ⎜ ⎜ a Λ 0 ⎟ ⎟ ⎜ ⎝ a Λ 0⎠ a Λ


147

here, a > 0 is a parameter and all matrix entries not mentioned explicitly are zero. As in 1.4.3, we have taken the “ordinary” lower Jordan blocks J 1 and carried out another transformation tε := CZtε , Z where C := diag(1, a, . . . , an−1 ) ; then pathwise (“ω-wise”) differentiation with respect to t yields ε = C dZ ε = CAµ(x,ζ) Z ε + CG(X ε,x )Z ε dZ t t t t t = CAµ(x,ζ) C −1 Z˜tε + CG(Xtε,x )C −1 Z˜tε ; hence, we can redefine Aµ(x,ζ) as CAµ(x,ζ) C −1 and G(Xtε,x ) as CG(Xtε,x )C −1 ; after this transformation the Jordan-blocks of Aµ(x,ζ) are of the form Jja . The “smallness” of G(Xtε,x ) on the chosen time scale in the sense of proposition 4.1.1 is maintained under this change of the coordinate sys1 log |ZTε (ε) | is tem, as already noted in 4.1.1. The investigated behavior of T (ε) invariant under this (invertible) transformation of variables. More precisely, the norm |C . | on Kn is equivalent to the “old” norm | . | as Kn is finite dimensional; but factors from converting these norms into each other end up ε for as vanishing summands, since T (ε) → ∞; therefore, we redefine Z ε as Z the following considerations. Hence, the underlying random differential equation for Z ε from (1) is of the form Λ1 Z ε;1 + (Gεt Z ε )1 dt d Z ε;1 = d Z ε;i = ai−1 Z ε;i−1 + Λi Z ε;i + (Gεt Z ε )i dt (i = 2, . . . , n) , where the superscript i in ( . )i again indicates the i-th coordinate, in particular, (i = 1, . . . , n) Z ε;i := (Z ε )i for the coordinates of Z ε ; furthermore, we used the notation Gεt := Gεt (ω) := G (Xtε,x (ω)) and ai := Aµ(x,ζ) i+1,i ∈ {0, a}

(i = 1, . . . , n − 1)

denoting the n − 1 entries of Aµ(x,ζ) below the diagonal. Also put a0 := 0 and Z ε;0 := 0 for definiteness. Furthermore, again let λ1 > λ2 > · · · > λp

148


denote the distinct numbers in the list Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) ; in particular, it again follows from these definitions that i := min{ i : Re(Λi ) = λk } =⇒ ai−1 = 0

(4.2)

for all k = 1, . . . , p. As in the proof of the Hartman-Wintner-Perron theorem 1.4.3, we now define auxiliary processes describing the norms of the projection of Z ε onto the subspaces belonging to the different real part of eigenvalues; more precisely, we define the following stochastic processes pathwise as ε;i := | Ztε;i | , t

i = 0, . . . , n ,

and hereby

Lε;k t (ω) :=

2 ε;i (ω) t

, k = 1, . . . , p ,

{i : Re(Λi )=λk }

Mtε;k (ω) :=

Ntε;k (ω) :=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

0

{i : Re(Λi )>λk } ⎪ n ⎪ 2 ⎪ ⎪ ε;i ⎪ (ω) ⎪ t ⎩ i=1

⎧ ⎪ ⎨

⎪ ⎩


0

,k = 1 ,

2 ε;i (ω) t ≡ | Ztε (ω) | 2 ε;i t (ω)

, k = 2, . . . , p , 2

,k = p + 1 , , k = 1, . . . , p , ,k = p + 1 ,

such that ε;p | Ztε |2 = Mtε;k + Ntε;k = Lε;1 . t + · · · + Lt

(4.3)

The above random differential equation for Z ε implies that the calculations as performed in the proof of Hartman-Wintner-Perron theorem 1.4.3 can be directly applied pathwise, i.e. for any realization ω ∈ Ω, in order to get the ε;k ε;k and Ntε;k . Since we are going to discuss what differentials of ε;i t , Lt , Mt can and what cannot be carried over from theorem 1.4.3 to the stochastic situation here and since we are also going to investigate the more tractable case that A( . ) only takes its values in the diagonal matrices (theorem 4.3.1), it seems indicated to recall the calculations in the notation used here such that these cases can be compared in full detail. Therefore (1.22) reads for the stochastic process ε;i t due to the above form of the random differential equation for Z ε pathwise as


149

d ¯ ε;i ε;i d ε;i 2 ( ) = (Z Z ) dt t dt

d ¯ ε;i d ε;i ε;i ε;i ¯ = Z Z +Z + 0 Z dt dt

d ε;i Z = 2 Re Z¯ ε;i dt ε;i ai−1 Z ε;i−1 + Λi Z ε;i + (Gεt Z ε )i = 2 Re Z¯ = 2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2

(4.4)

t

+ 2 Re(Z¯ ε;i (Gεt Z ε )i )

(4.5)

for all i = 1, . . . , n, where as usual ¯ denotes complex conjugation; hence, d( ε;i )2 ε;i 2 − 2 Re(Λi ) ( ) ≤ 2ai−1 ε;i ε;i−1 + 2 ε;i (Gεt Z ε )i . dt (4.6) Thus it follows for Lε;k that ε;k dL ε;k dt − 2λk L ≡

+

{i:Re(Λi )=λk }

(4.6)

≤ 2

≤ 2⎝

( ε;i )2 ⎠

{i:Re(Λi )=λk }

⎛

+2⎝

ε;i |(Gεt Z ε )i |

{i:Re(Λi )=λk }

⎞1/2 ⎛

C.-S.

ai−1 ε;i ε;i−1 + 2

{i:Re(Λi )=λk }

⎛

, d( ) − 2λk ( ε;i )2 dt ε;i 2

⎝

⎞1/2

a2i−1 ( ε;i−1 )2 ⎠

{i:Re(Λi )=λk }

⎞1/2 .1/2 n ε;i 2 ⎠ ε ε i 2 ( ) |(Gt Z ) |

{i:Re(Λi )=λk }

⎛

i=1

⎞ 1/2

= 2(Lε;k )1/2 ⎝

(4.2)

a2i−1 ( ε;i−1 )2 ⎠

{i:Re(Λi )=λk ,i−1≥i}

+ 2(L

ε;k 1/2

)

⎛

≤ 2a(Lε;k )1/2 ⎝

|(Gεt Z ε )|

⎞1/2 ( ε;i )2 ⎠

+ 2(Lε;k )1/2 Gεt |Z ε |

{i:Re(Λi )=λk } (4.3)

= 2aLε;k + 2Gεt (Lε;k )1/2 (M ε;k + N ε;k )1/2 . / M ε;k + N ε;k ε;k ε 2a + 2Gt =L Lε;k

(4.7)

150


for all k = 1, . . . , p, where “C.-S.” is an abbreviation for the Cauchy-Schwarz inequality. Furthermore,

d M ε;k = dt

{i : Re(Λi )>λk }

(4.5)

=

d ε;i 2 ( ) dt

2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2

{i : Re(Λi )>λk }

≥ −2

+ 2 Re(Z¯ ε;i (Gεt Z ε )i )

ai−1 ε;i ε;i−1 + 2λk−1

{i : Re(Λi )>λk }

−2

( ε;i )2

{i : Re(Λi )>λk }

ε;i | (Gεt Z ε )i |

{i : Re(Λi )>λk } (4.2)

≥ − 2 a M ε;k + 2 λk−1 M ε;k − 2 Gεt (M ε;k )1/2 |Z ε | √ (4.3) = 2 (λk−1 − a) M ε;k − 2 Gεt M ε;k M ε;k + N ε;k

(4.8)

for k = 2, . . . , p + 1; and d N ε;k = dt


(4.5)

=

d ε;i 2 ( ) dt


≤ 2

2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2 + 2 Re(Z¯ ε;i (Gεt Z ε )i ) ai−1 ε;i ε;i−1 + 2λk


+2

( ε;i )2


ε;i | (Gεt Z ε )i |

{i : Re(Λi )≤λk } (4.2)

≤ 2 a N ε;k + 2 λk N ε;k + 2 Gεt (N ε;k )1/2 |Z ε | √ (4.3) = 2 (λk + a) N ε;k + 2 Gεt N ε;k M ε;k + N ε;k for k = 1, . . . , p. From (4.8), (4.9) and (4.3) one gets for |Z ε |2 ≡ M ε;p+1 ≡ N ε;1 that for any ω, d |Z ε |2 ≤ 2 λ1 + a + Gεt |Ztε |2 2 λp − a − Gεt |Ztε |2 ≤ dt t

(4.9)


151

so that integration with respect to t ≥ 0 yields pathwise: exp (λp − a)t −

t 0

Gεu du |z| ≤ |Ztε | ≤ exp (λ1 + a)t +

t 0

Gεu du |z| .

Furthermore, due to equation (4.1) in proposition 4.1.1, 1 T (ε)

T (ε) 0

ε P Gu ( . ) du −−− −→ 0 . ε→0

Now if η > 0 is given, then fix the small parameter a as a ∈ (0, η) and assume without loss of generality that |z| = 1 (otherwise the summand ε→0 1 −−→ 0 deterministically vanishes). Then the above two statements T (ε) log |z| − which had been lastly obtained yield that 0 1 1 ε log ZT (ε) ( . , x, z) ∈ P / Re Λn (Kµ(x,ζ) ) − η, Re Λ1 (Kµ(x,ζ) ) + η T (ε) 1 1 ε ε log | ZT (ε) | < λp − η + P log | ZT (ε) | > λ1 + η =P T (ε) T (ε) ; T (ε) 1 Gεu du < λp − η ≤ P (λp − a) − T (ε) 0 ; T (ε) 1 ε Gu du > λ1 + η + P (λ1 + a) + T (ε) 0 ; T (ε) 1 Gεu du > η − a =2P T (ε) 0 ε→0

−−−→ 0. This is a reformulation of what had been claimed.

Corollary 4.1.3 (p = 1). Consider the situation of theorem 4.1.2. If p A Kµ(x,ζ) = 1, i.e. if

Re Λ1 (Kµ(x,ζ) ) = Re Λn (Kµ(x,ζ) ) =: λ ,

then

1 P log ZTε (ε) ( . , x, z) −−− −→ λ ε→0 T (ε)

for all z ∈ Kn \{0}. Remark 4.1.4 (n = 1). Consider the situation of corollary 4.1.3. Then the assumption p = 1 is trivially fulfilled if n = 1. In this case, A =: a is just a

152


continuous function on Rd taking its values in the scalar field K (R or C). The random vector differential equation for Z ε in (1) is now a scalar differential equation dZtε = a Xtε,x (ω) Ztε dt , Z0ε = z ∈ K which can be explicitly integrated pathwise as t

a Xuε,x (ω) du . Ztε (ω, x, z) = z · exp 0

Therefore the convergence assertion of corollary 4.1.3 can also be directly deduced by applying (the final statement of) theorem 2.5.10 to the continuous function f (x) := Re[a(x)] in the one-dimensional case n = 1. Due to corollary 4.1.3 it is left to consider the case that p ≡ p A Kµ(x,ζ) > 1 . More precisely, one needs to investigate the convergence behavior of the 1 random variable T (ε) log | ZTε (ε) ( . , x, z) | as ε → 0 in the case that there are several possible “candidates” λ1 , . . . , λp for this growth rate. Theorem 4.1.2 states that λ1 and λp serve as upper and lower bound in the sense of convergence in probability, respectively. In the light of the Multiplicative Ergodic Theorem 1.3.1 and the deterministic theorem 1.4.3 which enframe the sublimiting situation at hand, one might conjecture here (in the situation of theorem 4.1.2 with p > 1) that for any z ∈ Kn \{0}, there is an index J ∈ {1, . . . , p} such that 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λJ ; ε→0 T (ε)

(4.10)

another conjecture, motivated by the law of large numbers (1.16), is that under some hypoellipticity condition for any z ∈ Kn \{0}, 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λ1 . ε→0 T (ε)

(4.11)

For this purpose, we first continue the investigations of the proof of theorem 4.1.2 as in 1.4.3: Excluding the trivial case that Ztε = 0, we can consider the auxiliary processes vtε;k :=

Mtε;k Mtε;k ≡ ∈ [0, 1] | Ztε |2 Mtε;k + Ntε;k

for k = 1, . . . , p + 1; in particular v ε;1 ≡ 0 and v ε;p+1 ≡ 1. These processes further satisfy the pathwise relations d d ε;k N ε;k dt M ε;k − M ε;k dt N d ε;k v = , ε;k ε;k 2 dt (M + N )


153

where M ε;k , N ε;k and Z ε have rectifiable paths, and v ε;k ( 1 − v ε;k ) =

M ε;k N ε;k . (M ε;k + N ε;k )2

Due to (4.8) and (4.9), it follows for the pathwise derivatives with respect to t that d ε;k v ≥ 2 (λk−1 − λk − 2a) v ε;k ( 1 − v ε;k ) − 2 Gεt dt for any k = 1, . . . , p + 1; note that this estimate only contains information for k = 2, . . . , p, since it is trivially satisfied by v ε;1 ≡ 0 and v ε;p+1 ≡ 1. The full calculations can be found on p. 38f., where — in the current terminology — the pathwise derivatives are estimated for every fixed ε and ω. In other words, defining v ε := v ε;k and b := 2 (λk−1 − λk − 2a) which is strictly positive, b > 0, if we choose a > 0 sufficiently small, 2a
1, then one arrives at the random Riccati-type differential inequality d ε vt ≥ b vtε 1 − vtε − 2 Gεt . dt

(4.12)

For this situation, the following statement provides the stochastic analogue of 1.4.2: Lemma 4.1.5 (Random Riccati-type differential inequality (4.12)). Consider the situation of theorem 4.1.2 and the matrix function G( . ) := A( . ) − Aµ(x,ζ) as investigated in 4.1.1. Let (vtε )t≥0 be a family of stochastic processes on (Ω, F, P), ε > 0, taking their values in the interval [0, 1] such that the paths of (vtε )t≥0 are differentiable and satisfy the random Riccati-type differential inequality (4.12), i.e. d ε vt (ω) ≥ b vtε (ω) 1 − vtε (ω) − 2 G Xtε,x (ω) (t > 0 , ω ∈ Ω) , dt where b > 0 and Xtε,x is the underlying diffusion driving (1). Then it follows for any initial value v0 = v0ε and any η1 , η2 > 0 that 1 P L t ∈ [0, T (ε)] : vtε ∈ / [η1 , 1 − η2 ] −−− −→ 1 . ε→0 T (ε) Proof. For brevity, we define gtε (ω) := 2 G Xtε,x (ω) ≡ 2 Gεt (ω)

154


and β : [0, 1] → [0, b/4] ,

β(v) := b v (1 − v) .

Then one can rewrite the proposed inequality pathwise (for any ω ∈ Ω) as ε vT (ε) = v0 +

≥ v0 +

T (ε)

0 T (ε) 0

d ε v dt t

dt

β (vtε ) dt −

T (ε) 0

≥ v0 +

gtε dt β (vtε ) dt −

T (ε) 0

gtε dt

{ t∈[0,T (ε)] : vtε ∈[η1 ,1−η2 ] }

min β · L { t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ] } − ≥ v0 + = v0 +

[η1 ,1−η2 ]

min

[η1 ,1−η2 ]

β

· T (ε) · L

T (ε) 0

ε s ∈ [0, 1] : vs·T (ε) ∈ [η1 , 1 − η2 ]

Rearranging and dividing by T (ε) as well as by

min

gtε dt

−

T (ε) 0

gtε dt .

β > 0 now yields that

[η1 ,1−η2 ]

1 ε L {t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ]} = L s ∈ [0, 1] : vs·T ∈ [η , 1 − η ] 1 2 (ε) T (ε) .

−1 T (ε) ε vT (ε) − v0 1 ε min β gt dt + . ≤ T (ε) 0 T (ε) [η1 ,1−η2 ] 1 (v0 − However, v ε taking its values in [0, 1] is bounded and therefore T (ε) ε vT (ε) ) → 0 surely as ε → 0 (i.e. in the deterministic sense for any path); thus, as the time average T (ε) 1 gtε dt T (ε) 0

tends to zero in probability as ε → 0 by 2.5.10 and 4.1.1, the same holds true for the whole right hand side of the last inequality. Therefore, this estimate implies that also 0 ≤

1 P L { t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ] } −−− −→ 0 . ε→0 T (ε)

Hence the lemma is proven.

One might think that the proof of lemma 1.4.2 could be transferred to the stochastic situation above by means of a subsequence argument; this would entail to fix a sequence (tn )n∈N along which (gtε )t converges almost surely. However, the continuity of t → v(t) had been crucially used in 1.4.2; so by restricting this argument to the discrete time set {tn : n ∈ N}, it turns out that one cannot deduce a more refined convergence assertion on vtε here.


155

Now we consider the stochastic analogue of remark 1.4.6: Remark 4.1.6 (Kn = R2 ). Consider the two-dimensional real situation, Kn = R2 , for illustration. As before, p ≡ p(A(Kµ(x,ζ) )). If p = 1, i.e. if Re(Λ1 (Kµ(x,ζ) )) = Re(Λ2 (Kµ(x,ζ) )), then A(Kµ(x,ζ) ) either has two complex (conjugate) characteristic roots or it has one real eigenvalue with geometric multiplicity 1. This situation is covered by corollary 4.1.3. If p = 2, i.e. if Re(Λ1 (Kµ(x,ζ) )) > Re(Λ2 (Kµ(x,ζ) )), the situation is not included in corollary 4.1.3. Here, λ1 := Re Λ1 (Kµ(x,ζ) ) > Re Λ2 (Kµ(x,ζ) ) =: λ2 are two real eigenvalues of the matrix Aµ(x,ζ) =

λ1 0 0 λ2

(after the coordinate transformation C) and the canonical unit vector ei is the eigenvector corresponding to λi , i = 1, 2. Then vtε := vtε;2 :=

2 Mtε;2 ( ε;1 t ) ε 2 ≡ ε;2 2 = (cos αt ) , 2 | Ztε |2 ( ε;1 t ) + ( t )

where αεt again denotes the angle of Ztε in R2 . Figure 1.1 of remark 1.4.6 illustrates this situation for fixed ε and ω, if zt , vt and αt are replaced by Ztε , vtε and αεt , respectively. It follows from the Riccati differential inequality (4.12) for v ε and from lemma 4.1.5 that for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : vtε ∈ / [η1 , 1 − η2 ] −−− −→ 1 ; ε→0 T (ε) in other words, for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : αεt ∈ B(0, η1 ) or αεt ∈ B(π/2, η2 ) −−− −→ 1 , ε→0 T (ε) where (αε )t is again considered as process in [0, π), i.e. as the angle of the projective line Z ε ∈ P 1 ; more precisely, the angles 0 and π are identified as belonging to the same projective residue class as are π2 and 3π 2 . The preceding convergence in probability reflects the fact that the drift ¯ . , Kµ(x,ζ) ) of the angle process αε has two equilibria, namely at 0 and π . h( t 2 In section 4.3 we assume even stronger that A( . ) only takes its values in the diagonal matrices which means that ¯ h( . , x) has the same equilibria simultaneously for all x; in other words, the above coordinate transformation C simultaneously diagonalizes all values of A; vtε is then seen to obey a Verhulsttype differential equation (4.22) (Verhulst-type differential inequality (4.18)

156


in arbitrary dimension n) which is more precise than the previously obtained Riccati differential inequality (4.12). Note that by retransforming the coordinates via C −1 it generally follows for the angle process (αε )t from the preceding convergence in probability that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 , ε→0 T (ε) where α1 = α2 are the angles of the two eigendirections in [0, π) corresponding to the eigenvalues λ1 > λ2 of the (not necessarily diagonal) matrix Aµ(x,ζ) .

4.2 The local growth rate of the determinant In the situation of theorem 4.1.2 one can also obtain a local growth rate for the determinant of Z ε . This is due to the Jacobi equation and the consequence 2.5.10 of the metastability of X ε . Theorem 4.2.1. Consider a mapping A ∈ C(Rd , Kn×n ) and let X ε satisfy >l the assumptions 2.1.1. Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Furthermore, let Ztε (ω, x) be a matrix solution of (1) started in a fixed matrix Z0 ∈ GL(n, K). Then the exponential growth rate of det Ztε (ω, x) on the time scale T (ε) is trace A(Kµ(x,ζ) ) , where Kµ(x,ζ) again denotes the metastable state corresponding to x and ζ as given by definition 2.5.4 and the exposition preceding it. More precisely, 1 P log det ZTε (ε) ( . , x) −−− −→ trace A(Kµ(x,ζ) ) . ε→0 T (ε) Proof. From the Jacobi (Liouville) equation (1.2), it follows that t

det(Ztε (ω, x)) = (det Z0 ) · exp trace A Xuε,x (ω) du . 0

Therefore, log | det Z0 | 1 1 log det ZTε (ε) (ω, x) = + T (ε) T (ε) T (ε)

-

T (ε) 0

trace A Xuε,x (ω) du

.

and the claim follows from theorem 2.5.10, for the continuous function f := trace A. Note that the above theorem provides the sublimiting counterpart to the trace formula (1.13) for the sum of the Lyapunov exponents in the situation

4.3 Local Lyapunov exponents in the diagonal case

157

of Oseledets’ theorem, 1 log | det Ztε (ω, x) | t 1 t trace A Xuε,x ( . ) du = lim t→∞ t 0 trace A (x) pε (x) dx , =

Λε1 + · · · + Λεn = lim

t→∞

Rd

where the convergence holds almost surely.

4.3 Local Lyapunov exponents in the diagonal case In this section a special case is to be considered in detail: The conjecture (4.10) is to be verified in the situation that in the underlying differential system, dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + εσ (Xtε ) dWt ,

(1)

the continuous matrix function A ∈ C(Rd , Rn×n ) now solely takes its values in the diagonal, real matrices (n ∈ N); as before, d ∈ N denotes the dimension of the state space of X ε , ε ≥ 0 parametrizes the intensity of (Wt )t≥0 , a Brownian motion in Rd on a complete probability space (Ω, F , P), and X ε,x is a diffusion starting in x ∈ Rd , defined by the underlying SDE (2.1) such that the assumptions 2.1.1 hold. Again, Λj (x) := Λj (A(x))

(j = 1, . . . , n)

will denote the eigenvalues of the n×n matrix A(x), where x ∈ Rd . Since A(x) is assumed to be diagonal, the canonical j-th unit vector ej is an eigenvector corresponding to the (real) eigenvalue Λj (x) = (A(x))jj for each x. Theorem 4.3.1 (Local Lyapunov exponents). Let the assumptions 2.1.1 be satisfied and suppose that A ∈ C(Rd , Rn×n ) only takes its values in the diagonal matrices on Fx,ζ , where Fx,ζ denotes the subset of Rd as defined in corollary 2.5.7. >l Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x (see (2.20)). Furthermore, fix z ∈ Rn \ {0}. Then the stochastic process Ztε (ω, x, z), given by (1), has the following exponential growth rates on the time scale T (ε): 1 P log ZTε (ε) ( . , x, z) −−− −→ Λj Kµ(x,ζ) ε→0 T (ε)

158


for some j ∈ {1, . . . , n}, where Kµ(x,ζ) denotes the metastable state corresponding to x and ζ as given by definition 2.5.4 and the exposition preceding it. This stochastic limit Λj Kµ(x,ζ) will be called the local (j th ) Lyapunov exponent of Z ε with respect to x, ζ and z. Proof. As before, first rewrite the RDE for Z ε in (1) as dZtε = [ Aµ(x,ζ) + Gεt ] Ztε dt , where Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) and Gεt := Gεt (ω) := G(Xtε,x (ω)) := A(Xtε,x (ω)) − Aµ(x,ζ) . For brevity, again define Λj := Λj Kµ(x,ζ) , decreasingly indexed as Λ1 ≥ Λ2 ≥ · · · ≥ Λn , and the distinct numbers in this list are again denoted by λ1 > λ2 > · · · > λp . In the case that p = 1 the claim follows directly from corollary 4.1.3. Therefore, it is assumed that p ≥ 2 without loss of generality in the following analysis. Define ΩεΓ;F := ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)] for some parameter Γ > 0, for which it is known from corollary 2.5.7 that lim P ΩεΓ;F = 1 . ε→0

Then it follows from the diagonality assumption on A that the RDE under consideration now becomes d ε;i Zt (ω) = Λi Xtε,x (ω) Ztε;i (ω) dt i = Λi Ztε;i (ω) + Gεt (ω) Ztε (ω) ε;i = Λi Ztε;i (ω) + Gε;i t (ω) Zt (ω)

for all ω ∈ ΩεΓ;F and t ∈ (0, Γ T (ε)), where the superscript i ∈ {1, . . . , n} indicates the i-th coordinate as before and Gεt also takes its values in the


diagonal matrices,

159

⎛

⎞ Gε;1 0 t ··· ⎜ ⎟ .. Gεt = ⎝ ⎠ . . 0 · · · Gε;n t

(4.13)

Hence, due to the diagonality assumption, Z ε can be explicitly calculated as t

ε,x ε;i i Λi Xu (ω) du (i = 1, . . . , n) Zt (ω) = z exp 0

which by theorem 2.5.10 implies that 1 P log ZTε;i(ε) −−− Λi Kµ(x,ζ) ≡ Λi − → ε→0 T (ε)

(i = 1, . . . , n) .

However, this does not imply the claim yet and hence we aim at a more refined analysis. For this purpose, we again consider the auxiliary processes as in the proofs of the Hartman-Wintner-Perron theorem 1.4.3 and theorem 4.1.2. These processes now describe the norms of the projection of Z ε onto the subspaces spanned by the canonical unit vectors e1 , . . . , en of Rn which are the eigenvectors of Aµ(x,ζ) corresponding to Λ1 , . . . , Λn . Recalling the definitions, we consider more precisely the following stochastic processes ε;i ε;i i ε;i t (ω) ≡ | Zt (ω) | , starting in 0 = z

(i = 0, . . . , n),

and hereby

Lε;k t (ω) ≡

2 ε;i t (ω)

, k = 1, . . . , p ,

{i : Λi =λk }

Mtε;k (ω) ≡

Ntε;k (ω) ≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

0

2

ε;i t (ω)

{i : Λi >λk } ⎪ n ⎪ 2 ⎪ ⎪ ε;i ⎪ (ω) ⎪ t ⎩ i=1

⎧ ⎪ ⎨

⎪ ⎩

{i : Λi ≤λk }

0

≡ | Ztε (ω) |2

2 ε;i t (ω)

,k = 1 , , k = 2, . . . , p , ,k = p + 1 , , k = 1, . . . , p , ,k = p + 1 ,

ε;p such that |Ztε |2 = Mtε;k + Ntε;k = Lε;1 t + · · · + Lt , as has already been noted in (4.3). In contrast to the general calculations (4.5)–(4.9), the above form of the RDE for Z ε;i in the diagonal case then implies on ΩεΓ;F for t ∈ (0, Γ T (ε)) and i ∈ {1, . . . , n} that

160


d ε;i 2 d Ztε;i t = 2 Ztε;i dt dt

=

2 2 2Λi ε;i ε;i + 2 Gε;i . (4.14) t t t

Hence for k ∈ {1, . . . , p}, ε;k + , dL d( ε;i )2 ε;k ε;i 2 − 2λ − 2λ = L ( ) k k dt dt {i : Λi =λk } ε;i 2 ε;i (4.14) = 2 G {i : Λi =λk }

≤ 2 Lε;k Gε ,

(4.15)

and for k = 2, . . . , p + 1, d M ε;k dt

=

{i : Λi >λk }

(4.14)

=

{i : Λi >λk }

≥ 2λk−1

d ε;i 2 ( ) dt

2Λi ( ε;i )2 + 2 Gε;i ( ε;i )2

( ε;i )2 − 2 Gε

{i : Λi >λk }

ε;i 2

{i : Λi >λk }

= 2 ( λk−1 − G ) M ε

ε;k

,

(4.16)

as well as for k = 1, . . . , p + 1, d N ε;k dt

=

{i : Λi ≤λk }

(4.14)

=

d ε;i 2 ( ) dt

2Λi ( ε;i )2 + 2 Gε;i ( ε;i )2

{i : Λi ≤λk }

≤ 2λk

( ε;i )2 + 2 Gε

{i : Λi ≤λk }

= 2 ( λk + G ) N ε

ε;i 2

{i : Λi ≤λk } ε;k

.

(4.17)

Now we need to find for each solution Ztε ≡ 0 an index J ∈ {1, . . . , p} such that 1 P log | ZTε (ε) ( . , x, z)| −−− −→ λJ ε→0 T (ε) which had been formulated as conjecture (4.10).


161

As already mentioned on p.152, consider the auxiliary processes vtε;k :=

Mtε;k Mtε;k ≡ ∈ [0, 1] | Ztε |2 Mtε;k + Ntε;k

for k = 1, . . . , p + 1 for which in particular v ε;1 ≡ 0 and v ε;p+1 ≡ 1. Then one gets altogether on ΩεΓ;F for the pathwise derivatives with respect to t ∈ (0, Γ T (ε)) that d ε;k v dt

= (4.16),(4.17)

≥

= =

d d N ε;k ( dt M ε;k ) − M ε;k ( dt N ε;k ) (M ε;k + N ε;k )2 ε;k 1 N 2 ( λk−1 − Gε ) M ε;k ε;k ε;k 2 (M + N ) − M ε;k 2 ( λk + Gε ) N ε;k 1 2 M ε;k N ε;k (λk−1 − λk − 2 Gε ) ε;k ε;k 2 (M + N )

2 v ε;k ( 1 − v ε;k ) (λk−1 − λk − 2 Gε ) .

Hence, lemma 4.3.4 (see below) applies to the random Riccati-/Verhulsttype differential inequality d ε;k v ≥ 2 v ε;k ( 1 − v ε;k ) (λk−1 − λk − 2 Gε ) , dt

(4.18)

with v ε := v ε;k , b1 := 2 (λk−1 − λk ) > 0 and b2 := 4; note that the strict positivity of b1 is due to the assumption that p > 1. Therefore for all k ∈ {1, . . . , p + 1}, either v•ε;k (ω) ≡ 0 (∀ ε)

or

v•ε;k (ω) ≡ 1 (∀ ε)

or

P vTε;k(ε) −−− −→ 1. ε→0

Since v ε;1 ≡ 0 and v ε;p+1 ≡ 1, the following definition of the index J is thus well-defined and only depends on z ≡ Z0ε : J := max k = 1, . . . , p : v•ε;k ≡ 0 P = min k = 2, . . . , p + 1 : vTε;k(ε) −−− −→ 1 − 1 . ε→0

Zε

This index displays the direction eJ to which the projective line |Ztε | cont verges in probability; 0 and 1 are fixed points of the random Riccati-/ ε Z Verhulst-type differential inequality which means that |Ztε | is starting and t staying in the respective hyperplane of eigendirections of Aµ(x,ζ) ; any other P initial direction, i.e. any other v0ε;k ∈ (0, 1), implies that vTε;k(ε) −−− −→ 1; this

means that within the corresponding plane

Ztε |Ztε |

ε→0

is attracted towards the

162


eigendirection corresponding to the largest eigenvalue represented in this plane. Note the analogy to the definition of the J in theorem 1.4.3. This implies, as M ε;k = Lε;1 + · · · + Lε;k−1 and as k → vtε;k is increasing, that ⎧ ⎪ ε;k+1 ε;k ⎨ P = 0, k < J ; Lε;k − M M T (ε) T (ε) T (ε) ε;k+1 ε;k −− −−→ 1 , k = J ; = = vT (ε) − vT (ε) ε→0 ⎪ |ZTε (ε) |2 |ZTε (ε) |2 ⎩ P 0, k > J . −−−−→ ε→0

(4.19) Now (4.19) reduces the study of |ZTε (ε) | to

2

ε;J Lε;J T (ε) ; namely on {LT (ε) = 0},

2 Lε;J 1 T (ε) ε;J P ε log ε 2 −−− log LT (ε) − log |ZT (ε) | = −→ 0 ε→0 2 |ZT (ε) | and hence the above claim, (4.10), concerning the local Lyapunov exponent reduces to proving that 1 P log Lε;J − −−→ 2 λJ . T (ε) − ε→0 T (ε) But considering (4.15) for k = J , ε;J dL ε;J dt − 2λJ L

≤ 2 Lε;J Gε ,

ε;J yields on {Lε;J T (ε) = 0} after dividing by LT (ε) that for any fixed η > 0:

1 ε;J P log LT (ε) − 2λJ > η T (ε) 5 ;

6 T (ε) 1 d ε;J ε;J = P log Lu − 2λJ du > η log L0 + T (ε) du 0 5 6 ; T (ε) 1 ε;J d log Lε;J du > η − 2λ ≤ P log L0 + J u du T (ε) 0 6 ; 5 T (ε) (4.15) 1 ε;J ≤ P Gεu du > η log L0 + 2 T (ε) 0 ; T (ε) log(|z|2 ) 2 ε Gu du > η ≤ P + T (ε) T (ε) 0 ε→0

−−−→ 0,

due to proposition 4.1.1.


163

For proving the central lemma 4.3.4 on Riccati-type differential inequalities we will use the following comparison theorem from ODE theory; a proof of this theorem is e.g. given by Bainov and Simeonov [Bn-Si 92, p.57]. Proposition 4.3.2 (Comparison theorem). Let R(t, v) be a continuous, real-valued function on an open set O ⊂ R2 containing the point (t0 , w0 ), and suppose that the initial value problem r(t) ˙ = R(t, r(t)) , r(t0 ) = w0 has a maximal (minimal) solution r(t) with domain [t1 , t2 ] containing t0 . Let w(t) be a differentiable function on [t1 , t2 ] such that (t, w(t)) ∈ O for all t ∈ [t1 , t2 ] and (≥)

≤ R(t, w(t)) w(t) ˙

(t ∈ [t1 , t2 ]) ,

w(t0 ) = w0 . Then (≥)

w(t) ≤ r(t)

(t ∈ [t1 , t2 ]) .

In the proof of 4.3.1 the problem is reduced to considering a random differential inequality for the processes v ε;k which is of the type of the logistic Verhulst ODE. It is interesting to observe that this kind of differential equation is used to model randomly influenced population genetics (without “epistasis”); see the respective papers by Kliemann and Arnold: [Kl 80, p.112], [Kl 83b, p.491] and [Ar-Kl 83, p.29]. In these references the timedependent prefactor φ is chosen as (stationary) Ornstein-Uhlenbeck process. A pathwise solution can be given explicitly: Lemma 4.3.3 (Verhulst ODE). Consider the (logistic) Verhulst ODE w(t) ˙ = φ(t) w(t) 1 − w(t) (t ≥ t0 ) , w(t0 ) = w0 , where φ ∈ C([t0 , ∞), R) and w0 ∈ [0, 1]. Then if φ ≡ 0, any w ∈ [0, 1] is an equilibrium; if φ ≡ 0, w = 0 and w = 1 are the equilibria (fixed points). For all w0 ∈ (0, 1] the solution is given explicitly by

w(t) = 1+

1 w0

1

− 1 exp −

t t0

φ(u) du

(t ≥ t0 ) .

164


Proof. Computing the derivative of the specified function w(t) yields that t − φ(t) w10 − 1 exp − t0 φ(u) du w(t) ˙ =− 0 12 t 1 + w10 − 1 exp − t0 φ(u) du t 1 − 1 exp − φ(u)du w0 t0 φ(t) = t t 1 1 1 + w0 − 1 exp − t0 φ(u)du 1 + w0 − 1 exp − t0 φ(u)du = φ(t) w(t) 1 − w(t) and the initial condition is obviously fulfilled. The uniqueness of the solution follows from the local Lipschitz continuity of the drift R(t, w) := φ(t) w(1 − w). Figure 4.1 sketches the tendencies of the drift vector field R(φ, w) := φ w(1 − w) as considered in the Verhulst-type lemma 4.3.3; also see Kliemann [Kl 80, p.112]. w 6

r(0, 1)

?

?

6

?

6

6

r (0, 0)

-φ

Fig. 4.1 The tendencies of the drift vector field R(φ, w) := φ w(1 − w)

The following lemma is the crucial ingredient for the proof of theorem 4.3.1. It allows to sharpen the statement of lemma 4.1.5, the reason being that the diagonality assumption on A produced a random Riccati differential inequality which is also of Verhulst-type; see (4.18). In contrast, one had to deal with a less specific Riccati differential inequality in the previous situation of lemma 4.1.5. Also note that the diagonal structure of A is not going to be used explicitly in the subsequent lemma; it had come into play to deduce the random Riccati-/Verhulst-type differential inequality (4.18) which is now further treated: Lemma 4.3.4 (Random Riccati-/Verhulst-type differential inequality). Consider the situation of theorem 4.3.1 and the matrix function Gεt := Gεt (ω) := G(Xtε,x (ω)) := A(Xtε,x (ω)) − Aµ(x,ζ)


165

as above for Xtε,x being the underlying diffusion driving (1) under the assumptions 2.1.1. Furthermore, let (vtε )t≥0 be a family of stochastic processes on the fixed probability space (Ω, F, P), indexed by ε > 0, taking their values in the interval [0, 1] and starting in a common fixed initial value v0ε = v0 ∈ [0, 1]. Suppose that the paths (vtε )t>0 are differentiable for any ε > 0 and satisfy the random Riccati-/Verhulst-type differential inequality (RVDI) d ε vt (ω) ≥ vtε (ω) 1 − vtε (ω) b1 − b2 G Xtε (ω) , dt for all (t, ω) such that Xtε,x (ω) ∈ Fx,ζ ; here, b1 > 0 and b2 ≥ 0 are constants. Then v ≡ 0 and v ≡ 1 are fixed points of this differential equation and for any v0 ∈ (0, 1], the corresponding family satisfies P vTε (ε) −−− −→ 1 . ε→0

Proof. The statements for v0 ∈ {0, 1} follow immediately from the differential equation. Hence we choose v0 ∈ (0, 1); this open interval is invariant under the v ε motion (as will also follow from the arguments to come). If b2 = 0, then the statement is deterministic and follows from lemma 4.3.3 by the comparison assertion of proposition 4.3.2. Therefore, we will also suppose that b2 > 0 in the sequel. Fx,ζ denotes the subset of Rd as defined in corollary 2.5.7. As usual, let B(Kµ(x,ζ) , c) denote the open ball around Kµ(x,ζ) with radius c > 0. Since g(y) := G(y) is continuous on Rd and vanishes at Kµ(x,ζ) , also gt := g(Xtε,x ) ≡ Gεt depends continuously on Xtε,x . Therefore, theorem 2.5.6 implies that for any δ, Γ > 0 1 P L { t ∈ [0, Γ · T (ε)] : Gεt ≤ δ } −−− −→ Γ . ε→0 T (ε) In other words, it follows from theorem 2.5.6 that for any δ, Γ, η > 0, lim P ΩεδΓη = 1 , ε→0

where ΩεδΓη

:=

ω∈Ω :

1 L { t ∈ [0, Γ · T (ε)] T (ε) : b1 − b2 gt (ω) ∈ / [b1 − b2 δ, b1 ] } ≤ η

.

166


Since b1 , b2 > 0, we can choose δ > 0 sufficiently small such that b1 − b2 δ > 0 . Defining Cx,ζ := max g(y) ≡ max G(y) y∈Fx,ζ

y∈Fx,ζ

we get for any Γ > 0 that ΩεΓ;F ≡ ⊂

ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)]

ω ∈ Ω : b1 − b2 Gεt (ω) ∈ [b1 − b2 Cx,ζ , b] for all t ∈ [0, Γ · T (ε)]

,

where it is known from corollary 2.5.7 that lim P ΩεΓ;F = 1 . ε→0

First case: b1 − b2 Cx,ζ ≥ 0, i.e. g(y) ≤

b1 b2

for all y ∈ Fx,ζ : In this case the

right hand side v(1 − v){b1 − b2 g(y)} of the proposed RVDI is non-negative for all (v, y) ∈ (0, 1) × Fx,ζ and hence the drift Rω (t, v) := v(1 − v){b1 − b2 gt (ω)} is nonnegative for any ω ∈ ΩεΓ;F . Therefore, for any ω ∈ ΩεΓ;F the trajectory t → vtε (ω) starting in v0 ∈ (0, 1) is non-decreasing. Fixing δ < bb12 and ω ∈ Ωεδ1η implies (due to the continuity of gt ) that there are intervals [p, q] ⊂ [0, T (ε)] such that for all t ∈ [p, q] : b1 − b2 gt (ω) ≥ b1 − b2 δ (> 0) and the Lebesgue measure L of the union of such intervals is greater than (1 − η) T (ε). Such time intervals [p, q] will be called “good” in the remainder of this proof. Now we fix an ω ∈ Ωεδ1η ∩ Ωε1;F where δ < bb12 and such an interval [p, q] with the described property. Then we have ˜ ω (v) > 0 Rω (t, v) ≥ v (1 − v){b1 − b2 δ} =: R

(t ∈ [p, q], v ∈ (0, 1)) .

Hence, the comparison proposition 4.3.2 together with lemma 4.3.3 (for φ(t) := b1 − b2 δ > 0) implies that vtε (ω) ≥

1+

1 vpε (ω)

1

− 1 exp {−(b1 − b2 δ)(t − p)}

(t ∈ [p, q]) ,


167

since the function on the right hand side solves the comparison ODE w(t) ˙ = ˜ ω (w(t)) started in w0 := v ε (ω). Rewriting this estimate yields in other R p words (as has already been calculated in the proof of lemma 4.3.3) that 1 − 1 exp {−(b1 − b2 δ)(t − p)} ε vp (ω) 1 − vtε (ω) ≤ (t ∈ [p, q]) . 1 + vε1(ω) − 1 exp {−(b1 − b2 δ)(t − p)} p

Since the Lebesgue measure of the union of intervals [p, q] where this estimate holds true is larger than (1 − η) T (ε) and t → vtε (ω) is at least non-decreasing outside these intervals, it follows that for any κ > 0, 1 − 1 exp {−(b1 − b2 δ) (1 − η) T (ε)} v 0 < κ, 1 − vTε (ε) (ω) ≤ 1 + v10 − 1 exp {−(b1 − b2 δ) (1 − η) T (ε)} b1 b2

where the latter inequality is valid for ε small enough (for η < 1 and δ < fixed). Thus, we can conclude altogether that for any fixed κ > 0, ε→0 P 1 − vTε (ε) ≤ κ ≥ P Ωεδ1η ∩ Ωε1;F −−−→ 1 . Second case: b1 − b2 Cx,ζ < 0, i.e. Cx,ζ ≡ maxy∈Fx,ζ g(y) > case we fix ω ∈ Ωεδ1η ∩ Ωε1;F for some δ < small enough such that (b1 − b2 δ)

b1 b2 .

b1 b2 :

Also in this

Furthermore, we choose η

1−η > − (b1 − b2 Cx,ζ ) (>0) . η

(†)

On the “good” time intervals [p, q] (where the notation is as above) the same estimate vtε (ω) ≥

1+

1 vpε (ω)

1

− 1 exp {−(b1 − b2 δ)(t − p)}

(t ∈ [p, q])

follows. However, in this case the right hand side of the RVDI (i.e. the factor b1 − b2 gt (ω)) might become negative on certain intervals [h, k]; such time intervals will be called “bad” in the sequel. On “bad” time sets we have the following bound from below: ¯ ω (v) Rω (t, v) ≡ v(1 − v){ b1 − b2 gt (ω) } ≥ v(1 − v){b1 − b2 Cx,ζ } =: R AB C @ 0 .

This estimate describes the worst case possible. More precisely, due to the fact that ω ∈ Ωεδ1η and due to the definition of this set, the “bad” time set can at most constitute an interval of overall length η T (ε). Furthermore, the “bad” time set can be supposed to be connected and thus to be an interval indeed, since the two comparison ODEs, on the “good” and on the “bad” time sets, respectively, are autonomous. The latter fact implies that there is no other configuration of the “bad” time set which could result in a “worse” case, i.e. which would produce a value of v ε (ω) smaller than v¯.


169

In other words, the value v¯ can only be attained by the trajectory t → vtε (ω), if the “bad” time set is [h, k] = [0, η T (ε)] and we then have v¯ = vkε (ω). In this case, however, we get the “good” interval [p, q] = [k, T (ε)] = [η T (ε), T (ε)] by the definition of Ωεδ1η and therefore, estimating t → vtε (ω) on the latter interval from below, we can conclude altogether that 1 4.3.3 {−(b1 − b2 δ)(1 − η)T (ε)} ε v ¯ − 1 exp 1 − vT (ε) (ω) ≤ 1 1 + v¯ − 1 exp {−(b1 − b2 δ)(1 − η)T (ε)} 1 = exp {+(b1 − b2 δ)(1 − η)T (ε)} 1+ 1 −1 v¯ 1 Def.¯ v = exp {+(b . 1 − b2 δ)(1 − η)T (ε)} 1+ 1 − 1 exp {−(b1 − b2 Cx,ζ )ηT (ε)} v0 = 1+

v0 1−v0

1 exp T (ε) (b1 − b2 δ)(1 − η) + (b1 − b2 Cx,ζ )η

(†)

< κ,

where the latter inequality is true for ε small enough, since the parameters are related via (†). Therefore, we can conclude that given η and δ as in (†), then for any fixed κ > 0, ε→0 P 1 − vTε (ε) ≤ κ ≥ P Ωεδ1η ∩ Ωε1;F −−−→ 1 .

Hence the lemma is proved.

In the situation of lemma 4.3.4 the theorem 2.5.6 on the quasi-deterministic behavior of X ε can be “lifted” to a corresponding assertion concerning the proportion of the sojourn time of v ε . This subsequent corollary is not going to be used in the rest of the exposition. Corollary 4.3.5 (Proportion of the sojourn time of v ε• in 4.3.4). Consider the situation of 4.3.4. Then it follows, furthermore, for any v0 ∈ (0, 1] and Γ, β > 0 that P L t ∈ [0, Γ] : vtε T (ε) < 1 − β −−−−→ 0 . ε→0

Proof. The statement is trivial for v0 = 1, β ≥ 1 or b2 = 0; hence, fix v0 ∈ (0, 1), β ∈ (0, 1) and b2 > 0. We retain the same notation as before in 4.3.4. In particular, choose δ > 0 such that b1 − b2 δ > 0 and fix the set

170


ΩεδΓη ≡

ω∈Ω :

1 L { t ∈ [0, Γ · T (ε)] T (ε)

/ [b1 − b2 δ, b1 ] } ≤ η : b1 − b2 gt (ω) ∈ for which it is known from theorem 2.5.6 that for any δ, Γ, η > 0, limε→0 P(ΩεδΓη ) = 1. For Cx,ζ ≡ max g(y) ≡ max G(y) , y∈Fx,ζ

y∈Fx,ζ

one knows that for any Γ > 0, ΩεΓ;F ≡ ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)] ⊂ ω ∈ Ω : b1 − b2 Gεt (ω) ∈ [b1 − b2 Cx,ζ , b] for all t ∈ [0, Γ · T (ε)] , where limε→0 P(ΩεΓ;F ) = 1 due to corollary 2.5.7. First case: b1 − b2 Cx,ζ ≥ 0, i.e. g(y) ≤

b1 b2

for all y ∈ Fx,ζ : Fix ω ∈ ΩεδΓη ∩

ΩεΓ;F . In this case v•ε (ω) is greater than or equal to the trajectory vt which is exposed to the zero drift until time ηΓ T (ε) and is exposed to the drift v(1 − v){b1 − b2 δ} during the time interval [ηΓ T (ε), Γ T (ε)]. In other words, it is bounded below by ⎧ ⎨v0 , t ∈ [0, ηΓ T (ε)] , vt := 1 , t ∈ [ηΓ T (ε), Γ T (ε)] . ⎩ 1 1+

v0

−1 exp −(b1 −b2 δ) t−ηΓ T (ε)

Hence, it follows from solving for t that vtε (ω) ≥ vt ≥ 1 − β , (at least) if

t ≥ ηΓ T (ε) −

log

β v0 1−β 1−v0

b1 − b2 δ

.

Dividing the right hand side by T (ε) the resulting number converges to ηΓ as ε → 0. This proves that L t ∈ [0, Γ] : vtε T (ε) (ω) ≥ 1 − β β v0 log 1−β 1−v 1 0 ε→0 ≥ Γ − ηΓ + −−−→ Γ (1 − η) T (ε) b1 − b2 δ


171

for all ω ∈ ΩεδΓη ∩ΩεΓ;F . On the other hand, we also have that P(ΩεδΓη ∩ΩεΓ;F ) → 1 as ε → 0. Altogether, the desired convergence in probability follows in the first case. Second case: b1 − b2 Cx,ζ < 0, i.e. Cx,ζ ≡ maxy∈Fx,ζ g(y) > a fixed ω ∈

ΩεδΓη

∩

ΩεΓ;F .

In this case we estimate

v•ε

b1 b2 :

Again consider

by (compare with)

v˙ = v(1 − v){b1 − b2 δ} on a time interval of length (1 − η)Γ T (ε) and with v˙ = v(1 − v){b1 − b2 Cx,ζ } on a time interval of length ηΓ T (ε). Since both these comparison ODEs are autonomous, i.e. their drifts only depend on the state variable and not on time, we can again assume without restriction that the “bad” time set of length ηΓ T (ε), on which the second comparison ODE is valid, is connected and at the beginning of the whole time set under consideration. Therefore, vtε (ω) ≥ vt ≡

1+

1 v0

1

− 1 exp {−(b1 − b2 Cx,ζ )t}

for t ∈ [0, ηΓ T (ε)] and vtε (ω) ≥

1+

1 v0

1 − 1 exp {−(b1 − b2 Cx,ζ )ηΓ T (ε)} exp {−(b1 − b2 δ)(t − ηΓ T (ε))}

for t ∈ [ηΓ T (ε), Γ T (ε)]. Again we solve for t which amounts to requiring that !

1−β ≤

1+

1 v0

1 − 1 exp {−(b1 − b2 Cx,ζ )ηΓ T (ε)} exp {−(b1 − b2 δ)(t − ηΓ T (ε))}

or

exp − b1 − b2 Cx,ζ η Γ T (ε) exp − b1 − b2 δ t − η Γ T (ε)

−1 ! 1 1 −1 ≤ −1 1−β v0 β v0 = ; 1 − β 1 − v0 in other words, ! exp − b1 − b2 δ t − ηΓ T (ε) ≤ exp + b1 − b2 Cx,ζ ) η Γ T (ε)

β v0 1 − β 1 − v0

172


or, as −(b1 − b2 δ) < 0, equivalently !

t ≥ ηΓ T (ε) −

(b1 − b2 Cx,ζ )ηΓ T (ε) + log

β v0 1−β 1−v0

b1 − b2 δ

Again this implies for ω ∈ ΩεδΓη ∩ ΩεΓ;F that Tε L t ∈ [0, Γ] : vtε T (ε) (ω) ≥ 1 − β ≥ Γ − T (ε)

=: Tε .

v0 β log (b1 − b2 Cx,ζ ) 1 − β 1 − v0 ηΓ + = Γ(1 − η) + b1 − b2 δ (b1 − b2 δ)T (ε) which tends to

+

(b1 − b2 Cx,ζ ) η Γ (1 − η) + b1 − b2 δ

,

as ε → 0. Since η > 0 can be chosen arbitrarily small, the claim is proved as it again holds true that P(ΩεδΓη ∩ ΩεΓ;F ) → 1 as ε → 0, too. Remark 4.3.6 (n = 2). Consider the situation of theorem 4.3.1 and assume in addition that n = 2. Due to the above theorem, 1 P log ZTε (ε) ( . , x, z) −−− (j = 1, 2) . −→ Λj Kµ(x,ζ) ε→0 T (ε) This stochastic limit Λj Kµ(x,ζ) is called the local (j th ) Lyapunov exponent of Z ε with respect to x, ζ and z. The crucial step in the course of the proof had been to deduce the RVDI (4.18) d ε vt (ω) ≥ vtε (ω) 1 − vtε (ω) b1 − b2 G Xtε (ω) dt in the case that p > 1 which had then been dealt with in lemma 4.3.4. In the two-dimensional situation we would like to remark how this information can be sharpened; more precisely, one can deduce a random Riccati-/Verhulsttype differential equation (RVDE); see (4.22). For this purpose, we keep all notation as above; also see remark 4.1.6. Furthermore, we assume that p > 1, i.e. that p = 2 which entails that λ1 := Λ1 > Λ2 =: λ2 . In this case the auxiliary processes are given by Lε;k ≡ ( ε;k )2 , k ∈ {1, 2} ⎧ ⎪ ⎨0 ε;k M ≡ ( ε;1 )2 ⎪ ⎩42 ε;i 2 ε 2 i=1 ( ) ≡ | Z | ⎧42 ε;i 2 ε 2 ⎪ ⎨ i=1 ( ) ≡ | Z | ε;k N ≡ ( ε;2 )2 ⎪ ⎩ 0

,k = 1 ,k = 2 ,k = 3 ,k = 1 ,k = 2 ,k = 3


173

and describe the norms of the projection of Z ε onto the subspaces spanned by the canonical unit vectors e1 and e2 of R2 which are the eigenvectors of Aµ(x,ζ) corresponding to λ1 and λ2 ; also see remark 1.4.6. (4.14) and (4.15) now are expressed by ε;k dL d ε;k 2 ε;k ε;k 2 dt − 2λk L ≡ dt ( ) − 2λk ( ) 2 ε;k G = 2 ε;k ≡ 2 Lε;k Gε;k ; (4.16) for M ε := M ε;2 becomes d Mε dt

d ε;1 2 dt (4.14) = 2 ( ε;1 )2 λ1 + Gε;1 ≡ 2 M ε λ1 + Gε;1 ≡

(4.20)

and (4.17) for N ε := N ε;2 can be changed into d Nε dt

d ε;2 2 dt (4.14) = 2 ( ε;2 )2 λ2 + Gε;2 ≡ 2 N ε λ2 + Gε;2 . ≡

(4.21)

For Ztε ≡ 0, we again consider the auxiliary process vtε := vtε;2 ≡

2 Mtε;2 ( ε;1 t ) ≡ ≡ (cos αεt )2 ; ε ε;1 2 2 | Zt | ( t )2 + ( ε;2 ) t

see remark 4.1.6 and figure 1.1 in remark 1.4.6. The processes v ε;1 :=

Mtε;1 ≡ 0 | Ztε |2

and

v ε;3 :=

Mtε;3 ≡ 1 | Ztε |2

defined analogously do not contain any specific information. Then we get altogether from (4.20) and (4.21) that d d N ε ( dt M ε ) − M ε ( dt N ε) d ε v = dt (M ε + N ε )2 ε 1 N 2 M ε λ1 + Gε;1 = ε ε 2 (M + N ) − M ε 2 N ε λ2 + Gε;2

174


2 Mε Nε ( λ1 − λ2 ) + Gε;1 − Gε;2 (M ε + N ε )2 = 2 v ε (1 − v ε ) ( λ1 − λ2 ) + Gε;1 − Gε;2 , =

(4.22)

the RVDE which had been announced above. Of course, lemma 4.3.4 also applies here and therefore one gets the following definition P 1 , if vTε (ε) −−− −→ 1 , ε→0 J := 2 , else for the index function J such that the claim (4.10), 1 P log | ZTε (ε) | −−− −→ λJ , ε→0 T (ε) holds true. This means by lemma 4.3.4 that J = 2, if v0 = 0 and J = 1, if v0 ∈ (0, 1]. Again, this index J displays the direction eJ to which the Zε projective line |Ztε | converges in probability. t

Example 4.3.7. Consider for example the potential function from (1.34), U1 (x, y) :=

3 4 2 3 x − x3 − 3 x2 + c x y + y 4 , 2 3 2

where c ∈ R which had been discussed in 1.5.3 and 2.2.2: A sketch of the potential function U1 is given in figure 2.1 for c = 1. Note that in this example we use the less clumsy notation (x, y) for elements of R2 , instead of (x1 , x2 ) or (x1 , x2 ). Let X ε be the diffusion given by the SDE √ dXtε = −∇U (Xtε ) dt + ε σ dWt , where σ ∈ R2×2 is supposed to be a constant, invertible matrix, ε ≥ 0 and W is a Brownian motion in R2 . Considering the negative of the Hesse matrix of U1 ,

−c − 18x2 + 4 x + 6 , A1 (x, y) := −D∇U (x) ≡ − HU1 (x, y) = −c − 18 y 2 yields the linearized (“variational”) equation dZtε = A1 (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ dWt , √ since the diffusion coefficient ε σ does not depend on the state variable in this example; see examples 1.5.1 and 1.5.3.


175

Assume that c = 0. Also in this case, the function U1 is a two well potential as qualitatively described by figure 1; see example 2.6.2. However, the Hessian matrix now is everywhere diagonal and therefore remark 4.3.6 (or theorem 4.3.1 in other words) is applicable to this Jacobian system: The function U1 has a local minimum at K1 := (−0.847, 0), a global minimum at K2 := (1.180, 0) and a saddle point at K3 := (0, 0). Let Λ1 (Ki ) ≥ Λ2 (Ki ) denote the eigenvalues of A1 (Ki ) ≡ − HU1 (Ki ), i ∈ {1, 2}, which are non-positive due to the properties of the curvature at the minima. In this numerical example at hand (c = 0), Λ1 (K1 ) = Λ1 (K2 ) = 0 and −10.301 , i = 1 , Λ2 (Ki ) = −14.343 , i = 2 . Now we define the time scales T (ε) := eζ/ε which means that we connect the large parameters t = T (ε) and ε(t) := T −1 (t) =

1 ε

by

ζ log t

for a scaling parameter ζ > 0 in order to approach the sublimit distributions δK1 and δK2 . The object of our interest is thus the random variable ε(t) 1 1 log ZTε (ε) ( . , x, z) = log Zt ( . , x, z) T (ε) t and the assertion of 4.3.6 and 4.3.1 implies that it converges in probability to Λ0J (Ki ) as t → ∞ which is equivalent to ε → 0, where i ∈ {1, 2} depends on the ε,(x,y) and on the time scale parameter ζ of Xtε . initial position (x, y) = X0 More precisely, if (x, y) is in the K1 -regime and if also ζ < 2v, where v := 1 2 V (K1 , K3 ) denotes the height of the potential barrier (see example 2.6.2), then i = µ(x, ζ) = 1; otherwise, i = µ(x, ζ) = 2. Note that generally we also need to exclude the case that ζ = 2V , where V := 12 V (K2 , K3 ) is the height of the deep potential barrier (see example 2.6.2). The index J ∈ {1, 2} depends on the initial direction of Z ε . Then we call the possible different limits for the exponential growth rate of Z ε the local Lyapunov exponents of X ε . A final remark concerning the presumed diagonality of A1 ≡ − HU1 seems necessary: One might conjecture that the above procedure could also be

176


applied in case that c = 0 (which we assume from now on), if A1 only is transformed into diagonal form. However, the point here is that one has to simultaneously diagonalize A1 (x, y) for all (x, y) ∈ R2 . As is known from linear algebra, two diagonalizable matrices are simultaneously diagonalizable, if and only if they commute. Denoting the entries of A1 (x, y) by aij (x, y), this criterion implies the condition that

a11 (x, y) − c a11 (˜ x, y˜) − c x, y˜) − c a22 (x, y) − c a22 (˜

a11 (x, y) − c x, y˜) − c a11 (˜ = x, y˜) − c a22 (˜ − c a22 (x, y) for all (x, y), (˜ x, y˜) ∈ R2 . One then arrives at the requirement that a11 (x, y) − a22 (x, y) = a11 (˜ x, y˜) − a22 (˜ x, y˜) for all (x, y), (˜ x, y˜) ∈ R2 ; in other words, we get the constraint for the matrix function that the mapping R2 $ (x, y) → a11 (x, y) − a22 (x, y) needs to be constant. However, plugging in the specific values of A1 yields that a11 (x, y) − a22 (x, y) = − 18x2 + 18 y 2 + 4 x + 6 which clearly violates this condition. More precisely, A1 cannot be simultaneously diagonalized on any subset of R2 with non-void interior—such as Fx,ζ . Note that this fact can also be viewed from another perspective: As had been calculated in example 1.5.3, the angle αεt satisfies the pathwise ¯ ε , αε ), where the drift is given by differential equation α˙ εt = h(X t t ¯ (x, y), α = c ( sin2 α − cos2 α ) − a11 (x, y) − a22 (x, y) sin α cos α h ≡ c ( sin2 α − cos2 α ) + ( 18x2 − 18 y 2 − 4 x − 6 ) sin α cos α = − c cos 2α + ( 9x2 − 9 y 2 − 2 x − 3 ) sin 2α . If A1 was simultaneously diagonalizable, there would exist two linearly independent directions of eigenvectors, the angles of which are by definition fixed ¯ (x, y), . for all (x, y). However, the zero set of h ¯ (x, y), . is points of h characterized by the requirement tan 2α = −

2c a11 (x, y) − a22 (x, y)

≡ c ( 9x2 − 9 y 2 − 2 x − 3 )−1 ,

4.4 The two-dimensional, general case

177

where equality “=” is to be understood in R∪{±∞}. Hence, this presentation of the problem yields the same result as above; more precisely, we again get the constraint that (x, y) → a11 (x, y) − a22 (x, y) be constant. Thus, for (x, y) in a subset of R2 with non-void interior, such a fixed angle α cannot be found unless c = 0, since the right hand side of the condition last obtained is non-constant on any such subset of R2 . Therefore, the case c = 0 features a behavior of Z ε which is not captured by the situation c = 0 and the hitherto existing theorem 4.3.1 cannot be applied here. Thus a different argument is necessary which is the target of the following section.

4.4 Local Lyapunov exponents in the two-dimensional, general case In this section conditions are to be given which allow to deduce the claim (4.11), 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λ1 , ε→0 T (ε) in the general case p ≡ p(A(Kµ(x,ζ) )) = 2 in which the system matrix, however, also takes non-diagonal values, as opposed to the situation of the previous section. We investigate dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

where A ∈ C ∞ (Rd , R2×2 ) now denotes a 2 × 2-matrix function with smooth entries; in contrast to the hitherto investigations of this chapter, A is not supposed to be merely continuous, but differentiable the reason being that the results of the previous chapter are to be applied here. Furthermore, this section will exclusively deal with the two-dimensional situation, i.e. n = 2. All other data of (1) remain unchanged. As has already been remarked in 4.1.6, the above claim follows from corollary 4.1.3, if p = 1, i.e. if Re(Λ1 (Kµ(x,ζ) )) = Re(Λ2 (Kµ(x,ζ) )). Hence, we consider the case p = 2 here, i.e. the case that λ1 := Λ1 (Kµ(x,ζ) ) > Λ2 (Kµ(x,ζ) ) =: λ2

(4.23)

are two real eigenvalues of the matrix Aµ(x,ζ) ≡ A(Kµ(x,ζ) ); note that these values are indeed characteristic roots (not just real parts of such), since n = 2. Let the coordinate system be chosen such that

λ1 0 . Aµ(x,ζ) = 0 λ2

178


Then it is known from remark 4.1.6 that for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : αεt ∈ B(0, η1 ) or αεt ∈ B(π/2, η2 ) −−− −→ 1 , ε→0 T (ε) where αεt denotes the angle of Ztε in R2 as measured with respect to the coordinate system chosen. As had been seen, this behavior is due to the fact ¯ µ(x,ζ) , . ) of the angle process αε has two equilibria, namely that the drift h(K t π at 0 and 2 . For establishing the above claim (4.11), it needs to be shown that αεt has the preferential state 0 corresponding to the eigenvector e1 of Aµ(x,ζ) with respect to (the greater eigenvalue) λ1 ; cf. figure 1.1 of remark 1.4.6 which sketches this situation for fixed ε and ω, if zt , vt and αt are replaced by Ztε , vtε and αεt , respectively. The latter claim concerning the preferential state of the angle process is considered as an assertion on the sublimiting behavior of αεt and will be made precise in theorem 4.4.6. The effect on the local growth rate of Z ε , the local Lyapunov exponent, will be noted in theorem 4.4.8. In doing so, our reasoning leads via a sublimiting version of the FurstenbergKhasminskii formula: The coupled process (Xtε , αεt ) determines which values ¯ are sampled and which weight they get. Note that the final step in the of Q proof of the Hartman-Wintner-Perron theorem 1.4.3 cannot be transferred to the stochastic situation here to prove (4.11), since the corresponding estimate from above might become singular in the stochastic case when starting near π2 ; therefore, the bound from above thus obtained (see p.40) would be useless. Since all these issues pose the question of how the real noise driven angle process behaves in the sublimiting situation ε → 0, they can be considered more generally as problems concerning the qualitative theory of a certain nonlinear real noise driven system on time scales; these problems are tackled in the following subsection and will then be applied to αεt . Note that the terminology refers to the “qualitative theory of nonlinear stochastic [in the sense of ‘real noise driven’] systems” by Kliemann; see [Kl 80], [Kl 79] and [Kl 83b].

4.4.1 Qualitative theory of nonlinear real noise systems on time scales First, we investigate the situation in which the real noise system taking its values in a compact set is attracted solely to one switching curve (see definition 1.2.3). Such real noise driven systems have been prototypically investigated by many authors: Although the physical significance of this setting is “rather tenuous” (Horsthemke and Lefever [Hh-Lf 80, p.242]), it constitutes one of the sparse model classes of degenerate systems in which the probability distribution of the perturbed system can be calculated explicitly given that


179

the underlying noise is stationary; see Hongler [Hr 79], Horsthemke and Lefever [Hh-Lf 85, Sec.8.3], Arnold et al. [Ar-Hh-Lf 78], Kliemann [Kl 80, p.31,37f.,51ff.,82f.,132], Kliemann [Kl 83b, p.494ff.], Arnold and Kliemann [Ar-Kl 83, p.72] and Sancho and San Miguel [Sn-SM 80, p.362]. In these references the real noise case with one attracting switching curve is used to describe (noise induced) phase transitions (“bifurcations”); more precisely, if the stationary noise input is a stationary Ornstein-Uhlenbeck process, then the invariant probability distribution of the real noise system is unimodal or bimodal depending on the variance of the input noise. Another model class of real noise driven systems in which the stationary probability distribution can be calculated explicitly is discussed by Kitahara et al. [Kk-Hh-Lf 79] and [Kk-Hh-Lf-Ia 80]; also see Horsthemke and Lefever [Hh-Lf 85, p.259ff.]. These authors use stationary, dichotomous noise. However, all the results and references mentioned above which crucially rely on stationarity cannot be used here due to the special statistics of the driving noise process X ε , i.e. due to its sublimiting behavior on time scales. In order to clearly distinguish between the initial point of X ε and the space variable in Rd , the former will be denoted by x0 in the sequel while the latter is referred to by x from now on. Furthermore, since the results of this subsection shall be applied to the angle process αε of Z ε , the state space of the real noise driven system, called Y ε in the sequel, is taken to be one-dimensional. Theorem 4.4.1 (Qualitative sublimiting behavior in case of one attracting switching curve: Quasi-deterministic behavior of Y ε ). Consider the stochastic process (Ytε )t≥0 given as the solution of the RDE Y˙ tε = F (Xtε , Ytε ) ,

Y0ε = y0 ∈ R ,

where F : Rd × R → R is a differentiable function admitting pathwise unique solutions and where Xtε,x0 is given by the SDE (2.1) under the assumptions >l 2.1.1 with initial condition x0 . Suppose that x0 ∈ i=1 Di and fix T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x0 (see (2.20)). Suppose that F has exactly one attracting switching curve (see definition 1.2.3) on Fx0 ,ζ , where the latter set had been defined in corollary 2.5.7; more precisely, suppose that there is a continuous function Ysc : Fx0 ,ζ → R such that Ysc (x) is the only zero of F (x, . ) for any x ∈ Fx0 ,ζ and such that ∂F ∂y (x, y) < 0 at all points (x, y) ∈ graph(Ysc ). Then it follows for any initial value y0 ∈ R and all Γ, β > 0 that P L t ∈ [0, Γ] : YtεT (ε) − Ysc Kµ(x0 ,ζ) > β −−−−→ 0 ε→0

and

P YTε(ε) −−− −→ Ysc Kµ(x0 ,ζ) . ε→0

180


Proof. For the sake of brevity, we set Yµ(x0 ,ζ) := Ysc (Kµ(x0 ,ζ) ) . Due to the continuity of Ysc ( . ) we fix c > 0 such that Ysc (x) − Yµ(x ,ζ) < β 0 2

( ∀ x ∈ B(Kµ(x0 ,ζ) , c) ) .

Furthermore, define M ∗ := max Ysc (x) x∈Fx0 ,ζ

and

M∗ :=

min Ysc (x)

x∈Fx0 ,ζ

which are well defined real numbers, since Fx0 ,ζ is compact and Ysc is continuous. Also fix δ > 0 such that Yµ(x0 ,ζ) ± β ∈ (M∗ − δ, M ∗ + δ) . See figure 4.3 for a sketch of the situation at hand.

Fig. 4.3 The tendencies and the attracting switching curve of the drift vector field F (x, y) given the sublimiting statistics of X ε,x0

It is assumed in the sequel that y0 ∈ [M∗ − δ, M ∗ + δ]; this does not restrict generality, since there is no other switching curve of F on Fx0 ,ζ ; therefore, for any y0 ∈ [M∗ − δ, M ∗ + δ], there is some (deterministic) time t0 > 0 such that Ytε,y0 ∈ [M∗ − δ, M ∗ + δ] for all t ≥ t0 , where we already used the fact that only noise paths X•ε,x0 ⊂ Fx0 ,ζ will be considered due to corollary 2.5.7; see below. In the course of the proof we will make use of the following preliminary remark: The following quantities are well defined and not equal to zero: m1 := max F (x, y) : x ∈ B(Kµ(x0 ,ζ) , c) , y ∈ [Yµ(x0 ,ζ) + β, M ∗ + δ] < 0 and


m2 := min

181

F (x, y) : x ∈ B(Kµ(x0 ,ζ) , c) , y ∈ [M∗ − δ, Yµ(x0 ,ζ) − β]

> 0;

in particular, for a path Xt := Xtε,x0 (ω) which takes its values only in B(Kµ(x0 ,ζ) , c) we have for y0 ∈ [M∗ − δ, M ∗ + δ] , ≤ m1 < 0 , if y0 ≥ Yµ(x0 ,ζ) + β ε ε Y˙ t = F (Xt , Yt ) ≥ m2 > 0 , if y0 ≤ Yµ(x0 ,ζ) − β / B(Yµ(x0 ,ζ) , β) due to comparison 4.3.2. By virtue of these as long as Ytε ∈ bounds m1 and m2 for the function F on B(Kµ(x0 ,ζ) , c) × B(Yµ(x0 ,ζ) , β), comparison of Ytε,y0 with these two constant velocity ODEs yields that Ytε,y0 being subject to a driving trajectory X• ≡ X•ε,x0 (ω) ⊂ B(Kµ(x0 ,ζ) , c) hits B(Yµ(x0 ,ζ) , β) in finite time; due to the uniformity of the bounds m1 and m2 also the maximum of these hitting times with respect to the initial condition y0 ∈ [M∗ −δ, Yµ(x0 ,ζ) −β]∪[Yµ(x0 ,ζ) +β, M ∗ +δ] is finite; calling this maximum t1 further implies that Ytε ∈ B(Yµ(x0 ,ζ) , β)

(t ≥ t1 ) .

Let y˜ ∈ [M∗ − δ, M ∗ + δ] \ B(Yµ(x0 ,ζ) , β) denote an initial value for which this maximal time t1 is needed in order to approach B(Yµ(x0 ,ζ) , β). After this preliminary remark we now use the “preferences” of X ε as described by the Freidlin-Wentzell theory: Theorem 2.5.6 implies that for any Γ, η > 0, lim P ΩεcΓη = 1 , ε→0

where

ΩεcΓη

:=

ω∈Ω : Γ −

1 L { t ∈ [0, Γ · T (ε)] T (ε) :

Xtε,x0

∈ B(Kµ(x0 ,ζ) , c) } ≤ η

.

Fixing ω ∈ ΩεcΓη implies that there are intervals [p, q] ⊂ [0, Γ T (ε)] such that for all t ∈ [p, q] : Xtε,x0 (ω) ∈ B(Kµ(x0 ,ζ) , c) and the Lebesgue measure L of the union of such intervals is greater than or equal to (Γ − η) T (ε). Such intervals [p, q] will be called “good” in the remainder of the proof; note that this terminology is consistent with the one used in the proof of lemma 4.3.4. Furthermore, corollary 2.5.7 implies that for any Γ > 0, lim P ΩεΓ;F = 1 , ε→0

where we again use the notation ΩεΓ;F := { ω ∈ Ω : Xtε,x0 (ω) ∈ Fx0 ,ζ for all t ∈ [0, Γ · T (ε)] } .

182


Now fix η > 0 and ω ∈ ΩεcΓη ∩ ΩεΓ;F . In order to calculate L t ∈ [0, Γ] : YtεT (ε) (ω) − Yµ(x0 ,ζ) > β , we compare Y•ε with the “constant velocity” system Y˜ defined in the following: Let Y˜0 = y0 , if 0 is in a “good” time interval and let Y˜0 = y˜ otherwise, where the default point y˜ ∈ [M∗ − δ, M ∗ + δ] \ B(Yµ(x0 ,ζ) , β) had been fixed in the preliminary remark. On a “good” time interval let m1 if y˜ > Yµ(x0 ,ζ) + β , d ˜ Yt = dt m2 if y˜ < Yµ(x0 ,ζ) − β ; on all other (“bad”) times let Y˜t = y˜. These two comparison cases for “good” and “bad” time sets capture the possible “worst case scenarios”. Since both these comparison systems are autonomous, i.e. their velocities m1 , m2 and 0 do not depend on the time variable t themselves, we can again assume without restriction that the “bad” time set is of length η T (ε), connected (an interval) and at the beginning of the whole time set under consideration. Altogether it follows therefore that L t ∈ [0, Γ] : YtεT (ε) (ω) − Yµ(x0 ,ζ) > β ≤ η + L t ∈ [η, Γ] : Y˜ty˜T (ε) − Yµ(x0 ,ζ) > β 1 L t ∈ [ηT (ε), ΓT (ε)] : Y˜ty˜ − Yµ(x0 ,ζ) > β ≡η+ T (ε) t1 , =η+ T (ε) where the preliminary remark has been used in the final step. Since ω ∈ ΩεcΓη ∩ ΩεΓ;F and the probability of this set tends to 1 as ε → 0, this proves in other words that P L t ∈ [0, Γ] : YtεT (ε) − Yµ(x0 ,ζ) > β −−−−→ 0 . ε→0

The second claim concerning the stochastic convergence, P YTε(ε) −−− −→ Yµ(x0 ,ζ) , ε→0

follows in the same manner: As before fix an arbitrary β > 0, the corresponding parameter c ≡ c(β) > 0 and set Γ := 1; again let Y˜t (ω) denote the system which equals y˜ on the “bad” time set on which Xtε (ω) ∈ / B(Kµ(x0 ,ζ) , c) and which follows the ODE


d ˜ Y = m1 dt

183

or

d ˜ Y = m2 dt

depending on the position of Y˜ = y˜ on the “good” time set on which Xtε (ω) ∈ B(Kµ(x0 ,ζ) , c). Due to the preliminary remark, it follows that 1B(Yµ(x0 ,ζ) ,β) (Ytε ) ≥ 1B(Yµ(x0 ,ζ) ,β) (Y˜t ) and therefore P YTε(ε) − Yµ(x0 ,ζ) < β ≥ P Y˜T (ε) − Yµ(x0 ,ζ) < β ≡ P Xtε ∈ B(Kµ(x0 ,ζ) , c) for all t ∈ [T (ε) − t1 , T (ε)] ; however, the latter must tend to 1 as ε → 0, due to the sublimiting distribution characteristics of theorem 2.5.5 as X ε has continuous trajectories and t1 is a fixed number (not depending on ε). In other words, one can apply theorem 2.5.5 with the time scale T(ε) := T (ε) − t1 and observe that the deterministic lag of YTε(ε) is at most t1 . The following sample drift is taken from Kliemann [Kl 80, p.31,37f.,51ff., 82f.,132] : Example 4.4.2. Consider the system 1 − Ytε + Ytε (1 − Ytε ) Xtε =: F (Xtε , Ytε ) , Y˙ tε = 2

Y0ε = y0 ,

in which Xtε also takes its values in R, i.e where d = 1. The drift (right hand side) 1 − y + y (1 − y) x F (x, y) ≡ 2 of this RDE vanishes at the points (x, y) ∈ R2 for which x = (y − 12 )[ y (1 − y) ]−1 ; solving for y yields that (0, 12 ) is a zero, and for x = 0: 1 1 − ± y = y(x) = 2 2x

√

1 + x2 2x

.

Since F (x, 1) = − 12 < 0 for all x ∈ R, there is no trajectory of Y ε which can traverse the set {(x, y) : y = 1}, if started in y0 ≤ 1. In the controltheoretic terminology by Kliemann [Kl 80, Ch.6] and [Kl 79, p.461f.], this fact is summarized by calling the hypersurface {(x, y) : y = 1} strongly nonpenetrable from below or a strong no return plane from above; analogously, since F (x, 0) = 12 > 0, the set {(x, y) : y = 0} is strongly non-penetrable from above or a strong no return plane from below. In other words, the interval [0, 1] is invariant under any Ytε (irrespective of the time horizon) which is expressed by calling

184


y ∈R

1 x∈R

−2

2 −1

Fig. 4.4 The tendencies and switching curves of the drift F (x, y) =

1 2

−y +y(1−y)x

)0, 1( := R × [0, 1] a strong no return set. Hence, we restrict the state space of Y ε to [0, 1]. Note that this restriction is also reasonable from an applications’ point of view: The above differential equation can be interpreted as the kinetic equation of a chemical reaction in which “Y ” represents the reactant’s portion of the overall particle number; see Arnold et al. [Ar-Hh-Lf 78] for details who consider the system F (x, y) := 12 − y − y (1 − y) x which is basically a reflection of F at the ordinate. In the following, the switching curve in (0, 1) which is uniquely determined will be denoted by Ysc . It is this curve Ysc which has already been plotted in figure 4.3. If y0 ∈ [0, 1], theorem 4.4.1 is applicable to the system Y ε,y0 . Although 4.4.1 requires a global condition on F , namely that there is no other switching curve, the theorem is valid here, since Y ε,y0 only takes its values in [0, 1]. Either one redefines F on R2 \ R × [0, 1] such that the conditions of 4.4.1 are formally satisfied, where the behavior of Y ε,y0 and hence the result does not depend on the specific choice of this continuation of F ; or one recalls the proof of 4.4.1 which requires only a compact subset of R itself. The situation of theorem 4.4.1 and the previous example can be heuristically characterized by stating that the real noise driven system Y ε is “trapped in the neighborhood” of an attracting switching curve in each case. Besides, there is no other switching curve present near this neighborhood which might “interfere” in the sense that Y ε could be transferred to another domain . The following definition is motivated by this loose description:


185

Definition 4.4.3 (Separated switching surfaces). Let I ⊂ R denote some (open or closed) interval and consider a continuous function F : Rd ×I → R. Such a function F is thought of as a vector field of an RDE Y˙ t = F (Xt , Yt ) ,

Y0 = y0 ,

for a stochastic process (Yt )t≥0 , where X is a driving process (e.g. given by (2.1) under the assumptions 2.1.1). Suppose that there is a set M ⊂ Rd such that all the zero sets of F (x, . ), x ∈ M, can be precisely written as two 1 2 : M → R and Ysc : M → R , where continuous functions Ysc 1 2 Ysc (x) ≥ Ysc (x)

(x ∈ M)

1 2 and y → F (x, y) changes its sign at each Ysc (x) and Ysc (x). Then the function F (or the above RDE, respectively) is said to have separated switching surfaces on M (or separated switching curves on M, if d = 1), 1 2 ≥ supM Ysc . If the latter inequality is strict, if inf M Ysc 1 2 inf Ysc > sup Ysc , M

M

the system is said to have strictly separated switching surfaces (curves) on M. If a continuous function F : Rd × R → R has a restriction F : Rd × I → R which exhibits (strictly) separated switching surfaces on M in the above sense for some interval I ⊂ R and if in addition F (x, . ) is p-periodic for all x ∈ M, where p := L(I) denotes the length of I, then F is also said to have (strictly) separated switching surfaces on M. y 1 Ysc (x)

2 (x) Ysc

x ∈ Rd

Fig. 4.5 Strictly separated switching surfaces on Rd

Figure 4.5 illustrates strictly separated switching curves. Also see Arnold and Kliemann [Ar-Kl 83, p.38]; further note that the conditions which

186


Wihstutz [Wh 75, p.111ff.] assumes for the harmonic oscillator imply strict separatedness of the switching curves. For applying corollary 3.2.1 in the following theorems 4.4.6 and 4.4.7 to the real noise angle system αε we need to assure that the hypoellipticity assumption 3.1.3(c) can be met. Remark 3.1.4 yields that this can be achieved by means of a rank condition on the velocity vector field. Since our considerations are local in the sense of chapter 2, i.e. the relevant diffusive behavior takes place on M ∈ {Fx0 ,ζ , Hx0 ,ζ }, we are led to work with a localized version of 3.1.3(c). This is the contents of the new concept of “strong hypoellipticity on a set M” which we are now going to introduce. In order to compare the latter with a localized version of the hypoellipticity (1.15) for n = 2, we will also establish “hypoellipticity on a set M” . However, note that only “strong hypoellipticity on M” will come into play. Definition 4.4.4 (Local hypoellipticity). Let F : Rd × R → R be a differentiable function which is again considered as the velocity of (Yt )t≥0 given by the RDE Y˙ t = F (Xt , Yt ) ,

Y0 = y0 ,

where X is a driving process (e.g. given by (2.1) under the assumptions 2.1.1). Furthermore, let M ⊂ Rd be a fixed set. (a) The drift function F will be called hypoelliptic on M, if for any y ∈ R there is some x ∈ M such that F (x, y) = 0 . (b) The drift function F will be called strongly hypoelliptic on M, if for all (x, y) ∈ M × R ,

∂F (x, y) = 1. rank ∂xj 1≤j≤d

4.4.2 The local Lyapunov exponent Definition 4.4.5 (Value function of the angle process). The angle αε of the stochastic process Z ε is given by the SDE ¯ X ε , αε dt dαεt = h t t √ dXtε = b Xtε dt + ε σ Xtε dWt in the case that the state space of Z ε is two-dimensional, n = 2. Here, the ¯ which formally is a real-valued function on Rd ×[0, 2π) can be drift function h ¯ : Rd × R → R which is π-periodic with respect equivalently read as function h to the second argument. Therefore, αε can also be considered as process in R. Hence, it is in accordance with notation 3.1.2 (s := 0) to define α0 as the


187

solution of the ODE ¯ dα0 (t) = h(u(t), α0 (t)) dt , where u is a “control” function u ∈ C 1 (R+ , Rd ) and where the initial condition is (u(0), α0 (0)) = (x, α) ∈ Rd × R ; furthermore, /D ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ denotes the corresponding exit time of α0 from some set D ⊂ R and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. As in assumption 3.1.3(e), UD (0; T ; x, α) needs to be nonempty. Then the value function of αε on Fx0 ,ζ is defined as 0 ID,F (0; T ; x, α) x0 ,ζ

:=

ϑD (0;x,α;u)

inf

u∈UD (0;T ;x,α)∩{u• ∈Fx0 ,ζ }

K u(t), u(t) ˙ dt ,

0

where x ∈ Fx0 ,ζ and {u• ∈ Fx0 ,ζ } denotes the set of control functions which only take their values in the set Fx0 ,ζ which had been defined in corollary 2.5.7 and where 2 1 K(x, v) := a(x)−1/2 [v − b(x)] . 2 In the same manner, the value function of αε on Hx0 ,ζ is defined as 0 ID,H (0; T ; x, α) x0 ,ζ

:=

inf

u∈UD (0;T ;x,α)∩{u• ∈Hx0 ,ζ }

ϑD (0;x,α;u)

K u(t), u(t) ˙ dt ,

0

where x ∈ Hx0 ,ζ and {u• ∈ Hx0 ,ζ } now denotes the set of control functions which only take their values in the set Hx0 ,ζ which had been defined in corollary 2.5.8 under the assumptions imposed there. Note, in particular, that the value functions of αε on Fx0 ,ζ and Hx0 ,ζ only ¯ b, σ on the respective set. depend on the behavior of h, The following theorem contains our fundamental result concerning the asymptotic behavior of the angle process αε on time scales. As will become clear from the proof, a further condition is needed to hold involving the value function as introduced in definition 4.4.5. This is requirement (4.24); see below. Intuitively it says that the cost of controlling the angle system away from the unstable switching curve A2 needs to be smaller than the scale parameter ζ. This constraint is plausible, since on the one hand the cost

188


of controlling the angle system is calculated by means of the L2 -deviation ϑ K(u(t), u(t))dt ˙ of a control function u from X 0 in definition 4.4.5 . On the 0 other hand, ζ is a cost parameter for the diffusion X ε itself; see corollary 2.5.7; it provides a bound on the values of the quasipotential which X ε can overcome on the time scale T (ε) ! eζ/ε and the quasipotential is also calculated via the L2 -deviation which had been called the “rate function” I[0,T ],x0 (u) ≡ T K(u(t), u(t))dt ˙ ; see theorem 2.3.4 . 0 As will be proven afterwards in remark 4.4.12 this condition (4.24) is satisfied, if the SDE for X ε is of gradient type (2.2). Theorem 4.4.6 (Proportion of the occupation time of the angle process αε near A1 (K µ(x0 ,ζ) ) = 0 becomes full on Fx0 ,ζ ). Consider the stochastic process Z ε solving the SDE (1), where X ε is given by the SDE (2.1) under the assumptions 2.1.1. Furthermore, suppose that n = 2 and that A ∈ C ∞ (Rd , R2×2 ). Let αε be the angle process of Z ε which is given as the solution of the random differential equation (1.6), ¯ ε , αε ) , α˙ εt = h(X t t

αε0 = α0 .

>l Fix an initial value x0 ∈ i=1 Di of X ε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x0 ; see (2.20). Let Fx0 ,ζ be the set defined in corollary 2.5.7 according to the underlying diffusion X ε,x0 and the time scale T (ε) ! eζ/ε . ¯ has strictly separated switching surfaces Suppose that the drift function h A1 and A2 on Fx0 ,ζ ; see definitions 4.4.3 and 1.2.4; according to the latter definition, A1 is supposed to be attracting whereas A2 is repelling. ¯ is strongly hypoelliptic on Fx0 ,ζ ; see definition Furthermore, suppose that h 4.4.4. If α0 ∈ A2 (Fx0 ,ζ ), then it is assumed in addition that there exists T¯ > 0 such that 0 ¯ ; x, α) : (x, α) ∈ Fx0 ,ζ × A2 (Fx0 ,ζ ) , (4.24) (0; T ζ > max ID,F x0 ,ζ 0 (0; T ; x, α) is where D ⊃ A2 (Fx0 ,ζ ) denotes some open set and where ID,F x0 ,ζ ε the value function for α as defined in 4.4.5. Then it follows for any η > 0 that

P 1 L t ∈ [0, T (ε)] : αεt ∈ B A1 (Kµ(x0 ,ζ) ), η −− −−→ 1 . ε→0 T (ε) Proof. If α0 ∈ / A2 (Fx0 ,ζ ), the statement follows from theorem 4.4.1 in the same way as example 4.4.2. Hence, we consider the case that α0 ∈ A2 (Fx0 ,ζ ) for which the additional assumption (4.24) is valid. Note that the separatedness assumption on the switching curves necessarily implies that λ1 > λ2 , for otherwise λ1 = λ2 ¯ µ(x ,ζ) , α) = 0 for all α. Thus the separatedness of the would imply that h(K 0


189

switching curves automatically enforces the condition p = 2 which characterizes the case yet to be solved; see the discussion at the beginning of this section. Define α1 := A1 (Kµ(x0 ,ζ) )

α2 := A2 (Kµ(x0 ,ζ) )

and

for brevity. Note that no confusion with the values of the stochastic process (αεt )t≥0 is possible, since the latter object is always written with its superscript ε; furthermore, this notation is consistent with definition 1.2.4 as αi ≡ Ai (Kµ(x0 ,ζ) ) = arctan ξi (A(Kµ(x0 ,ζ) )) for i ∈ {1, 2}, if the numeration is chosen such that α1 is attracting and α2 is repelling. However, this decomposition by means of ξ-variables will not be used in the sequel; for a discussion see remark 4.4.9 nonetheless. E.g. if A is diagonal at the metastable state,

λ1 0 , Aµ(x0 ,ζ) ≡ A(Kµ(x0 ,ζ) ) = 0 λ2 then α1 ≡ A1 (Kµ(x0 ,ζ) ) = 0 and α2 ≡ A2 (Kµ(x0 ,ζ) ) = π2 ; considering Aµ(x0 ,ζ) in diagonal form here amounts to applying a constant transformation of the system A which leaves all its properties unchanged; note that the separatedness of the switching curves is not necessarily preserved under such a coordinate transformation; however, the diagonality of Aµ(x0 ,ζ) is not crucial but only used for illustrative purposes, since it has been used in the previous proofs of this chapter. In the general case the eigendirections of Aµ(x0 ,ζ) are represented by angles α1 = α2 in [0, π) corresponding to λ1 > λ2 . From 4.1.6 it is already known that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 ; ε→0 T (ε) hence, it is left to be shown that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α2 , η) −−− −→ 0 ε→0 T (ε) for arbitrary η > 0. In the sequel we will prove the following slightly stronger claim: P 1 L t ∈ [0, T (ε)] : αεt ∈ A2 (Fx0 ,ζ ) −−− −→ 0 . ε→0 T (ε) ¯

∂h , the strict separatedness of the Due to the continuity of A1 , A2 and ∂x j switching curves and the strong hypoellipticity necessarily also hold true on an open enlargement of Fx0 ,ζ , i.e. on some open set

D x ,ζ ⊃ Fx ,ζ , F 0 0

190


D x ,ζ \ Fx ,ζ can be arbitrarily small. For proving the where the difference F 0 0 D x0 ,ζ ) , i.e. an open preceding claim we fix an open neighborhood D of A2 (F set D ⊂ R such that D x ,ζ , D ⊃ A2 F 0 for which we also have

Dx ,ζ > 0 , dist D, A1 F 0

where dist( . , . ) denotes the distance function. The former condition concerning the choice of D can be trivially satisfied, whereas the latter requirement can be met, since the separation of the switching surfaces A1 and A2 is D x0 ,ζ as well). After taking the interassumed to be strict on Fx0 ,ζ (and on F section, we can assume that this set has the prescribed property (4.24) and it will be denoted by D further on for the investigations to come. The rest of the proof is organized in five steps: In the first step the coefficients of the angle SDE are modified such that they satisfy the assumptions 3.1.3. This allows to apply 3.2.1 in the second step. Afterwards this result is used to estimate the exit time of the coupled process from the set Fx0 ,ζ × A2 (Fx0 ,ζ ) “around” the repelling switching curve by means of the strong Markov property. In the fourth step this is used to derive that, due to the control-theoretic structure of the posed situation, the system αε cannot reenter D after leaving this set on the event ΩεΓ;F on which the paths X•ε take their values exclusively in Fx0 ,ζ ; hence, the exit time of αε from A2 (Fx0 ,ζ ) as calculated before can be used to estimate the length of the time interval until αε leaves A2 (Fx0 ,ζ ); this step yields that the decisive probability can be estimated from above by a function of ε of exponential type. The latter bound is then verified to converge (in the deterministic sense) to zero in the final step. ¯ of the angle SDE be modified on 1) Let the coefficient functions b, σ and h D x ,ζ and on (F D x ,ζ ) × R, respectively, as smooth functions ˆb, σ F ˆ and F such 0 0 that the set of assumptions 3.1.3 on the SDE (3.1) is satisfied; more precisely, the functions ˆb, σ ˆ , F are chosen to satisfy 3.1.3 and ˆb, σ ˆ shall coincide with ¯ on F D x0 ,ζ × R : D b, σ on Fx0 ,ζ as well as F coincides with h In particular, the (global) hypoellipticity assumption 3.1.3(c) on F can be ¯ is strongly hypoelliptic on F Dx0 ,ζ ; see definition 4.4.4 and remark met, since h D ¯ can 3.1.4. Furthermore, since dist D, A1 Fx0 ,ζ > 0 , the continuation F of h be chosen such that F has strictly separated switching surfaces (also called A1 and A2 ) on Rd which do not intersect ∂D. Especially this implies that 3.1.3(d) holds true. Finally, 3.1.3(e) is met since A2 is repelling (definition 1.2.4) and non-constant due to the strong hypoellipticity assumption; hence, there are trajectories which can leave D within a prescribed time horizon; also see below.


191

ˆ ε, α In order to avoid any confusion, the new system is denoted by (X ˆ ε ), i.e. ε ε ˆ ,α dˆ αεt = F X t ˆ t dt ε √ ε ε ˆ t = ˆb X ˆ t dt + ε σ ˆ t dWt , dX ˆ X satisfying the fixed initial condition ε ε ε ε ˆ ,α X 0 ˆ 0 = (X0 , α0 ) ≡ (x0 , α0 ) ; for abbreviation we also denote this system by ε,(x0 ,α0 )

ˆ α (X, ˆ )t

0 ˆ tε,x0 , α := (X ˆ ε,α ) . t

As in 3.1.1 (s := 0), let 0 T ε ≡ T ε (0, x0 , α0 ) ≡ TDε (0, x0 , α0 ) := inf { t ≥ 0 : α ˆ ε,α ∈ / D} t 0 denote the first exit time of (ˆ αε,α )t≥s from the set D which had been fixed t previously. According to 3.1.2, let u ∈ C 1 (R+ , Rd ) be a control function and let α0 be the path defined by dα0 (t) = F (u(t), α0 (t)) dt

under the initial condition (u(0), α0 (0)) = (x, α) ∈ Rd × R (s := 0). As before, ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ /D denotes the corresponding exit time of α0 from D and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. Note that UD (0; T ; x, α) is nonempty for all ¯ (x, α) ∈ Rd × R, since A2 is repelling (definition 1.2.4) and non-constant (h is strongly hypoelliptic) as has been already mentioned above in connection with assumption 3.1.3(e). More formally, here we use the control-theoretic fact that Rd × D is a weak control set for the above control system α˙ 0 (t) = F (u(t), α0 (t)); see Kliemann [Kl 80, p.31f.].

192


Using this notation, the value function according to 3.1.2 is defined as 0 (0; T ; x, α) := I 0 ≡ ID

where

ϑD (0;x,α;u)

inf u∈UD (0;T ;x,α)

K u(t), u(t) ˙ dt ,

0

0 1 2 1 ˆ(x)−1/2 v − ˆb(x) . a 2 we have

K(x, v) := Note that for x ∈ Fx0 ,ζ

0 ID (0; T ; x, α) ≤

ϑD (0;x,α;u)

inf

u∈UD (0;T ;x,α)∩{u• ∈Fx0 ,ζ }

0 = ID,F x

0

K u(t), u(t) ˙ dt

0

(0; T ; x, α) , ,ζ

where {u• ∈ Fx0 ,ζ } again denotes the set of control functions which take 0 their values in Fx0 ,ζ only. Furthermore, the latter quantity ID,F (0; T ; x, α) x0 ,ζ which had been defined in definition 4.4.5 only depends on the behavior of 0 ¯ b, σ (and thus also of F, ˆb, σ h, ˆ ) on Fx0 ,ζ ; in other words, ID,F (0; T ; x, α) x0 ,ζ ¯ does not depend on how h, b, σ are redefined outside Fx0 ,ζ . ˆ α) 2) Applying corollary 3.2.1 to the (open) set D and the process (X, ˆ ε defined in step 1), it follows that for any fixed η˜ > 0, there is an ε0 > 0 such that for all ε ∈ (0, ε0 ], the exit probability QεD (0; T ; x, α) ≡ P0,x,α {TDε ≤ T } ≡ P TDε (0, x, α) ≤ T satisfies

1 0 − max ID (0; T ; x, α) − η˜ > exp ε (x,α)∈G

1 0 − max ID,Fx ,ζ (0; T ; x, α) − η˜ ≥ exp 0 ε (x,α)∈G 0 I (T ) + η˜ , ≡ exp − max ε

min (x,α)∈G

QεD (0; T ; x, α)

where G denotes the compact set G := Fx0 ,ζ × A2 (Fx0 ,ζ ) ⊂ Rd × D and 0 Imax (T ) :=

0 max ID,F x

(x,α)∈G

0 ,ζ

(0; T ; x, α)

for abbreviation. Due to assumption (4.24) there is some time horizon T¯ > 0 such that 0 (T¯ ) ; in other words, it is possible to fix η˜ sufficiently small such ζ > Imax that


193

0 ζ > Imax (T¯) + η˜

for the investigations to come. In other words, it follows for the inclusion probability ε PD (0; T ; x, α) ≡ 1 − QεD (0; T ; x, α) ≡ P TDε (0, x, α) > T from 3.2.1 and by assumption (4.24) that 0 Imax (T¯ ) + η˜ < 1 − exp − ε ζ . < 1 − exp − ε

max (x,α)∈G

ε PD (0; T¯ ; x, α)

3) Let ε,(x,α)

ε τG ≡ τG

:= inf

ε ˆ ε, α t ≥ 0 : (X /G t ˆt ) ∈

ε,(x,α) ˆ α from G; since G is closed and the denote the first exit time of (X, ˆ )t ε underlying filtration is complete, τG is a stopping time.1 Since G ⊂ Rd × D, it follows from step 2) that for any ε ≤ ε0 , ε,(x,α) max P τG > T¯ ≤ max P TDε (0, x, α) > T¯ (x,α)∈G

(x,α)∈G

≡

ε max PD (0; T¯; x, α)

(x,α)∈G

< 1 − exp

−

< 1 − e−ζ/ε

0 Imax (T¯ ) + η˜ ε

.

By induction, this implies that for any ε ≤ ε0 , k I 0 (T¯ ) + η˜ ε,(x,α) max P τG > k T¯ < 1 − exp − max (x,α)∈G ε

(k ∈ N) .

More precisely, one obtains for any initial value (x, α) ∈ G and ε ≤ ε0 that ε,(x,α) P τG > k T¯ ; ˆ ε,x ε, X ˆ ε,α ¯ ,α ¯ ε,(x,α) (k−1) T (k−1) T > (k − 1) T¯ , τG > T¯ = P τG

1


194


+ = E 1 τ ε,(x,α) >(k−1)T¯ · G

1 − EF(k−1)T¯

1 τ

ˆ ε,x X ˆ ε,α ¯ +• , α ¯ +• (k−1)T (k−1)T

≤T¯

⎤

⎥ ⎦

, + ε,(x,α) ˆ α) ˆ (k−1)T¯ +• , ≡ E 1 τ ε,(x,α) >(k−1)T¯ · 1 − EF(k−1)T¯ H ◦ (X, G

where H is given on the path space C(R+ , Rd+1 ) by H := 1{τ ≤T¯} and τ is defined on the paths h ∈ C(R+ , Rd+1 ) by τ (h) := inf{t ≥ 0 : ht ∈ / G} ε,(x,α)

such that τG

ˆ •ε,x , α ≡ τ ◦ (X ˆ ε,α • ). With

ˆ α (T H)(p) := E(H ◦ (X, ˆ )ε,p • )

(p ∈ Rd × R)

the strong Markov property2 for the stochastic system ε,(x,α)

ˆ α (X, ˆ )t

ˆ ε,x , α ≡ (X ˆ ε,α t t )t

implies that ε,(x,α) ε,(x,α) ˆ α) ˆ α EF(k−1)T¯ H ◦ (X, ˆ (k−1)T¯ +• = (T H) ◦ (X, ˆ )(k−1)T¯ ε,p ˆ ≡ E 1{τ ≤T¯} ◦ (X, α ˆ )•

.

ε,(x,α) ˆ α) p=(X, ˆ (k−1)T¯

Plugging this in the previous string of equations, we get (as we look only at ε,(x,α) ˆ α the case that p = (X, ˆ )(k−1)T¯ is still in G): ε,(x,α) P τG > k T¯ 5 = E 1 τ ε,(x,α) >(k−1)T¯ G

5 =E

2

1

ε,(x,α)

τG

>(k−1)T¯

6 ε,(x,α) ˆ α) ˆ (k−1)T¯+• · 1 − EF(k−1)T¯ H ◦ (X, ˆ α · 1 − E 1{τ ≤T¯} ◦ (X, ˆ )ε,p •

6 ε,(x,α) ˆ α) p=(X, ˆ (k−1)T¯

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.32 & 6.41].


5 ≤E

1

5 ≡E

1

ε,(x,α)

τG

ε,(x,α)

τG

>(k−1)T¯

>(k−1)T¯

ˆ α · 1 − inf E 1{τ ≤T¯} ◦ (X, ˆ )ε,p •

195

6

p∈G

6 ε,p ¯ · sup P τG > T p∈G

0 Imax (T¯ ) + η˜ ε,(x,α) · P τG > (k − 1)T¯ < 1 − exp − ε k 0 Imax (T¯ ) + η˜ , < 1 − exp − ε

where in the last two steps the case “k = 1” shown above and the induction assumption have been used. 4) Now, fix an arbitrary δ > 0 and define I H δ · T (ε) −1 , k ≡ k(ε) := T¯ where . denotes the integer part function. Furthermore, for the set ε,x0 Ωε := Ωε1;F ≡ ω ∈ Ω : XtT (ω) ∈ F for all t ∈ [0, 1] , x0 ,ζ (ε) it is known from corollary 2.5.7 (Γ := 1) that lim P ( Ωε ) = 1 .

ε→0

D x ,ζ × ˆ ε, α ˆ ε ) coincide on the open set F Since the SDEs for (X ε , αε ) and (X 0 R, it follows that these processes are necessarily equal to each other (nonD x ,ζ × R. More precisely3 , there is a unique distinguishable) before exiting F 0 Dx0 ,ζ × R ; maximal solution of the SDE with respect to the “state space” F this fact will be used in the subsequent calculation. Furthermore, it will now be used that on Ωε , i.e. on the event that the path map X•ε only takes its values in Fx0 ,ζ , the system αε cannot reenter A2 (Fx0 ,ζ ) after leaving this set. Formally, this fact is expressed in control-theoretic terminology by noting that Fx0 ,ζ × ([0, π) \ A2 (Fx0 ,ζ )) is a strong control ¯ set of the control system α˙ 0 (t) = h(u(t), α0 (t)) with control functions u ∈ d C(R+ , R ) ∩ {u• ∈ Fx0 ,ζ } ; see Kliemann [Kl 80, p.31f.]. Due to our previous construction, this is equivalent to the fact that the set Rd × ([0, π) \ A2 (Rd )) is a strong control set of the control system α˙ 0 (t) = F (u(t), α0 (t)) with control functions u ∈ C(R+ , Rd ) . The reason behind this argument is that 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.22]. The maximality of the solution mentioned above refers to the maximal choice of its life time, i.e. the first exit D x ,ζ ×R ; see e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.19]. What will time from F 0 D x ,ζ × R . ˆ ε, α ˆ ε ) coincide before exiting F be used here is the fact that (X ε , αε ) and (X 0

196


¯ has strictly separated switching curves A1 and A2 by assumption. More h precisely, in the following it will be used that for all ω ∈ Ωε , 0 L t ∈ [0, T (ε)] : αε,α (ω) ∈ A2 (Fx0 ,ζ ) ≥ (k(ε) + 1)T¯ t if and only if 0 αε,α (ω) ∈ A2 (Fx0 ,ζ ) ; (k(ε)+1)T¯

we denote this argument by (∗); of course the time (k(ε) + 1)T¯ which is relevant for the subsequent estimate can be replaced by any other element of [0, T (ε)] here. Thus it follows altogether for any initial condition (x0 , α0 ) of (Xtε , αεt ) that

1 0 ∈ A (F ) ≥ δ L t ∈ [0, T (ε)] : αε,α 2 x0 ,ζ t T (ε)

ε ε ε,α0 ≤ P Ω + P Ω ∩ L t ∈ [0, T (ε)] : αt ∈ A2 (Fx0 ,ζ ) ≥ δT (ε)

0 ¯ ∈ A (F ) ≥ (k(ε) + 1) T ≤ P Ωε + P Ωε ∩ L t ∈ [0, T (ε)] : αε,α 2 x ,ζ t 0

(∗) 0 = P Ωε + P Ωε ∩ αε,α ∈ A2 (Fx0 ,ζ ) (k(ε)+1)T¯

0 Xtε,x0 ∈ Fx0 ,ζ for all t ∈ [0, T (ε)] ∩ αε,α ∈ A (F ) ≡ P Ωε + P 2 x ,ζ ¯ 0 (k(ε)+1)T ε ε,x0 ε,α0 ∈ G for all t ∈ 0, (k(ε) + 1)T¯ ≤ P Ω + P Xt , αt 0 ˆ tε,x0 , α = P Ωε + P X ˆ ε,α ∈ G for all t ∈ 0, (k(ε) + 1)T¯ t ε,(x ,α ) ≡ P Ωε + P τG 0 0 ≥ (k(ε) + 1)T¯ ε,(x ,α ) ≤ P Ωε + P τG 0 0 > k(ε)T¯ k(ε) I 0 (T¯ ) + η˜ , < P Ωε + 1 − exp − max ε

P

where in the final estimate the inductive consequence of 3.2.1 obtained in step 3) has been used. As already mentioned, the first summand of the hence obtained upper bound tends to zero as ε → 0 due to corollary 2.5.7. 5) Therefore, the claim has been reduced in step 4) to verifying that

1 − exp

Since

−

k(ε) 0 Imax (T¯ ) + η˜ ε→0 −−−→ 0 . ε

δ δ T (ε) − 1 ≥ k(ε) > ¯ T (ε) − 2 , T¯ T


197

it is clear that also k(ε) ! eζ/ε ; in particular, as 0 ζ > Imax (T¯ ) + η˜ ,

which had been assured by assumption (4.24), see step 2), it is possible to fix an arbitrary 0 η˜ ˜ ∈ 0 , ζ − Imax (T¯ ) − η˜ , for which we have

k(ε) > K exp

ζ − η˜˜ ε

for all ε ≤ ε0 and some constant K > 0 (depending only on η˜˜ and ε0 ). Using this preliminary note, the auxiliary function h(ε) :=

1

1 − exp −

0 Imax (T¯ ) + η ˜ ε

−1 =

exp +

1 0 Imax (T¯ ) + η ˜ ε

−1

∈ (0, ∞)

and the Bernoulli inequality4 , one gets altogether that

1 − exp

−

0 (T¯) + η˜ Imax ε

k(ε) ≡ ≤
0 is not contained in a finite exceptional set depending on x0 ; see (2.20). Assume that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) , where again V denotes the matrix of mutual quasipotential values and J is the map expressing the “following-behavior” of 0-cycles; let Hx0 ,ζ be the set hence defined in corollary 2.5.8 according to the underlying diffusion X ε,x0 and the time scale T (ε) ! eζ/ε . ¯ has strictly separated switching surfaces Suppose that the drift function h A1 and A2 on Hx0 ,ζ ; see definitions 4.4.3 and 1.2.4. ¯ is strongly hypoelliptic on Hx ,ζ ; see definition Furthermore, suppose that h 0 4.4.4. Assume in addition that there exists T¯ > 0 such that 0 ¯ ; x, α) : (x, α) ∈ Hx0 ,ζ × A2 (Hx0 ,ζ ) , (4.25) (0; T ζ > max ID,H x0 ,ζ 0 (0; T ; x, α) is where D ⊃ A2 (Hx0 ,ζ ) denotes some open set and where ID,H x0 ,ζ ε the value function for α as defined in 4.4.5. Then it follows for any α0 and for any η > 0 that P 1 L t ∈ [0, T (ε)] : αεt ∈ B A1 (Kµ(x0 ,ζ) ), η −− −−→ 1 . ε→0 T (ε)

Proof. As in the proof of the previous theorem 4.4.6, let α1 := A1 (Kµ(x0 ,ζ) )

and

α2 := A2 (Kµ(x0 ,ζ) )


199

¯ µ(x ,ζ) , . ) in [0, π) corresponding to denote the two distinct zeros of h(K 0 λ1 > λ2 . Recall from 4.1.6 that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 ε→0 T (ε) which is why it needs to be shown that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α2 , η) −−− −→ 0 ε→0 T (ε) for arbitrary η > 0 also in this situation. In the sequel we will prove the following slightly stronger claim: P 1 L t ∈ [0, T (ε)] : αεt ∈ A2 (Hx0 ,ζ ) −−− −→ 0 . ε→0 T (ε) In the same manner as in the proof of the previous theorem 4.4.6, the strict separatedness of the switching curves and the strong hypoellipticity necessarily also hold true on an open enlargement of Hx0 ,ζ , i.e. on some open set D x0 ,ζ ⊃ Hx0 ,ζ H ¯ ∂h D x ,ζ \ Hx ,ζ due to the continuity of A1 , A2 and ∂x ; again, the difference H 0 0 j might be arbitrarily small. For verifying the preceding claim fix an open D x0 ,ζ ) in the situation at hand, i.e. an open set D ⊂ R neighborhood D of A2 (H such that D x0 ,ζ , D ⊃ A2 H

for which we also have

D x ,ζ > 0 . dist D, A1 H 0

As in the previous proof, the former condition concerning the choice of D can be trivially satisfied; the latter requirement can be met since the separation of the switching surfaces A1 and A2 is assumed to be strict on Hx0 ,ζ (and D x0 ,ζ ). After taking the intersection, this set also has the prescribed also on H property (4.25) and it will also be denoted by D for the further investigations. As has already become clear from these introductory remarks, this proof follows the same lines as the one of the previous theorem 4.4.6; nonetheless, all arguments will be provided in the sequel for definiteness. This proof is organized in four steps: In the first step the coefficients of the angle SDE are modified such that they satisfy the assumptions 3.1.3. This allows to apply 3.2.1 in the second step. Afterwards this result is used to estimate the exit time of the coupled process from the set Hx0 ,ζ × A2 (Hx0 ,ζ ) around the repelling switching curve by means of the strong Markov property. In the fourth step this bound is used together with the control-theoretic

200


structure of the posed situation; more precisely, the system αε cannot reenter D after leaving this set on the event Ωεc,Γ;H on which the paths X•ε take their values exclusively in Hx0 ,ζ during the time interval [c T (ε), Γ T (ε)] ; hence, the exit time of αε from A2 (Hx0 ,ζ ) can then be used to estimate the length of the time interval until αε leaves A2 (Hx0 ,ζ ); this step yields that the decisive probability can be estimated from above by a function which is exponential in ε. The resulting bound then converges to zero by what has been shown in step 5) in the proof of theorem 4.4.6. ¯ of the angle SDE be modified on 1) Let the coefficient functions b, σ and h D D Hx0 ,ζ and on (Hx0 ,ζ ) × R, respectively, as smooth functions ˆb, σ ˆ and F such that the set of assumptions 3.1.3 on the SDE (3.1) is satisfied; more precisely, the functions ˆb, σ ˆ , F are chosen to satisfy 3.1.3 and ˆb, σ ˆ shall coincide with ¯ on H D x ,ζ × R : D x ,ζ as well as F coincides with h b, σ on H 0 0 ¯ is The hypoellipticity assumption 3.1.3(c) on F can be assured, since h D strongly hypoelliptic on Hx0 ,ζ ; again see definition 4.4.4 and remark 3.1.4. ¯ can be chosen such D x0 ,ζ As dist D, A1 H > 0 , the continuation F of h that F has strictly separated switching surfaces (also denoted by A1 and A2 ) on Rd which do not intersect ∂D. In particular, this implies that 3.1.3(d) holds true. Finally, 3.1.3(e) is met, since A2 is repelling and non-constant ¯ is strongly hypoelliptic) and hence, there are trajectories which can leave (h D within a prescribed time horizon; also see below. The resulting system is ε,(x,α) ˆ ε,x , α ˆ α ≡ (X ˆ ε,α again denoted by (X, ˆ )t t t ), i.e. ˆ tε , α dˆ αεt = F X ˆ εt dt √ ε ˆ tε = ˆb X ˆ tε dt + ε σ ˆ t dWt , dX ˆ X satisfying the fixed initial condition ε ε d ˆ ,α X 0 ˆ 0 = (x, α) ∈ R × R . ε,(x,α) ˆ α Note that neither this process (X, ˆ )t≥0 nor the set D (nor the terms to be 0 ε ε derived thereof in the sequel such as TDε , UD , ID , QεD , PD , G, τG , H and Ωε ) in the present proof do coincide with the homonymous objects of the previous proof (theorem 4.4.6). However, since each definition is made precise and no confusion is possible, there is no need to introduce a clumsy notation such as adding tildes. Furthermore, using the respective symbols again highlights the parallels between the two proofs. As in 3.1.1 (s := 0), let

T ε ≡ T ε (0, x, α) ≡ TDε (0, x, α) := inf { t ≥ 0 : α ˆ ε,α ∈ / D} t denote the first exit time of (ˆ αε,α t )t≥s from D.


201

According to 3.1.2 (s := 0), let u ∈ C 1 (R+ , Rd ) be a control function and let α0 be the path defined by dα0 (t) = F (u(t), α0 (t)) dt under the initial condition (u(0), α0 (0)) = (x, α) ∈ Rd × R . As before, ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ /D denotes the corresponding exit time of α0 from D and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. The set UD (0; T ; x, α) is nonempty for all (x, α) ∈ Rd × R also in this case, since A2 is repelling and non-constant; see the above discussion of assumption 3.1.3(e) and Kliemann [Kl 80, p.31f.]. Using this notation, the value function according to 3.1.2 is 0 I 0 ≡ ID (0; T ; x, α) :=

where K(x, v) :=

ϑD (0;x,α;u)

inf u∈UD (0;T ;x,α)

K u(t), u(t) ˙ dt ,

0

0 1 2 1 ˆ(x)−1/2 v − ˆb(x) . a 2

For x ∈ Hx0 ,ζ we have 0 ID (0; T ; x, α)

≤

u∈UD (0;T ;x,α)∩{u• ∈Hx0 ,ζ }

0 = ID,H x

ϑD (0;x,α;u)

inf

0

K u(t), u(t) ˙ dt

0

(0; T ; x, α) , ,ζ

where {u• ∈ Hx0 ,ζ } is the set of control functions which take their values in 0 (0; T ; x, α) which had been defined in 4.4.5 only depends Hx0 ,ζ only; ID,H x0 ,ζ ¯ b, σ (and thus also of F, ˆb, σ on the behavior of h, ˆ ) on Hx ,ζ . 0

2) In this situation apply corollary 3.2.1 to the (open) set D and the process ˆ α) (X, ˆ ε defined in step 1). Then it follows that for any fixed η˜ > 0, there is an ε0 > 0 such that for all ε ∈ (0, ε0 ], the exit probability QεD (0; T ; x, α) ≡ P0,x,α {TDε ≤ T } ≡ P TDε (0, x, α) ≤ T

202


satisfies

1 0 − max ID (0; T ; x, α) − η˜ ε (x,α)∈G

1 0 − max ID,H (0; T ; x, α) − η ˜ ≥ exp x0 ,ζ ε (x,α)∈G I 0 (T ) + η˜ , ≡ exp − max ε

min QεD (0; T ; x, α) > exp

(x,α)∈G

where in the present situation, G denotes the compact set G := Hx0 ,ζ × A2 (Hx0 ,ζ ) ⊂ Rd × D and accordingly 0 Imax (T ) :=

0 max ID,H x

(x,α)∈G

0 ,ζ

(0; T ; x, α) .

Due to assumption (4.25) one can fix T¯ > 0 and η˜ > 0 such that 0 (T¯ ) + η˜ . ζ > Imax

In other words, it follows for the inclusion probability ε PD (0; T ; x, α) ≡ 1 − QεD (0; T ; x, α) ≡ P TDε (0, x, α) > T from 3.2.1 and by assumption (4.25) that 0 Imax (T¯ ) + η˜ < 1 − exp − ε ζ . < 1 − exp − ε

max (x,α)∈G

ε PD (0; T¯ ; x, α)

3) Let ε,(x,α)

ε τG ≡ τG

:= inf

ε ˆ ε, α t ≥ 0 : (X /G t ˆt ) ∈

ε,(x,α) ˆ α from the set G as defined in step 2). denote the first exit time of (X, ˆ )t d Since G ⊂ R × D, it follows from step 2) that for any ε ≤ ε0 , ε,(x,α) max P τG > T¯ ≤ max P TDε (0, x, α) > T¯ (x,α)∈G

(x,α)∈G

ε max PD (0; T¯; x, α) I 0 (T¯ ) + η˜ < 1 − exp − max ε −ζ/ε k T¯ < 1 − exp − max ε (x,α)∈G

(k ∈ N) .

The proof of this fact again uses the strong Markov property: More precisely, the notation has been chosen such that the argument given in step 3) of the proof of theorem 4.4.6 literally holds true in the situation at hand. We refrain from repeating it verbatim. 4) Now, fix an arbitrary δ > 0, some c ∈ (0, δ ∧ 1) and define I H (δ − c) · T (ε) −1 . k ≡ k(ε) := T¯ First, it follows for the set ε,x0 Ωε := Ωεc,1;H := ω ∈ Ω : XtT (ω) ∈ H for all t ∈ [c, 1] x0 ,ζ (ε) from corollary 2.5.8 (Γ := 1) that lim P ( Ωε ) = 1 .

ε→0

ˆ ε, α Secondly, by construction the SDEs for (X ε , αε ) and (X ˆ ε ) again coinD x0 ,ζ × R here; hence, it follows5 cide on the relevant open set which is H D that when started in a point (x, α) ∈ Hx0 ,ζ × R these processes are nondistinguishable before exiting this set. Furthermore, it will be used that on Ωε , i.e. on the event that the paths X•ε take their values in Hx0 ,ζ on the time interval [c T (ε), T (ε)] , the system αε cannot reenter A2 (Hx0 ,ζ ) during the time interval [c T (ε), T (ε)] after leaving this set A2 (Hx0 ,ζ ). Again, the control-theoretic reason is that Hx0 ,ζ × ([0, π) \ ¯ A2 (Hx0 ,ζ )) is a strong control set of the control system α˙ 0 (t) = h(u(t), α0 (t)) with control functions u ∈ C(R+ , Rd ) ∩ {u• ∈ Hx0 ,ζ }; see Kliemann [Kl 80, D x0 ,ζ × p.31f.]. Due to the continuation F of the coefficient function ¯h outside H d d R, this is equivalent to saying that the set R × ([0, π) \ A2 (R )) is a strong control set of the control system α˙ 0 (t) = F (u(t), α0 (t)) with control functions ¯ has strictly separated u ∈ C(R+ , Rd ). Here we made use of the fact that h switching curves A1 and A2 on Hx0 ,ζ by assumption. More precisely, in the following this remark will be used in the form that for all ω ∈ Ωε , 0 L t ∈ [c T (ε), T (ε)] : αε,α (ω) ∈ A (H ) ≥ (δ − c) T (ε) 2 x ,ζ 0 t

5


204


if and only if 0 αε,α (ω) ∈ A2 (Hx0 ,ζ ) for all t ∈ [c T (ε), δ T (ε)] ; t

again we denote this control-theoretic conclusion by (∗). D be given on the path space C(R+ , Rd+1 ) by Finally, let H D := 1 H

τ 0, (Xtε,x0 , αε,α t

1 0 L t ∈ [0, T (ε)] : (Xtε,x0 , αε,α ) ∈ B Kµ(x0 ,ζ) , c × B A1 (Kµ(x0 ,ζ) ), c t T (ε) P −− −−→ 1 ; ε→0

in other words, one obtains for any c > 0 that P L Tεc ( . ) −−− −→ 1 , ε→0


where

207

ε,x0 ε,α0 Tεc (ω) := t ∈ [0, 1] : Xt·T (ω) , α (ω) (ε) t·T (ε) ∈ B Kµ(x0 ,ζ) , c × B A1 (Kµ(x0 ,ζ) ), c .

It needs to be proven that this implies that 1 T (ε)

0

T (ε)

P 0 ¯ Xtε,x0 , αε,α dt −−− Q −→ Λ1 Kµ(x0 ,ζ) , t ε→0

since the convergence claimed above then follows from (1.8): 1 log |z0 | 1 log |ZTε (ε) | = + T (ε) T (ε) T (ε)

0

T (ε)

P 0 ¯ Xtε,x0 , αε,α dt − Λ1 Kµ(x0 ,ζ) . Q → t ε→0

Define Kµ,i :=

≡ Kµ(x0 ,ζ) , αi ∈ Rd+1 Kµ(x0 ,ζ) , Ai Kµ(x0 ,ζ)

for i = 1, 2. Since by construction, ¯ µ,i ) ≡ Q ¯ Kµ(x ,ζ) , Ai Kµ(x ,ζ) Q(K = Λi Kµ(x0 ,ζ) 0 0

(i = 1, 2),

we have to show that T (ε) 1 P 0 ¯ Xtε,x0 , αε,α ¯ µ,1 ) dt −−− Q −→ Q(K t ε→0 T (ε) 0 in this notation. A statement of this type without the angle coordinate has already been verified in theorem 2.5.10; the same rationale now also applies here. For an arbitrary η > 0 one gets that P

; 1 T (ε) ¯ (Kµ,1 ) > η ¯ X ε,x0 , αε,α0 dt − Q Q t t T (ε) 0 1 ¯ (Kµ,1 ) > η ¯ X ε,x0 ( . ) , αε,α0 ( . ) dt − Q = P Q t·T (ε) t·T (ε) 0 1

≤P

0

≤ P Ωε

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt > η (.) − Q Q Xt·T (ε) ( . ) , αt·T (ε)

1

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt > η (.) − Q , + P Ωε ∩ Q Xt·T (ε) ( . ) , αt·T (ε) 0 AB C @ =:B ε

where the set Ωε is taken as ε,x0 Ωε := Ωε1;F ≡ ω ∈ Ω : XtT (ω) ∈ F for all t ∈ [0, 1] , x ,ζ 0 (ε)

208


for which it is known from corollary 2.5.7 (which holds true in either of the posed situations) that lim P Ωε = 0 . ε→0

ε→0

Therefore, it needs to be shown that B ε −−−→ 0 , which is done in the sequel: For estimating B ε , we fix ω ∈ Ωε and c > 0 for which we get

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε) 0 ¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt = (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε) 1

[0,1]\Tε c (ω)

+

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε)

Tε c (ω)

¯ ¯ (Kµ,1 ) : (x, α) ∈ Fx ,ζ × [0, 2π] α) − Q ≤ L [0, 1] \ Tεc (ω) · max Q(x, 0 ¯ ¯ (Kµ,1 ) : (x, α) ∈ B(Kµ(x,ζ) , c) × B(A1 (Kµ(x,ζ) ), c) α) − Q + max Q(x,

¯ and its periodicity in the angle varidue to the continuity of the function Q able, as well as due to the defining properties of the sets Ωε and Tεc (ω). Now if we choose c sufficiently small such that the latter summand is bounded by ¯ is continuous), η/2 (Q ¯ α) − Q ¯ (Kµ,1 ) : (x, α) ∈ B(Kµ(x,ζ) , c) × B(A1 (Kµ(x,ζ) ), c) < η max Q(x, 2 and define

¯ α) − Q ¯ (Kµ,1 ) : (x, α) ∈ Fx0 ,ζ × [0, 2π] , m := max Q(x,

then it follows from the previously obtained estimate that 1

¯ ε,x0 ε,α0 ε ε ¯ ≡ P Ω ∩ B Q Xt·T (ε) ( . ) , αt·T (ε) ( . ) − Q (Kµ,1 ) dt > η 0 η ε ε > η ≤ P ω ∈ Ω : m · L [0, 1] \ Tc (ω) + 2 η ≤ P L [0, 1] \ Tεc ( . ) > 2m ε→0

−−−→ 0 , since the stochastic convergence stated in the beginning of this proof says in P other words that L [0, 1] \ Tεc ( . ) − → 0 as ε → 0.


209

This theorem and its proof provide the counterpart to the FurstenbergKhasminskii formula (1.17) in the situation that X ε has sublimiting distributions on time scales: Under the prescribed assumptions, we obtained ¯ α) δKµ,1 (dx, dα) = Λ1 Kµ(x ,ζ) as the limit in the subQ(x, d 0 R ×[0,π) limiting situation; hence, we consider the above result as a sublimiting Furstenberg-Khasminskii formula. Further note the analogy with the law of large numbers (1.16): The statement of (1.16) is often described by saying that “one only observes the top Lyapunov exponent Λε1 in the hypoelliptic case”. In contrast, our findings of theorem 4.4.8 can be paraphrased by stating that “one only observes the top eigenvalue Λ1 (Kµ(x0 ,ζ) ) in the sublimiting, strongly hypoelliptic, strictly separated case”. In the following the assumptions of this section are going to be discussed. Remark 4.4.9 (Strictly separated switching curves entail distinct eigenvalues (4.23)). As in the previous theorems, consider a coefficient matrix A ∈ C ∞ (Rd , R2×2 ) of (1) and the thus defined angle velocity function ¯ h(A(x), α) ≡ − a12 (x) sin2 α + a21 (x) cos2 α + (a22 (x)−a11 (x)) sin α cos α . ¯ has strictly separated switching curves A1 and A2 on If it is assumed that h M := Fx0 ,ζ (theorem 4.4.6) or on M := Hx0 ,ζ (theorem 4.4.7), respectively, it follows in particular that the characterizing inequality (4.23) of this section (see p.177) is automatically satisfied; moreover, it does not only hold at Kµ(x0 ,ζ) , but on all of M, Λ1 (x) > Λ2 (x)

for all x ∈ M ;

the reason is that due to definitions 4.4.3 and 1.2.4 there are exactly two switching curves: A1 (A(x)) is the attracting and A2 (A(x)) is the repelling branch of arctan ξ1,2 (A(x)) , where ξ1,2 (A(x)) =

(a22 − a11 )2 + 4a12 a21 , 2a12 0 and (a22 − a11 )2 > −4a12 a21 , =

(a22 − a11 ) ± if a12

and ξ1 (A(x)) = ∞ , ξ2 (A(x)) =

a21 , if a12 = 0 and a11 = a22 . a11 − a22

The eigenvalues of A(x) are calculated as Λ1,2 (x) =

1 1 0 (a11 + a22 ) ± (a11 − a22 )2 + 4a12 a21 2

and are distinct in either of the two cases for the matrix A(x).

210


Next, we would like to take a closer look at the assumptions of theorems 4.4.6 and 4.4.7 in order to discuss criteria under which the conditions of theorem 4.4.8 are valid. First, we pose the question in which situation strict separatedness of the switching curves A1 and A2 can be read off the matrix function A. As Kliemann [Kl 80, p.150–153] and [Kl 79, p.467f.] shows, there are two “strong ¯ has exactly two switching control sets” for the angle motion in case that h curves and a12 is constant. We translate Kliemann’s findings to our situation for which we assume for convenience that a12 ( . )

and

a21 ( . )

are both constants, again denoted by a12 and a21 , respectively, which also satisfy that a12 = 0 ,

a21 = 0

and

sign(a12 ) = sign(a21 ) .

The latter line of conditions is e.g. fulfilled in the symmetric case a12 = a21 = 0 ; furthermore note that the case of constant entries a12 = a21 = 0 obviously yields strictly separated switching curves 0 and π2 and had been solved in section 4.3 . Remark 4.4.10 (Strictly separated switching curves of the angle ¯ in (1)). Let A ∈ C ∞ (Rd , R2×2 ) be the coefficient matrix of (1) and drift h suppose that its secondary diagonal elements a12 and a21 are both constant such that a12 = 0 ,

a21 = 0

and

sign(a12 ) = sign(a21 ) .

¯ are strictly separated on any compact set. Then the switching curves of h Proof. As follows from definition 1.2.4 (the relevant part being recalled above in remark 4.4.9) under these assumptions there are exactly two switching surfaces represented by the zeros 2 [a22 (x) − a11 (x)] ± [a22 (x) − a11 (x)]2 + 4a12 a21 ξ1,2 (A(x)) = 2a12 1 −b(x) ± b(x)2 + a = a12 of the drift of tan αεt , where b(x) :=

a11 (x) − a22 (x) 2

and

a := a12 a21 > 0 .


211

Now define ξ1,2 (A(x)) := a12 · ξ1,2 (A(x)) ≡ −b(x) ± b(x)2 + a , where the positive root is defined as ξ1 and the negative root is defined as ξ2 ; note that this enumeration does not necessarily correspond to the one of A1 (attracting) and A2 (repelling). Furthermore, let M ⊂ Rd be some compact set as for example M := Fx0 ,ζ in theorem 4.4.6 or M := Hx0 ,ζ in theorem 4.4.7, respectively. We have to prove that inf ξ1 (A(x)) > sup ξ2 (A(x)) ,

x∈M

x∈M

since it then follows from the continuity of arctan on R that ¯h has strictly separated switching surfaces A1and A2 on M in the sense of definition 4.4.3. However, since a > 0, |b(x)| < b(x)2 + a and therefore ξ1 (A(x)) ≡ −b(x) + b(x)2 + a >0 > −b(x) − b(x)2 + a ≡ ξ2 (A(x)) ; the strict inequality asserted above now follows from the continuity of ξ1 and ξ2 as well as from the compactness of M. Remark 4.4.11. In the light of the previous remark 4.4.10 one is tempted to look for generic examples of theorem 4.4.8 in the class of coefficient matrices A ∈ C ∞ (Rd , R2×2 ) of (1) with constant nonzero secondary diagonal elements a12 and a21 of the same sign. However, suppose that some coefficient matrix function A( . ) with constant elements a12 , a21 ∈ R is given. Then it follows from remark 1.2.1 that the angle-drift ¯ h is calculated as ¯ α) = h ¯ A(x), α h(x, = − a12 sin2 α + a21 cos2 α + a22 (x) − a11 (x) sin α cos α 1 a22 (x) − a11 (x) sin 2α = − a12 sin2 α + a21 cos2 α + 2 implying that ¯ α) 1 ∂ h(x, = ∂xj 2

∂ a11 (x) ∂ a22 (x) − ∂xj ∂xj

sin 2α

(j = 1, . . . , d),

212


since a12 and a21 do not depend on x. As this term vanishes for α ∈ ¯ cannot be strongly hypoelliptic on any set M k π2 : k ∈ Z it follows that h in the sense of definition 4.4.4 . In other words, the assumption of the previous remark 4.4.10 that A ∈ C ∞ (Rd , R2×2 ) shall have constant (nonzero) secondary diagonal elements a12 and a21 (of the same sign) prevents the set of conditions in the theorems 4.4.6 and 4.4.7 and in theorem 4.4.8, respectively, from being satisfied. What would remain to be done here in order to obtain such a result nevertheless is ¯ outside the to truncate and appropriately redefine the coefficient function h D D set M × A2 (M) in the proof of 4.4.6 and 4.4.7 instead of only redefining it D × R , where M D denotes the respective set F D x ,ζ or H D x ,ζ ; see the outside M 0 0 beginnings of the steps 1) in the proofs of 4.4.6 and 4.4.7, respectively. Note in particular that the above class of coefficient matrices of (1) with constant nonzero secondary diagonal elements of the same sign contains the linearized gradient SDEs, A = −HU , as discussed in 1.5.3 with potential functions U ∈ C ∞ (R2 , R) for which there is a constant c such that ∂U (x) = c = 0 ∂x1 ∂x2

(for all x ∈ R2 )

due to the symmetry of the Hesse matrix. An example for this situation is given by the potential function U1 (x) =

3 4 2 3 x − x31 − 3 x21 + c x1 x2 + x42 2 1 3 2

(x ∈ R2 )

as defined in (1.34) in case that c = 0; again see figure 2.1 for c = 1. Here, the coefficient matrix of the linearized system had been calculated as

−c − 18x21 + 4 x1 + 6 (x ∈ R2 ) . A1 (x) ≡ − HU1 (x) = −c − 18 x22 The reason for concentrating on the case of nonzero secondary elements above is that if the matrix has constant zero secondary elements, a12 = a21 = 0, then our findings of section 4.3 already apply. The target of the following remark is to prove that in case that b derives from a potential function the conditions (4.24) and (4.25) are redundant given the other assumptions of theorems 4.4.6 and 4.4.7 (and hence also of theorem 4.4.8), respectively. If the underlying set of assumptions is changed, e.g. if the switching curves are not supposed to be strictly separated, the arguments can be suitably modified. Remark 4.4.12 (on the assumptions (4.24) and (4.25)). As before let the function A ∈ C ∞ (Rd , R2×2 ) be the coefficient matrix of the SDE (1) and ¯ of the angle process αε of Z ε in the again suppose that the drift function h RDE (1.6), ¯ ε , αε ) , α˙ εt = h(X αε0 = α0 , t t


213

¯ is has strictly separated switching surfaces A1 and A2 on M and that h d strongly hypoelliptic on M, where the compact set M ⊂ R either represents >l M := Fx0 ,ζ (theorem 4.4.6) or M := Hx0 ,ζ (theorem 4.4.7), x0 ∈ i=1 Di being the initial value of X ε in SDE (2.1) (under the assumptions 2.1.1) and ζ > 0 parametrizing the time scale T (ε) ! eζ/ε . Additionally assume for the diffusion X ε that its SDE (2.1) is of gradient form (2.2), √ dXtε,x0 = − ∇U (Xtε,x0 ) dt + ε dWt , X0ε,x0 = x0 ∈ Rd , i.e. that the drift b = −∇U is given by a potential U ∈ C ∞ (Rd , R), and where σ = idRd . Then the conditions (4.24) and (4.25) are already satisfied, i.e. there exists T¯ > 0 such that 0 (0; T¯ ; x, α) : (x, α) ∈ M × A2 (M) , ζ > max ID,M 0 where D ⊃ A2 (M) denotes some open set and where the terms ID,F x0 ,ζ 0 ε (0; T ; x, α) and ID,Hx ,ζ (0; T ; x, α) are the value functions of α as defined in 0 4.4.5 corresponding to the respective choice of M ∈ Fx0 ,ζ , Hx0 ,ζ .

Proof. As in the proofs of theorems 4.4.6 and 4.4.7 let D ⊂ R be some open set such that A2 (M) ⊂ D and D ∩ A1 (M) = ∅ ; due to the latter two properties, the separatedness of the switching curves A1 and A2 and the π¯ α) it follows that L(D) < π . This set D will be fixed periodicity of α → h(x, throughout the following argument and can also be used in the proofs of the previous theorems. According to definition 4.4.5 it is to be proven that there are T¯ > 0 and η ∈ (0, ζ) such that for all (x, α) ∈ M × A2 (M), 0 ID,M (0; T¯ ; x, α) ≡

ϑD (0;x,α;u)

inf

u∈UD (0;T¯ ;x,α)∩{u• ∈M}

K u(t), u(t) ˙ dt < ζ −η;

0

unwinding all relevant notation, the latter assertion means that there exists some function u ∈ C 1 (R+ , Rd ) taking all its values in M (which is abbreviated as {u• ∈ M}), starting in u(0) = x such that ϑD (0; x, α; u) ≤ T¯ , where

ϑ ≡ ϑD (0; x, α; u) ≡ inf t ≥ 0 : α0 (t) ∈ /D

denotes the first exit time of the solution α0 of the controlled initial value problem d 0 ¯ α (t) = h(u(t), α0 (t)) , α0 (0) = α , dt

214


from D and such that

ϑ

K u(t), u(t) ˙ dt < ζ − η ,

0

where K(q1 , q2 ) =

1 2 | q2 + ∇U (q1 ) | , 2

due to the special gradient form (2.2) of the SDE (2.1). For proving this claim we will use the following notation: Given two continuous functions Pi : [0, Zi ] −→ Rd

(i ∈ {1, 2})

(“paths”), where Z1 , Z2 ≥ 0 are two time horizons and where P1 (Z1 ) = P2 (0), we define the juxtaposition of P1 and P2 as the continuous function , t ∈ [0, Z1 ], P1 (t) d P1 P2 : [0, Z1 +Z2 ] −→ R , (P1 P2 )(t) := P2 (t − Z1 ) , t ∈ [Z1 , Z1 + Z2 ]. The proof is now organized in six steps. Firstly, after fixing the relevant notation we construct an absolutely continuous path, called g2x , which connects x with one of the local minima Ki of U such that the “cost function” K(g2x (t), g2x (t))dt can be reasonably estimated. In the second step this function g2x is used as a control input for the angle system. Two cases now arise: In the first situation the resulting angle, called α , is bounded away from the ¯ can be uniunstable equilibrium A2 (Ki ) and hence the velocity function h formly bounded away from 0 which forces the controlled angle process to leave D; this is elaborated in step 3). In the second case α can be arbitrarily close ¯ can become arbitrarily small; this situato A2 (Ki ) and hence the function h tion is coped with in 4). The fifth step concludes from 3) and 4) that we have altogether constructed a control input g which has all the desired properties with the exception that g is absolutely continuous instead of being continuously differentiable. The final step then explains that the latter drawback can be overcome by means of a Sobolev argument. 1) According to the set of assumptions 2.1.1, there are finitely many points K1 , . . . , Kl which are the local minima of U in the gradient case and points Kl+1 , . . . , Kl which are the saddles and the local maxima of U (see remark 2.1.2 on (K) ). Let k ∈ N denote the number of all critical points {K1 , . . . , Kl } in M and let k ≤ k denote the number of local minima {K1 , . . . , Kl } in M ; we relabel these points such that {K1 , . . . , Kk } are the local minima in M and such that {Kk+1 , . . . , Kk } are all the other critical points in M .


215

Let U ⊂ Rd denote the union of all those lower dimensional subspaces which are attracted to {Kk+1 , . . . , Kk } under the deterministic system (2.8), X˙ 0 = −∇U (X 0 ); further define Uκ := M ∩ v ∈ Rd : dist(v, U) < κ (κ > 0) , the set of points in M whose distance to U is at most κ, where the parameter κ > 0 will be fixed later; see step 5). Fix ρ > 0 sufficiently small such that B(Ki , ρ) ⊂ M\Uκ for all i = 1, . . . , k , where B(Ki , ρ) again denotes the open ball around Ki with radius ρ. Then there is a constant T1 ≡ T1 (ρ) < ∞ such that for all x ∈ M \ Uκ , : B(Ki , ρ) (t ≥ T1 ) , f1x (t) := Xt0,x ∈ i=1,...,k

where as usual X 0,x denotes the solution of the deterministic differential equation (ε = 0) starting in x. As in the proof of lemma 2.4.9, this follows since all trajectories starting in M \ Uκ are attracted towards a unique element of {K1 , . . . , Kk }; the universality of the constant T1 follows from the Gronwall argument (b ≡ −∇U > is Lipschitz on the compact set M \ Uκ ) and the compactness of (M \ Uκ ) \ i=1,...,k B(Ki , ρ). Moreover, if x ∈ Uκ , let y ≡ y(x) denote its proximum in M \ Uκ , i.e. a (not necessarily unique) element y ∈ M \ Uκ such that dist(x, M \ Uκ ) = |y − x| =: T0 (x) ≤ κ ; such an element y exists due to the compactness of M \ Uκ ; in particular, y(x) the previous discussion yields a well-defined function f1 , since y ≡ y(x) ∈ M \ Uκ . Moreover, for f0x (t) := x + t

y−x , |y − x|

it follows that f0x (0) = x , f0x (T0 (x)) = y and 0

T0 (x)

K f0x (t), f˙0x (t) dt ≡ I[0,T0 (x)],x (f0x ) 1 ≡ 2

0

T0 (x)

2 ˙x f0 (t) + ∇U (f0x (t)) dt

2 1 1 + max |∇U | ≤ T0 (x) ¯κ 2 U ≤ LM κ , where LM :=

2 1 1 + max |∇U | M 2

216


and where I denotes the rate function of X ε ; see theorem 2.3.4 . This argument is an adapted and corrected version of Lemma 4.2.3 by Freidlin and Wentzell [Fr-We 98, p.112]. Now juxtapose f0x and f1y , if necessary; more precisely, define y(x) , if x ∈ Uκ , f0x f1 x d x g1 := g1 : [0, T1 (ρ) + κ] −→ R , x f1 , if x ∈ M \ Uκ ; y(x)

note that formally f0x has been defined till time T0 (x) ≤ κ ; therefore, f1 is supposed to have the domain [0, T1 (ρ) + κ − T0 (x)] in order to make the juxtaposition well-defined; furthermore, f1x is taken as defined on [0, T1 (ρ) + κ] ; since both the latter intervals contain [0, T1 (ρ)] , g1x hits one of the balls y(x) B(Ki , ρ) by definition of f1x and f1 . All in all, for any x ∈ M we have a piecewise differentiable (absolutely continuous, in particular) path g1x which is defined on [0, κ + T1 (ρ)] such that : B(Ki , ρ) g1x κ + T1 (ρ) ∈ i=1,...,k

and 0

κ+T1 (ρ)

K (g1x (t), g˙ 1x (t)) dt ≡ I[0,κ+T1 (ρ)],x (g1x ) ≤ I[0,T0 (x)],x (f0x ) + 0 ≤ LM κ ; y(x)

are trajectories of the this estimate holds true, since the paths f1x and f1 deterministic dynamical system X 0 and hence do not contribute to the rate function. By (2.9) of lemma 2.4.7, applied with O := Ki for any i ∈ {1, . . . , k} , there exists some absolutely continuous function f2 : [0, 1] → Rd such that f2 (0) = g1x κ + T1 (ρ) , f2 (1) = Ki and I[0,1],f2 (0) < δ ; here, δ > 0 denotes an arbitrary parameter only determining the specific choice of ρ. Just as κ, we will choose δ (and hence ρ in other words) in step 5). Now we prolong g1x by means of f2 . For this purpose define the absolutely continuous function g2x := g1x f2 : [0, κ + T1 (ρ) + 1] −→ Rd . Then g2x (0) = x , g2x κ + T1 (ρ) + 1 = Ki for some i ∈ {1, . . . , k} and (again due to the additivity of I)


κ+T1 (ρ)+1 0

217

K g2x (t), g˙ 2x (t) dt ≡ I[0,κ+T1 (ρ)+1],x (g2x ) ≤ LM κ + δ .

2) Using g2x as input function for the controlled initial value problem under investigation, d 0 ¯ g x (t), α0 (t) , α (t) = h 2 dt

α0 (0) = α ,

this system reaches some angle α := α0 (κ + T1 (ρ) + 1) ∈ [0, 2π) within the prescribed time interval [0, κ + T1 (ρ) + 1]. We need to distinguish two cases: Firstly, we will consider the situation in which α has some pre¯ i , . ) is bounded scribed distance to A2 (Ki ); this is the easier case, since h(K away from 0 here. Secondly, we investigate the more intricate situation that ¯ i , . ) has its (unstable) zero. More precisely, α is “near” A2 (Ki ), where h(K let γ > 0 be some small parameter. The first case is defined by the condition that α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ which will be considered in step 3). The second case is given by α ∈ (A2 (Ki ) − γ, A2 (Ki ) + γ) which is the object of step 4). 3) Assume that α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ . ¯ i , α) Since A2 is the repelling switching surface, the mapping α → h(K changes its sign from “−” to “+” at A2 (Ki ); see definition 1.2.4. Since furthermore D ∩ A1 (M) = ∅ (see the beginning of this proof) and A1 and A2 ¯ on any interval of length π, it follows that describe all the zeros of h ¯ i , α) > 0 max h(K {α∈D : α ≥ A2 (Ki )+γ }

and

¯ i , α) < 0 . min h(K {α∈D : α ≤ A2 (Ki )−γ }

Due to the compactness of α ∈ D : α ≥ A2 (Ki ) + γ ∪ α ∈ D : α ≤ A2 (Ki ) − T2 ≡ T2 (γ) > 0 such that for any γ}, there is hence some α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ , the system d 0 ¯ f3 (t), α0 (t) , α (t) = h dt

α0 (0) = α

218


exits D within time T2 , where f3 (t) := Ki for all t ≥ 0. Applying this fact to the original system by considering the control function g3x := g2x f3 : [0, κ + T1 (ρ) + 1 + T2 (γ)] −→ Rd , then

α0 κ + T1 (ρ) + 1 + T2 (γ) ∈ D

for the system α0 with control input g3x . In particular, ϑD (0; x, α; g3x ) ≤ T¯1 , where

T¯1 ≡ T¯1 κ, ρ, γ := κ + T1 (ρ) + 1 + T2 (γ) .

Hereby we have 0

T¯1

K g3x (t), g˙ 3x (t) dt ≡ I[0,κ+T1 (ρ)+1+T2 (γ)],x(g3x ) ≤ LM κ + δ ,

since f3 remains in the equilibrium Ki . 4) Now we concentrate on the second case that α ∈ (A2 (Ki ) − γ, A2 (Ki ) + γ) . ¯ is strongly hypoelliptic on M meaning that By assumption the function h

¯ ∂ h(x, α) = 1 rank ∂xj 1≤j≤d for all (x, α) ∈ M × R due to definition 4.4.4. In particular, applying this property at x = Ki , it follows that it is possible to choose some v1 ≡ vi1 ∈ B(Ki , ρ) \ {Ki } such that ¯ r1 (t) > 0 for all t ∈ (0, |v1 − Ki |] , h where r1 (t) := Ki + t and such that

v1 − Ki |v1 − Ki |

(t ∈ [0, |v1 − Ki |] )

¯ h Xt0,v1 > 0 for all t ≥ 0 .

Note that for the last assertion the gradient assumption b = −∇U is needed which implies that there are no rotations of X 0 around Ki . Correspondingly, it follows that it is possible to choose some v2 ≡ vi2 ∈ B(Ki , ρ) \ {Ki } such


that

219

¯ r2 (t) < 0 for all t ∈ (0, |v2 − Ki |] , h

where r2 (t) := Ki + t and such that

v2 − Ki |v2 − Ki |

(t ∈ [0, |v2 − Ki |] )

¯ h Xt0,v2 < 0 for all t ≥ 0 .

The points v1 ≡ vi1 and v2 ≡ vi2 will be chosen sufficiently close to Ki for all i ∈ {1, . . . , k} in step 5). Consider the juxtaposition rj X 0,vj for j ∈ {1, 2} . By comparison it follows that these two input functions achieve to force the angle to leave D in some finite time T3 > 0 when started at α = A2 (Ki ) (“worst case scenario”). Again by comparison and choosing r1 X 0,v1 or r2 X 0,v2 as control function (depending on α ≥ A2 (Ki ) or α < A2 (Ki ) ), this implies that for any α ∈ A2 (Ki ) − γ, A2 (Ki ) + γ , there is some control function called r3 , r3 ∈ r1 X 0,v1 , r2 X 0,v2 , such that the controlled system d 0 ¯ r3 (t), α0 (t) , α (t) = h dt

α0 (0) = α

has left D at the time T3 > 0 and the corresponding “costs” can be estimated by the previous argument and the additivity of I as 0

T3

K r3 (t), r˙3 (t) dt ≡ I[0,T3 ],Ki (r3 ) (j ∈ {1, 2}) = I[0,|vj −Ki |],Ki (rj ) |vj −Ki | 1 2 ≡ | r˙j (t) + ∇U (rj (t)) | dt 2 0 .2 1 1 + max |∇U | ≤ |vj − Ki | 2 B(Ki ,ρ) ≤ LM |vj − Ki | .

g2x

Juxtaposing the paths g2x and r3 yields an absolutely continuous path g4x := r3 on the time interval [0, T¯2 ], where

220


T¯2 ≡ T¯2 κ, ρ, γ, (vij ) := κ + 1 + T1 (ρ) +

max

i=1,...,k ; j=1,2

|vij − Ki | ;

here, the dependence on γ is implicitly given via the distinction of cases in the second step; furthermore, note that formally the domain of r3 is not bounded; this definition of g4x by juxtaposition is hence to be understood in the sense that r3 ≡ rj X 0,vj follows the trajectory X 0,vj for a sufficiently long time such that g4x is well-defined on [0, T¯2 ]. By the previous construction it follows that ϑD (0; x, α; g x ) ≤ T¯2 4

for the controlled initial value problem d 0 ¯ g x (t), α0 (t) , α (t) = h 4 dt

α0 (0) = α .

The associated “costs” are T¯2 K g4x (t), g˙ 4x (t) dt ≡ I[0,T¯2 ],x (g4x ) 0

= I[0,κ+T1 (ρ)+1],x (g2x ) + I[0,T3 ],Ki (r3 ) ≤ LM κ + δ + LM

max

i=1,...,k ; j=1,2

|vij − Ki | .

5) Using the definitions and results of the previous two cases 3) and 4) we can define T¯ ≡ T¯ κ, ρ, γ, (vij ) := max T¯1 , T¯2

and g :=

g3x g4x

in case 3) in case 4)

for which ϑD (0; x, α; g) ≤ T¯ by definition and the following “worst case estimate” holds

ϑD (0;x,α;g)

K g(t), g(t) ˙ dt

-

0

≤ max

T¯1

K

0

≤ L M κ + δ + LM

g3x (t), g˙ 3x (t) max

dt ,

i=1,...,k ; j=1,2

T¯2

K

0

g4x (t), g˙ 4x (t)

. dt

|vij − Ki |

l drift b := −∇U satisfies the assumptions 2.1.1. Fix an initial value x0 ∈ i=1 Di of Xtε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x0 ; see (2.20). Let M ⊂ R2 either denote the set Fx0 ,ζ as defined in corollary 2.5.7 or the set Hx0 ,ζ as given in corollary 2.5.8 under the additional condition that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) , where the quasipotential values V can be calculated as twice the potential barriers. Assume that the drift

∂2U 1 ∂2U ∂2U ¯ h(x, α) = − sin 2α cos 2α + − ∂x1 ∂x2 2 ∂x21 ∂x22 ¯ ε , αε ) dt , has strictly separated switching for the angle αε of Z ε , dαεt = h(X t t surfaces A1 and A2 on M in the sense of the definitions 4.4.3 and 1.2.4 and ¯ is strongly hypoelliptic on M meaning that suppose that h ¯ ¯ α)

∂ h(x, α) ∂ h(x, = 1 , rank ∂x1 ∂x2


223

for all (x, α) ∈ M × R ; see definition 4.4.4. Then it follows for any initial value z0 ∈ Rn \ {0} of Z ε that P 1 logZTε (ε) ( . , x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) − → ε→0 T (ε) ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) ≤ 0, =Q 0 0 where 2 2 2 ¯ α) = − ∂ U cos2 α − ∂ U sin2 α − 2 ∂ U sin α cos α Q(x, 2 2 ∂x1 ∂x2 ∂x1 ∂x2

and where Kµ(x0 ,ζ) denotes the metastable state corresponding to x0 and ζ ; see definition 2.5.4. This limit Λ1 Kµ(x0 ,ζ) is the local Lyapunov exponent of Z ε with respect to x0 , ζ and z0 . This number is non-positive here, since the curvatures of U at the local minimum Kµ(x0 ,ζ) are non-negative; in other words, the eigenvalues of the Hesse matrix of U at Kµ(x0 ,ζ) are non-negative and hence the eigenvalues Λi Kµ(x0 ,ζ) , i ∈ {1, 2}, of the negative Hessian A(Kµ(x0 ,ζ) ) ≡ −HU (Kµ(x0 ,ζ) ) are non-positive at Ki . 2 Example 4.4.14 (Diagonal matrix plus small skew-symmetric perturbation). Consider the system (1) with coefficient matrix of example 1.5.4, . . . 0 1 λ1 x λ1 0 +x = , A(x) = 0 λ2 −x λ2 −1 0 where λ1 > λ2 , n = 2, d = 1 and with X ε being a one-dimensional diffusion defined by (2.2), √ dXtε = − U (Xtε ) dt + ε dWt . > Again suppose that b = −U satisfies 2.1.1; fix an initial value x0 ∈ li=1 Di of Xtε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a finite exceptional set depending on x0 ; see (2.20). Let M ⊂ R either denote the set Fx0 ,ζ as defined in corollary 2.5.7 or the set Hx0 ,ζ as given in corollary 2.5.8 under the additional condition that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) . Assume that the metastable state corresponding to x0 and ζ is Kµ(x0 ,ζ) = 0

and that M ⊂

λ1 − λ2 λ2 − λ1 , 2 2

224


Fig. 4.6 The drift vector field ¯ h(x, α) = − x + 12 (λ2 −λ1 ) sin 2α and the sublimiting statistics of X ε,x0

entailing that the drift ¯ α) = − x + λ2 − λ1 sin 2α h(x, 2 for the angle αεt has strictly separated switching curves A1 and A2 on M; see ¯ is a priori strongly hypoelliptic, since figure 4.6 . Note that h ¯ α) ∂ h(x, = −1 . ∂x As ¯ α) = λ1 + (λ2 − λ1 ) sin2 α Q(x, and hence ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) = Q ¯ 0, A1 (0) = Q(0, ¯ 0) = λ1 , Λ1 Kµ(x0 ,ζ) ≡ Q 0 0 it follows altogether that for any initial condition z0 ∈ Rn \ {0} of Z ε , 1 P log ZTε (ε) ( . , x0 , z0 ) −−− −→ λ1 . ε→0 T (ε)


225

This limit is the local Lyapunov exponent of Z ε with respect to x0 , ζ and z0 . 2 Remark 4.4.15. The stochastic convergence P 1 ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) log ZTε (ε) (., x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) = Q − → 0 0 ε→0 T (ε) which had been verified under certain assumptions in theorem 4.4.8 and which had been applied to the previous two examples can again be rewritten as in section 4.3: Let the time scales be explicitly given by T (ε) := eζ/ε and then define ε(t) := T −1 (t) =

ζ log t

(t > 0)

for a scaling parameter ζ > 0 which connects the large parameters t = T (ε) and 1ε . The above convergence towards the local Lyapunov exponent can thus be rewritten as ε(t) 1 P log Zt ( . , x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) − → t→∞ t for any initial value z0 ∈ Rn \ {0} of Z ε .

2

Remark 4.4.16 (Stability). In this exposition we are concerned with the asymptotic behavior of the linear system Z ε solving (1) and the conclusion of our work is that on certain time scales T (ε) ! eζ/ε this system realizes growth rates (local Lyapunov exponents) given by the spectrum of A(Kµ(x0 ,ζ) ). It is therefore natural to call Z ε (un)stable at x0 with respect to the time scale parameter ζ, if the real parts of the eigenvalues of A(Kµ(x,ζ) ) are strictly negative (positive). Due to the standing assumption (K) the local Lyapunov exponents of the linearized systems (1.33) dZtε = Db (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ dWt are non-positive as has already been mentioned in the penultimate example of a linearized two well potential. Therefore, such systems cannot be unstable with respect to a time scale parameter ζ. Note that this concept of stability is different from the measure of stability as introduced by Freidlin and Wentzell [Fr-We 98, p.364] who do not consider the linearized system, but measure stability by means of a quasipotential barrier thus calculating the exit time asymptotics.

226


4.5 Concluding remarks Finally we would like to comment on possible generalizations, extensions and limitations of the previously obtained results: Remark 4.5.1. It might be interesting to consider instead of dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

the more general system dZtε = A (Xtε ) Ztε dt +

√ i ε B (Xtε ) Ztε ◦ dWti i

dXtε = b (Xtε ) dt +

√ ε σ (Xtε ) ◦ dWt

(4.26)

where B i are further n × n matrices; note that such a system arises for example when linearizing the SDE (2.1) for X ε as in (1.33) in case that σ is not a constant matrix. An appropriate assumption on B i might be that all these mappings are bounded and the question is whether the previous results are still valid then. Certainly our argumentation fails, since it crucially uses and copes with the degenerate SDE (1). In the more general case there would appear more stochastic terms instead of “0” in equation (4.4), which need to be taken care of; this “0” is a consequence of the rectifiability of Z ε , due to (1). It is remarkable that if b ≡ 0 and also σ ≡ 0 in this case, then the “time-scale-diffusion-part” of the original stochastic dynamical system is not observable any more, but only produces the constant solutions X ε,x = x, and setting A := A(x) and B := B(x) the system reduces to the linear white noise SDE √ i dZt = A Zt dt + ε B Zt ◦ dWti . i

This is the SDE which has been investigated e.g. by Khasminskii [Kh 80], Nishioka [Nk 76], B¨ ohme [Bm 80], Auslender and Milshtein [Al-Mi 82], Pardoux and Wihstutz [Pd-Wh 88] and Imkeller and Lederer [Im-Ld 99] and [Im-Ld 01]. These authors mostly consider cases where A is given in normal form and expand the Lyapunov exponent around the exponent of the t corresponding deterministic system dz dt = Azt in orders of ε. In this setting it seems challenging to investigate, if there is a “wide band limit of local Lyapunov exponents”: More precisely, it is known that white noise SDEs can be considered as limit in law, the wide band noise limit, of a suitably scaled sequence of solutions of a real noise SDE; see e.g. Pardoux [Pd 86], Blankenship and Papanicolaou [Bls-Pp 78], Kushner [Ks 82b] and [Ks 82a] and the papers by Wong and Zakai. The question then is what can

4.5 Concluding remarks

227

be inferred in this limit for the growth rate on time scales, if such an object exists in the respective setting. Remark 4.5.2 (exponential time scales). In the preceding exposition we exclusively used exponential time scales T (ε) ! eζ/ε . It would be interesting to consider the case of polynomial scales T (ε) := ε−p ,

p>0;

here, one gets that ζ := limε→0 ε log T (ε) = 0 which is not an admissible parameter ζ; see (2.20). Hence, the results by Freidlin and Wentzell are not available. Furthermore, the final step 5) in the proofs of theorems 4.4.6 and 4.4.7 fails in case that “ζ = 0”. Remark 4.5.3. A worthwhile task would be to try to get rid of the assump¯ shall have strictly separated switching curves tion that the drift function h on M (= Fx0 ,ζ or Hx0 ,ζ ). E.g. consider figure 4.6 for the situation of example 1.5.4 and suppose that

λ1 − λ2 λ2 − λ1 M ⊂ , . 2 2 Then X ε can leave M on the time scales under consideration thus enabling αε to perform rotations. It is not certain what happens in this case, but one then needs to prove generalizations of theorems 4.4.6 and 4.4.7 for the sojourn times of αε . At best this would lead to results on sublimiting distributions for the angle process. However, such assertions are not in sight for two reasons: Firstly, it is not sure how Hern´ andez-Lerma’s theorem 3.1.7 can then be applied, since one additionally needs to take care of its assumption 3.1.3 (d) which had been eluded by appropriately truncating the drift in the proofs of theorems 4.4.6 and 4.4.7. This problem might be tackled by means of a Markov chain argument. But secondly, also the step 4) in the proofs of theorems 4.4.6 and 4.4.7 break down: Here, the crucial argument is to identify the length of the sojourn time of αε in the set A2 (M) with its exit time from D ⊃ A2 (M); this reasoning is not true, if αε can reenter A2 (M) after leaving this set. Furthermore, note that the condition of separated switching curves resembles the controltheoretic findings of Kliemann [Kl 80]7 , if one assumes that the noise process X takes its values exclusively in M: For strictly separated switching curves A1 and A2 , the set A2 (M) is a “weak control set” implying transience and A1 (M) is a “strong control set” entailing recurrence. Therefore, one might also interpret the assertions of our theorems 4.4.6 and 4.4.7 as transience of αε on the time scale T (ε). Likewise we call αε 7 Kliemann [Kl 80, Ch.6 & 14] focuses on linear systems in dimension n = 2 as we do in this exposition; control-theoretic investigations of linear systems in dimension n = 3 are undertaken by Sommer [So 81].

228


recurrent on the time scale T (ε), if the switching curves are not separated. However, even in the familiar context of recurrence (as t → ∞) it is not possible to calculate the expected average of the sojourn time of a process in T a certain set S, T1 0 P (t, x, S)dt, since the latter integral diverges as T → ∞ without further information on the speed of this divergence available; see e.g. the description of the dichotomy of recurrence and transience by Kliemann [Kl 87, Prop.3.1] and [Kl 83a] and also cf. Arnold and Kliemann [Ar-Kl 87b] as well as Roynette [Ry 75]. Therefore, one is also tempted to be pessimistic, if such an average of the sojourn time can be calculated on time scales for a process αε which is recurrent on the time scale in the above sense. Hence, a generalization to the case of non-separated switching curves seems to be an intricate task. It might be instructive to start with performing numerical simulations in this general situation. For a discussion of the invariant measure for degenerate diffusions and the emerging problems see Stettner [Ste 89]. The invariant measure of the harmonic oscillator with real Markovian noise in the domain where the oscillator has no switching curve at all is calculated by R¨ umelin [Rm 78] and [Rm 79]. Remark 4.5.4. Consider the real noise driven linear differential system dZˆt = A (Xt ) Zˆt dt dXt = − ∇U (Xt ) dt +

ε(t) dWt ,

where A ∈ C(Rd , Rn×n ), ε(t) → 0 is a time dependent noise intensity, W is a Wiener process in Rd and U is a potential function. In other words, the process X ε in (1) has been replaced by a simulated annealing process; see remark 2.5.12. The task here is to give conditions under which the exponential ˆ growth rate of Z, 1 log Zˆt ( . , x0 , z0 ) t converges (e.g. in probability) to Λ1 A(S) , where S denotes the global minimum of U . Since S is necessarily a metastable state for the process X ε defined by (2.2) for this choice of the potential U , on might then prove that “the local Lyapunov exponent is the Lyapunov exponent of Zˆ ”. Remark 4.5.5. This exposition has been concerned with exponential growth rates on time scales. A different task is to investigate the convergence of the Lyapunov spectrum Λε1 , Λε2 , . . . , Λεn as given by the Multiplicative Ergodic Theorem 1.3.1 in the limit as ε → 0. Such a convergence result for Lyapunov exponents has been obtained by Ledrappier and Young [Le-You 91] for the system (4.26) in the linearized i i case, A = Db and B = (Dσ) , under the assumptions that the εvector fields b( . ) and σ( . )ei i=1,...,d are divergence free, the state space of X is compact

4.5 Concluding remarks

229

and the equation for Z ε in the system (4.26) is elliptic. It should be interesting to bridge the gap to their arguments. However, difficulties will likely appear, since our derivation of the local Lyapunov exponent is not based on a modification of the proof of the Multiplicative Ergodic Theorem.

Notations

K Z R+ R>0 N N0 s∧t d x x˙ = dt U =

d dx

U

the set R of real numbers or the set C of complex numbers set of integers {t ∈ R : t ≥ 0} {t ∈ R : t > 0} set of strictly positive integers N ∪ {0} min(s, t) ; minimum of two real numbers s and t differentiation of a function x with respect to the time variable t differentiation of a function U (x) with respect to the space variable x ⎛ ∂U ⎞ ∂x1

∇U , ∇x U

⎜ .. ⎟ ⎝ . ⎠; gradient of the function U with respect to the ∂U ∂xd

∆f , ∆x f

a := b , b =: a a≡b f (x) ≡ c !

multi-dimensional space variable x = (x1 , . . . , xd ) ∂2 f ∂2f + · · · + ∂x 2 ; Laplacian of the real-valued function f ∂x21 d with respect to the multi-dimensional space variable x = (x1 , . . . , xd ) a is defined by b a equals b by definition the function f attains the constant value c for all x T (ε) ! eζ/ε denotes logarithmic equivalence of a function T (ε) (which is defined for small values of ε, i.e. T : (0, ε0 ) → R>0 ) to eζ/ε in the sense that lim ε log T (ε) = ζ; see p.100 ε→0

stochastic independence of two objects with respect to a prescribed measure P

−−−−→ w

−−−−→

convergence in probability weak convergence (convergence in distribution) 231

232

(α → β) , | | ∞ , [T1 ,T2 ]

1M ∂M M M◦ t

Notations

arrow with initial point α and endpoint β; see p.92 scalar product of the underlying space Kn (Rn or Cn ) norm of the underlying space Kn (Rn or Cn ) operator norm supremum (maximum) norm on the space of (continuous) functions f : [T1 , T2 ] → Rn ; f ∞ ≡ f [T1 ,T2 ] := supt∈[T1 ,T2 ] |f (t)| indicator function of the set M topological boundary of the set M topological closure of the set M topological interior of the set M max{k ∈ N0 : k ≤ t}; integer part of t ∈ R+ (“Gaußbracket” or “floor” of t)

Im z Re z z¯

imaginary part of the complex number z real part of the complex number z complex conjugate of the number z ¯ of αε ) (exception: drift function h t

a

σ σ ∗ ; εa is the covariance matrix of X ε in the SDE (2.1) σ ˆσ ˆ ∗ corresponding to the SDE (3.1) for X ε small parameter in the Jordan-form matrix calculations; see the proofs of theorems 1.4.3 and 4.1.2 angle of the solution Ztε of the linear differential system (1); characterized by the differential equation (1.6) attracting switching curve of the angle processes under consideration; see p.18f. repelling switching curve of the angle processes under consideration; see p.18f. system matrix of the SDE (1) given by a continuous function A : Rd → Kn×n adjoint operator (matrix) for an operator A

a ˆ a>0 αεt A1 A2 A A∗ b ˆb B(X) B(x, r) Br (x)

drift coefficient of the SDE (2.1) for (Xtε )t≥0 (component of the) drift coefficient of the SDE (3.1) for (Xtε , Ytε )t≥0 Borel-σ-algebra on the topological space X { y ∈ Rd : |x − y| < r}; open ball with center x and radius r { y ∈ Rd : |x − y| ≤ r}; closed ball with center x and radius r

Notations

M C k (M, N )

C(M, N ) C b (Rd , R) C c (Rd , R) Cx (J, M ) C (k) d D D Di δij δx (B) det(A) dist(x, B) div b e 1 , . . . , ed ε≥0 E(f ) EF Ex E(C) F F FX Fx0 ,ζ

233

E \ M ; set-theoretical complement of M ⊂ E {f : M → N continuous, all derivatives of f of order up to k exist and are continuous } for differentiable manifolds M, N and k ∈ N0 C 0 (M, N ) {f ∈ C(Rd , R) : f bounded} {f ∈ C(Rd , R) : supp(f ) compact} {f : J → M continuous, f (0) = x} for an interval J ⊂ R, 0 ∈ J, a differentiable manifold M and x ∈ M set of k-cycles; see p.93ff. dimension of the state space of (Xtε )t≥0 as defined in (2.1) bounded, open domain in Rd to which the exit time investigations in the non-degenerate case apply; see 2.4.1ff. bounded, open domain in Rm to which the exit time investigations in the degenerate case apply; see 3.1.1ff. t→∞ {x ∈ Rd : Xt0,x −−−−→ Ki }; the domain of attraction of Ki under the deterministic motion X 0 ; see p.79 1 if i = j and 0 otherwise; Kronecker symbol 1 if x ∈ B and 0 otherwise; Dirac measure at x determinant of the matrix A inf y∈B |x − y| ; distance between the point x ∈ Rd and the set B ⊂ Rd ∂bd ∂b1 ∂x1 + · · · + ∂xd ; divergence of the vector-valued function b canonical unit vectors in Rd parameter of the noise intensity in SDEs (1), (2.1) and (3.1) f dP; expected value of the function f with respect to the probability measure P conditional expectation with respect to the σ-algebra F expected value with respect to the probability measure Px exit rate of the cycle C; see p.93ff. (component of the) drift coefficient of the SDE (3.1) for (Xtε , Ytε )t≥0 continuous mapping between separable metric spaces “pushing forward” the LDP; see p.68 σ-algebra generated by the random variable X { x ∈ Rd : V (x0 , x) ≤ ζ }; the set to which X ε is asymptotically constrained on the time scale T (ε); see corollary 2.5.7

234

g GW (L) Gi (L) Γ Gε G, Gε G0 , G+ , G+ T GL(n, K)

h(x, ψ) ¯ α) h(x,

Hx0 ,ζ

HU

H1 ≡ H1 ([0, T ], Rd)

H(s, x, p)

Notations

graph on L, i.e. a set of arrows; see p.92 the set of W -graphs on L; see p.92f. the set of i-graphs on L; see p.92f. parameter for the (scaled) time horizon; see p.102ff. 4d 4d ε ∂ ∂2 i=1 bi ∂xi + 2 i,j=1 aij ∂xi ∂xj ; generator of the diffusion process (Xtε )t≥0 ; see p.53 perturbations of the system matrix A point sets corresponding to the data of the SDE (3.1); see p.128ff. general linear group, consisting of the invertible n × n matrices with entries in K A(x)ψ − A(x)ψ , ψ ψ ; drift function of the direction ψtε of the solution Ztε of the linear differential system (1); see RDE (1.4) 2 2 −a 12 (x) sin α + a 21 (x) cos α + a22 (x) − a11 (x) sin α cos α ; drift function of the angle αεt of the solution Ztε of the linear differential system (1); see RDE (1.6) { x ∈ Rd : V (Kµ(x0 ,ζ) , x) ≤ ζ } ; the set where X ε gets asymptotically stuck on the time scale T (ε); see 2.5.8 corollary ∂2 U ∂xi ∂xj

; Hesse matrix of U , i.e. the i,j=1,...,d

symmetric matrix of second derivatives of the potential function U ∈ C 2 (Rd , R) T 0

g(s) ds : g ∈ L2 ([0, T ], Rd) ; the space of absolutely continuous functions starting in 0 with L2 -derivative 4d ˆ 4d 1 îj (s, x) pi pj ; i=1 bi (s, x) pi − 2 i,j=1 a see p.135

In = idKn I : E → [0, ∞]

identity operator on Kn (unit matrix) rate function, defined on a separable metric space (mostly a function space) E; see p.68ff.

J

index singling out the (local) Lyapunov exponent in the list of eigenvalues; see p.38ff. and p.160ff. cycle following after the cycle C; see p.93ff.

J(C) K 1 , . . . , Kl Kl+1 , . . . , Kl Kµ(x0 ,ζ)

stable attractors of X 0 unstable attractors of X 0 metastable state for the initial value x0 and the time scale T (ε) ! eζ/ε ; see definition 2.5.4

Notations

K(s, x, q)

l L L L(x) Lp ([a, b], Rd )

Lε Lˆε

Λ1 (A) , . . . , Λn (A)

M (C) m(C) µ

n N (x), N (y) O (Ω, F , P) p ≡ p(A) Ps,z

235

2 [ q − ˆb(s, x)] ; dual function of H(s, x, p) ; see 3.1.2 and p.135 1 ˆ(s, x)−1/2 2 a

number of the stable attractors Ki of X 0 {1, . . . , l} ; enumeration representing the set of stable attractors {K1 , . . . , Kl } of X 0 ; see p.91 Lebesgue measure vector field such that b(x) = −∇U (x) + L(x) and L(x) ⊥ ∇U (x) , where such a decomposition exists space of (equivalence classes of the) Rd -valued, Lebesgue-measurable functions on [a, b] such that their norm is p−integrable with respect to the Lebesgue measure L ∂ G ε + h ∂s ; generator of the coupled (Markov) process (Xtε , ψtε ) ; see p.14 4m 4d ˆ ∂ 2 ε 4d ∂ îj ∂x∂i ∂xj + i=1 Fi ∂y ; i=1 bi ∂xi + 2 i,j=1 a i generator of the coupled (Markov) process (Xtε , Ytε ) ; see p.128 eigenvalues (characteristic roots, i.e zeros of the characteristic polynomial in the complex plane) of the n × n-matrix A main state of the cycle C; see p.93ff. stationary distribution rate of the cycle C; see p.93ff. the index function which assigns the respective metastable state Kµ(x0 ,ζ) to the initial value x0 and to the time scale parameter ζ ; see (2.21) dimension of the state space of (Ztε )t≥0 as defined in (1) outer normal vector to ∂D at x or to ∂D at y, respectively stable attractor of X 0 in the first exit time investigations (element of {K1 , . . . , Kl }); see p.78ff. underlying probability space number of distinct values in the set of real parts of eigenvalues {Re Λ1 , . . . , Re Λn } of the matrix A law of the stochastic process Z conditioned to start in z at time s ≥ 0; Ps,z {Z ∈ M } = P{Zs = z ∧ Z ∈ M }

236

Pz P (t, x, A), Ptε (x, A) pεt (x, y) pε (x) P n−1 pr[0,T ] φ Φε ψtε

Ψ P ε (s, x, y) Qε (s, x, y) Q 0 , Q , Q1 Q(x, ψ) ¯ α) Q(x,

Notations

P0,z ; law of Z conditioned to start in z at time 0 Markov transition probabilities density of Ptε (x, A) density of ρε ; see section 2.2 {s ∈ Rn : |s| = 1}/{s = −s}; projective space in Rn pr[0,T ] : (Rd )[0,∞) → (Rd )[0,T ] , pr[0,T ] (f ) := f [0,T ] ; restriction map on the function space prefactor of the Verhulst comparison ODE; see 4.3.3f. cocycle solution of (1.1); see section 1.3 Ztε n−1 spherical component (direction) of the |Ztε | ∈ S ε solution Zt of the linear differential system (1); characterized by the differential equation (1.4) set on which large deviation estimate is supposed to hold; see p.68, p.74 inclusion probability; see p.130 exit probability; see p.130 for the SDE (3.1); see p.128ff. parameter domains A(x)ψ , ψ ; integral function identifying the modulus of the solution Ztε of the linear differential system (1); see (1.3) and (1.5) a11 (x)cos2 α + a22 (x) sin2 α + a12 (x) + a21 (x) sin α cos α ; integral function identifying the modulus of the solution Ztε of the linear differential system (1) in dimension n = 2; see (1.7) and (1.8)

R(C) ρε εt

rotation rate of the cycle C; see p.93ff. stationary distribution of X ε ; see section 2.2 | Ztε | ∈ (0, ∞) radial component (modulus) of the solution Ztε of the linear differential system (1); characterized by the differential equation (1.3)

s S n−1 Sr (x)

time parameter { ψ ∈ Rn : |ψ| = 1}; unit sphere of Rn { y ∈ Rd : |x − y| = r}; sphere with center x and radius r

σ σ ˆ

noise coefficient of the SDE (2.1) for (Xtε )t≥0 noise coefficient of the SDE (3.1) for (Xtε , Ytε )t≥0

Notations

237

σ(M) σρε,x

σ-algebra generated by a family M of sets or functions inf{t ≥ 0 : Xtε,x ∈ Bρ (O) ∪ ∂D}; first hitting time of X•ε,x of either ∂D or a small neighborhood of O; see 2.4.9

t ε,x τ ε,x ≡ τD

trace(A)

time parameter inf{t ≥ 0 : Xtε,x ∈ / D}; first exit time of X•ε,x from a bounded, open domain D in Rd with smooth boundary ∂D; see 2.4.1 inf{t ≥ 0 : Ytε,x ∈ / D}; first exit time of Y•ε,y from a bounded, open domain D in Rm with smooth boundary ∂D; see 3.1.1 stopping times in the proof of the exit time law; see p.86 T ≥ 0; fixed time horizon T (ε) ! eζ/ε ; time scale on which the respective sublimiting distribution is observed; see theorem 2.5.5 trace of the matrix A

U

U ∈ C ∞ (Rd , R); potential function

(Wt )t≥0

Brownian motion in Rd over a complete probability space (Ω, F , P)

x, x0

ξtε

space variable (element in Rd ) ; initial point of the stochastic processes X, X ε and X ε ; for X ε in (2.1), the >l initial condition is taken from i=1 Di , a set of full Lebesgue measure, in order to have the metastability results available tan αεt ; characterized by the differential equation (1.9)

y0

initial point of the stochastic processes Y and Y ε

ζ

parameter which determines the time scale T (ε) ! eζ/ε ; for the index function µ(x0 , ζ) to be well defined, ζ must not be contained in a finite exceptional set depending on x0

X ≡ (Xt )t≥0 X ε ≡ (Xtε )t≥0

real noise process diffusion process of Freidlin-Wentzell type as defined in (2.1) diffusion process driving the SDE (3.1) solution process of a real noise SDE driven by X solution process of the real noise SDE (3.1) driven by X ε

T ε,y ≡ TDε,x ε,x ε,x , θm+1 τm T T (ε)

X ε ≡ (Xtε )t≥0 Y ≡ (Yt )t≥0 Y ε ≡ (Ytε )t≥0

238

Z ε ≡ (Ztε )t≥0 X•ε , Z•ε , . . .

Notations

solution process of the linear real noise SDE (1) driven by X ε t → Xtε or Ztε , . . . ; path mappings of the respective processes

LDP ODE PDE RDE RDS RVDE RVDI SDE

large deviation principle ordinary differential equation partial differential equation random differential equation random dynamical system random Riccati-/Verhulst-type differential equality random Riccati-/Verhulst-type differential inequality stochastic differential equation

cf. e.g. et al. i.e. p.

confer for example (exempli gratia) and others (et alii) that is (id est) page(s)

Bibliography

[Ab-Brw-Ke 91]

[Ab-Brw-Ke 92]

[Ah-Mr 78] [A¨ı 96] [A¨ı 99]

[Ad-Br 04]

[And-Pon-Vi 33]

[Ar 79]

[Ar 84]

[Ar 88]

H. Abarbanel, R. Brown & M. Kennel. Variation of Lyapunov exponents on a strange attractor. J. Nonlinear Sci., 1(2):175–199, 1991. H. Abarbanel, R. Brown & M. Kennel. Local Lyapunov exponents computed from observed data. J. Nonlinear Sci., 2(3):343–365, 1992. G. Ahmadi & A. Morshedi. On the stability of stochastic chemical systems. J. Franklin Inst., 306(1):77–86, 1978. Y. A¨ıt-Sahalia. Testing Continuous-Time Models of the Spot Interest Rate. Rev. Fin. Stud., 9(2):385–426, 1996. Y. A¨ıt-Sahalia. Transition Densities for Interest Rate and Other Nonlinear Diffusions. J. Finance, 54(4):1361–1395, 1999. J. Anderluh & S. Borovkova. Commodity volatility modelling and option pricing with a potential function approach. In Second European Deloitte Conference in Risk Management Research, pages 1-16. University of Antwerp, Antwerp, 2004. A. Andronov, L. Pontryagin & A. Vitt. On the statistical treatment of dynamical systems (in Russian). Zh. Eksper. Teoret. Fiz., 3(3):165–180, 1933. English translation: On the statistical treatment of dynamical systems. In Noise in nonlinear dynamical systems, Vol. 1, Theory of continuous Fokker-Planck systems (Eds: F. Moss & P. McClintock), pages 329–348. Cambridge University Press, Cambridge, 1989. L. Arnold. A new example of an unstable system being stabilized by random parameter noise. Plenary lecture at the Bremen Conference on Chemistry 1978, Inform. Communication of Math. Chemistry, 7:133–140, 1979. L. Arnold. A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math., 44(4):793–802, 1984. L. Arnold. Lyapunov exponents of nonlinear stochastic systems. In Nonlinear stochastic dynamic engineering systems. Proceedings IUTAM symposium, Innsbruck/Igls 1987 (Eds: F. Ziegler and G. Schuëller), pages 181–201. Springer-Verlag, Berlin, Heidelberg, 1988.

239

240 [Ar 98]

Bibliography

L. Arnold. Random Dynamical Systems. Springer-Verlag, Berlin, Heidelberg, 1998. [Ar-Hh-Lf 78] L. Arnold, W. Horsthemke & R. Lefever. White and coloured external noise and transition phenomena in nonlinear systems. Z. Phys. B, 29:367–373, 1978. [Ar-Im 00] L. Arnold & P. Imkeller. The Kramers oscillator revised. In Stochastic Processes in Physics, Chemistry, and Biology (Eds: J. Freund and T. P¨ oschel), pages 280–291. SpringerVerlag (Lect. Notes Phys. 557), Berlin, Heidelberg, New York, 2000. [Ar-Kl 83] L. Arnold & W. Kliemann. Qualitative theory of stochastic systems. In Probabilistic analysis and related topis, Vol. 3 (Ed: A.T. Bharucha-Reid), pages 1–79. Academic Press, New York, 1983. [Ar-Kl-Oe 86] L. Arnold, W. Kliemann & E. Oeljeklaus. Lyapunov exponents of linear stochastic systems. In Lyapunov exponents. Proceedings, Bremen 1984 (Eds: L. Arnold and V. Wihstutz), pages 85–125. Springer-Verlag (Lect. Notes Math. 1186), Berlin, Heidelberg, New York, 1986. [Ar-Kl 87a] L. Arnold & W. Kliemann. Large deviations of linear stochastic differential equations. In Stochastic differential systems. Proceedings, Eisenach 1986 (Eds: H. Engelbert and W. Schmidt), pages 117–151. Springer-Verlag (Lect. Notes Control Inf. Sci. 96), Berlin, Heidelberg, New York, 1987. [Ar-Kl 87b] L. Arnold & W. Kliemann. On unique ergodicity for degenerate diffusions. Stochastics, 21:41–61, 1987. [Ar-Kd 89] L. Arnold & P. Kloeden. Lyapunov exponents and rotation number of two-dimensional systems with telegraphic noise. SIAM J. Appl. Math., 49(4):1242–1274, 1989. [Ar-Oe-Pd 86] L. Arnold, E. Oeljeklaus & E. Pardoux. Almost sure and moment stability for linear Ito equations. In Lyapunov exponents. Proceedings, Bremen 1984 (Eds: L. Arnold and V. Wihstutz), pages 129–159. Springer-Verlag (Lect. Notes Math. 1186), Berlin, Heidelberg, New York, 1986. [Ar-Pp-Wh 86] L. Arnold, G. Papanicolaou & V. Wihstutz. Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications. SIAM J. Appl. Math., 46(3):427–450, 1986. [Ar-Wh 78] L. Arnold & V. Wihstutz. On the stability and growth of real noise parameter-excited linear systems. In Measure theory and its applications. Proceedings, Oberwolfach 1977 (Eds: G. Kallianpur and D. K¨ olzow), pages 211–227. Springer-Verlag (Lect. Notes Math. 695), Berlin, Heidelberg, New York, 1978. [Au-Bo-Cr-Pn-Vu 96] E. Aurell, G. Boffetta, A. Crisanti, G. Paladin & A. Vulpiani. Predictability in systems with many characteristic time scales: The case of turbulence. Phys. Rev. E, 53(3):2337–2349, 1996. [Au-Bo-Cr-Pn-Vu 97] E. Aurell, G. Boffetta, A. Crisanti, G. Paladin & A. Vulpiani. Predictability in the large: An extension of the concept of Lyapunov exponent. J. Phys. A, 30(1):1–26, 1997. [Al-Mi 82] E. Auslender & G. Milshtein. Asymptotic expansions of the Lyapunov index for linear stochastic systems with small noise. J. Appl. Math. Mech., 46:277–283, 1983.

Bibliography [Ba-El-Ny 97]

[Bn-Si 92]

[Bar-Bln 97]

[Bh-Lg-Jh 98]

[Be 69]

[Be-Pt 75]

[Bz-Ma-Mz-Tt 99]

[Bg-Gen 06]

[Bs 33] [Bis 81]

[Bla-Fr 61]

[Bls-Pp 78]

[Bm 80]

[Bo-Gi-Pn-Vu 98]

[Boh-Pet 01]

[Br-Dh-Re-Tu 03]

[Bd-Ct 98]

[Bv-Ec-Gd-Kn 04]

241 B. Bailey, S. Ellner & D. Nychka. Chaos with confidence: Asymptotics and applications of local Lyapunov exponent. Fields Inst. Commun., 11:115–133, 1997. D. Bainov & P. Simeonov. Integral Inequalities and Applications. Mathematics and Its Applications Vol. 57, Kluwer Academic Publishers, Dordrecht, Boston, London, 1992. G. Barles & A.-P. Blanc. Large deviations estimates for the exit probabilities of a diffusion process through some vanishing parts of the boundary. Adv. Differ. Equ., 2(1):39–84, 1997. U. Behn, A. Lange & T. John. Electrohydrodynamic convection in liquid crystals driven by multiplicative noise: Sample stability. Phys. Rev. E, 58(2):2047–2060, 1998. M. Benderskii. Determination of the stability region for a second-order linear differential equation with random coefficients of special form. Differential Equations, 5:1403–1406, 1969. M. Benderskii & L.Pastur. The asymptotic behavior of the solutions of a second order equation with random coefficients (in Russian). Teorija Funkcij Funkcionalnyi Analiz i ich Prilozenija (Charkov), 22:3–14, 1975. R. Benzi, M. Marrocu, A. Mazzino & E. Trovatore. Characterization of the long-time and short-time predictability of low-order models of the atmosphere. J. Atmospheric Sci., 56:3495–3507, 1999. N. Berglund & B. Gentz. Noise-induced phenomena in slow-fast dynamical systems: A sample-paths approach. Springer-Verlag, London, Berlin, 2006. S. Bernstein. On the Fokker-Planck differential equation (in French). C. R. Acad. Sc. Paris, 196:1062–1064, 1933. J.-M. Bismut. A generalized formula of Ito and some other properties of stochastic flows. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 55:331–350, 1981. Y. Blagovescenskii & M. Freidlin. Certain properties of diffusion processes depending on a parameter. Soviet Math. Dokl., 2(3):633–636, 1961. G. Blankenship & G. Papanicolaou. Stability and control of stochastic systems with wide-band noise disturbances I. SIAM J. Appl. Math., 34(3):437–476, 1978. ¨ hme. On the asymptotic behavior of a solution of a sysO. Bo tem of stochastic linear differential equations of second order. Theory Probab. Math. Stat., 20:9–20, 1980. G. Boffetta, P. Giuliani, G. Paladin & A. Vulpiani. An extension of the Lyapunov analysis for the predictability problem. J. Atmospheric Sci., 55(23):3409–3416, 1998. M. Bohner & A. Peterson. Dynamic Equations on Time Scales - An Introduction with applications. Birkh¨ auser, Boston, Basel, Berlin, 2001. S. Borovkova, H. Dehling, J. Renkema & H. Tulleken. A potential-field approach to financial time series modelling. Computational Economics, 22:139–161, 2003. J.-P. Bouchaud & R. Cont. A Langevin approach to stock market fluctuations and crashes. Europ. Phys. J. B, 6:543– 550, 1998 A. Bovier, M. Eckhoff, V. Gayrard & M. Klein. Metastability in reversible diffusion processes I. Sharp asymptotics for

242

[Bv-Gd-Kn 05]

[Bro-Wil 72]

[vB-Ni 93]

[Bu 72] [Ca-Gv-Xu 00]

[Cm-Fr 03]

[Cv 85a]

[Cv 85b] [Cv-Cl-Ew 86]

[Cs-Ga-Ol-Va 84]

[Ce 94] [Cha-Kac-Smo 86]

[Cg-Hw-Sh 87]

[Cu-Kl 93]

[Cu-Kl 96]

[Cu-Kl 97]

Bibliography capacities and exit times. J. Eur. Math. Soc., 6(4):399–424, 2004. A. Bovier, V. Gayrard & M. Klein. Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc., 7(1):69–99, 2005. R. Brockett & J. Willems. Average value criteria for stochastic stability. In Stability of stochastic dynamical systems. Proceedings, Warwick 1972 (Ed: R. Curtain), pages 252–272. Springer-Verlag (Lect. Notes Math. 294), Berlin, Heidelberg, New York, 1972. C. van den Broeck & G. Nicolis. Noise-induced sensitivity to the initial conditions in stochastic dynamical systems. Phys. Rev. E, 48(6):4845–4846, 1993. H. Bunke. Ordinary differential equations with random parameters (in German). Akademie-Verlag, Berlin, 1972. J. Callen, S. Govindaraj & L. Xu. Large time and small noise asymptotic results for mean reverting diffusion processes with applications. Econ. Theory, 16(2):401–419, 2000. S. Carmona & M. Freidlin. On logarithmic asymptotics of stochastic resonance frequencies. Stoch. and Dyn., 3(1):55–71, 2003. A. Carverhill. A formula for the Lyapunov numbers of a Stochastic flow. Application to a perturbation theorem. Stochastics, 14:209–226, 1985. A. Carverhill. Flows of stochastic dynamical systems: Ergodic theory. Stochastics, 14:273–317, 1985. A. Carverhill, M. Chappell & K. Elworthy. Characteristic exponents for stochastic flows. In Stochastic processes - mathematics and physics. Proceedings 1st BiBoSSymposium, Bielefeld 1984 (Eds: S. Albeverio, P. Blanchard and L. Streit), pages 52–80. Springer-Verlag (Lect. Notes Math. 1158), Berlin, Heidelberg, New York, 1986. M. Cassandro, A. Galves, E. Olivieri & M. Vares. Metastable behavior of stochastic dynamics: A pathwise approach. J. Stat. Phys., 35(5/6):603–634, 1984. P. Cessi. A simple box model of stochastically forced thermohaline flow. J. Phys. Oceanogr., 24:1911–1920, 1994. S. Chandrasekhar, M. Kac & R. Smoluchowski. Marian Smoluchowski: his life and scientific work. Polish Scientific Publishers, Warsaw, 1986. T.-S. Chiang, C.-R. Hwang & S.-J. Sheu. Diffusion for global optimization in Rn . SIAM J. Control Optim., 25(3):737–753, 1987. F. Colonius & W. Kliemann. Linear control semigroups acting on projective space. J. Dynam. Differential Equations, 5(3):495–528, 1993. F. Colonius & W. Kliemann. The Lyapunov spectrum of families of time-varying matrices. Trans. Amer. Math. Soc., 348(11):4389–4408, 1996. F. Colonius & W. Kliemann. Nonlinear systems with multiplicative and additive perturbation under state space constraints. In Active/Passive Vibration Control and Nonlinear Dynamics of Structures. Proceedings, Dallas (TX) 1997 (Eds: W. Clark, X. Xie, D. Allaei, Y. Hwang, N. Sri Namachchivaya

Bibliography

[Cu-Kl 00] [Cp 65]

[Cra 84] [Day 90]

[Day-Dar 85]

[De-Zt 98] [Du-Ks 89]

[Ea 89]

[Ed-Fo-Tm 91]

[Ev 02]

[Fl 77]

[Fl 78] [Fl 80]

[Fl-Ris 75]

[Fr 68]

[Fr 69] [Fr 73a]

[Fr 73b]

243 and O. O’Reilly), pages 131–142. Amer. Soc. Mech. Engineering ASME (ASME DE-Vol. 95, AMD-Vol. 223), New York, 1997. F. Colonius & W. Kliemann. The Dynamics of Control. Birkh¨ auser, Boston, 2000. W.A. Coppel. Stability and asymptotic behavior of differential equations. Heath Mathematical Monographs, Boston, 1965. H. Crauel. Lyapunov numbers of Markov solutions of linear stochastic systems. Stochastics, 14:11–28, 1984. M. Day. Large deviations results for the exit problem with characteristic boundary. J. Math. Anal. Appl., 147(1):134– 153, 1990. M. Day & T. Darden. Some regularity results on the VentcelFreidlin Quasi-Potential Function. Appl. Math. Optim., 13(1):259–282, 1985. A. Dembo & O. Zeitouni. Large Deviations Techniques and Applications. 2nd edition, Springer-Verlag, New York, 1998. P. Dupuis & H. Kushner. Minimizing exit probabilities: A large deviations approach. SIAM J. Control Optim., 27(2):432–445, 1989. M. Eastham. The asymptotic solution of linear differential systems: Applications of the Levinson theorem. Clarendon Press, Oxford, 1989. A. Eden, C. Foias & R. Temam. Local and Global Lyapunov Exponents. J. Dynamics and Diff. Equations, 3(1):133–177, 1991. L. Evans. Partial Differential Equations. Graduate Studies in Mathematics Vol. 19, 2nd edition, American Mathematical Society, Providence (RI), 2002. W. Fleming. Inclusion probability and optimal stochastic control. Analyse et contrˆ ole de syst` emes (IRIA Sem., Rocquencourt), 6:53–61, 1977. W. Fleming. Exit probabilities and optimal stochastic control. Appl. Math. Optim., 4(4):329–346, 1978. W. Fleming. Exit probabilities for diffusions depending on a small parameter. In Analysis and optimization of stochastic systems. Proceedings, Oxford 1978 (Eds: O. Jacobs, M. Davis, M. Dempster, C. Harris and P. Parks), pages 141–145. Academic Press, London, 1980. W. Fleming & R. Rishel. Deterministic and Stochastic Optimal Control. Springer-Verlag, Berlin, Heidelberg, New York, 1975. M. Freidlin. On the Factorization of non-negative definite matrices. Th. of Probability and its Appl., 13(2):354–356, 1968. M. Freidlin. On degenerating elliptic equations. Th. of Probability and its Appl., 14(1):137–141, 1969. M. Freidlin. On stability in case of random perturbations (in German). Wiss. Z. Techn. Univ. Dresden, 22(2):283–286, 1973. M. Freidlin. Probabilities of large deviations for randomly perturbed dynamic systems and stochastic stability. Th. of Probability and its Appl., 18(4):779–784, 1973.

244 [Fr 77]

[Fr 00] [Fr-Sv 76]

[Fr-We 98]

[Fri 75]

[Fri 76]

[Fs 66]

[Fu 83]

[Ga-Ol-Va 87]

[Gem-Hw 86] [Ge-Ll 90] [Gen 03]

[Gs-Lh-Pr-Ta 99] [Gs-Lh-Ta 01]

[Gy 84]

[Gb 99]

[Gn 75]

[Gr-Ta 96]

[Gu-Hl 83]

Bibliography M. Freidlin. Sublimiting distributions and stabilization of solutions of parabolic equations with a small parameter. Soviet Math. Dokl., 18(4):1114–1118, 1977. M. Freidlin. Quasi-deterministic approximation, metastability and stochastic resonance. Physica D, 137:333–352, 2000. M. Freidlin & V. Svetlosanov. Effects of small stochastic perturbations on the stability of ecological systems (in Russian). J. General Biol., 37(5):715–721, 1976. M. Freidlin & A. Wentzell. Random Perturbations of Dynamical Systems. 2nd edition, Springer-Verlag, Berlin, 1998. A. Friedman. Stochastic differential equations and applications. Vol. 1. Probability and Mathematical Statistics Series Vol. 28, Academic Press, New York, 1975. A. Friedman. Stochastic differential equations and applications. Vol. 2. Probability and Mathematical Statistics Series Vol. 28, Academic Press, New York, 1976. U. Frisch. On the resolution of stochastic differential equations with Markovian coefficients (in French). C. R. Acad. Sc. Paris S´ er. A, 262:762–765, 1966. H. Fujisaka. Statistical Dynamics generated by fluctuations of local Lyapunov eponents. Progr. Theoret. Phys., 70(5):1264–1275, 1983. A. Galves, E. Olivieri & M. Vares. Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab., 15(4):1288–1305, 1987. S. Geman & C.-R. Hwang. Diffusions for global optimization. SIAM J. Control Optim., 24(5):1031–1043, 1986. G. Gennotte & H. Leland. Market Liquidity, Hedging and Crashes. American Econ. Review, 80(5):999–1021, 1990. B. Gentz. Random Perturbations of dynamical systems. Lecture notes, University of Leipzig and MPI Mathematics in the Sciences Leipzig, 2003. R. Gibson, F-S. Lhabitant, N. Pistre & D. Talay. Interest Rate Model Risk. The Journal of Risk, 1(3):37–62, 1999. R. Gibson, F-S. Lhabitant & D. Talay. Modeling the Term Structure of Interest Rates: A Review of the Literature. Working Paper Nr. 9801, Ecole des HEC, Lausanne University, June 2001. W. Gray. Atlantic seasonal hurricane frequency. Part I: El Ni˜ no and 30 mb quasi-biennial oscillation influences. Monthly Weather Rev., 112(9):1649–1668, 1984. R. Griesbaum. On the stability of dynamical systems with stochastic parameter excitation (in German). PhD thesis, Universit¨ at Karlsruhe, Fortschr.-Ber. VDI Reihe 11 Nr. 281, VDI Verlag, D¨ usseldorf, 1999. A.G. Grin. On perturbations of dynamical systems by regular Gaussian processes. Th. of Probability and its Appl., 20(2):442–443, 1975 and 22(4):881, 1978 (erratum). A. Grorud & D. Talay. Approximation of Lyapunov exponents of nonlinear stochastic differential equations. SIAM J. Appl. Math., 56(2):627–650, 1996. J. Guckenheimer & P. Holmes. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. SpringerVerlag, Berlin, Heidelberg, New York, 1983.

Bibliography [Hg-Wd-Mn 85]

[Hb-Th 94] [Hn-Hd-Jo 98]

[Harr-Lutz 77] [Ha-Wi 55]

[HL 79]

[HL 80]

[HL 81]

[Hi 88]

[H¨ o 67] [Hr 79]

[Hh-Lf 80]

[Hh-Lf 85] [Hui-Mey-Sch 04]

[Hw-Sh 90]

[Ic-Ku 74]

[Il 64] [Il-Kh 65]

245 G. Haag, W. Weidlich & G. Mensch. A macroeconomic potential describing structural change of the economy. Th. and Decision., 19(3):279–299, 1985. W. Hackenbroch & A. Thalmaier. Stochastic Analysis (in German). Teubner Verlag, Stuttgart, 1994. J. Hansen, A. Hodges & J. Jones. ENSO Influences on Agriculture in the Southeastern United States. J. Climate, 11(3):404–411, 1998. W. Harris & D. Lutz. A unified theory of asymptotic integration. J. Math. Anal. Appl., 57:571–586, 1977. P. Hartman & A. Wintner. Asymptotic integrations of linear differential equations. Amer. J. Math., 77(1):45–86 and 404 and 932, 1955. ńdez-Lerma. Lyapunov criteria for stability of difO. Herna ferential equations with Markov parameters. Bol. Soc. Mat. Mexicana (Segunda Serie), 24(1):27–48, 1979. ńdez-Lerma. Exit probabilities for degenerate sysO. Herna tems. In Proceedings of the Ninth IFIP Conference on Optimization Techniques. Proceedings, Warsaw 1979, Part 1, pages 164–168. Springer-Verlag (Lect. Notes Control Inf. Sci. 22), Berlin, New York, 1980. ńdez-Lerma. Exit probabilities for a class of O. Herna perturbed degenerate systems. SIAM J. Control Optim., 19(1):39–51, 1981. S. Hilger. A measure chain calculus with application to center manifolds (in German). PhD thesis, Universit¨ at W¨ urzburg, W¨ urzburg, 1988. ¨ rmander. Hypoelliptic second order differential equaL. Ho tions. Acta Math., 119:147–171, 1967. M.-O. Hongler. Exact time dependent probability density for a nonlinear non-Markovian stochastic process. Helv. Phys. Acta, 52(2):280–287, 1979. W. Horsthemke & R. Lefever. A perturbation expansion for external wide band Markovian noise: Application to transitions induced by Ornstein-Uhlenbeck noise. Z. Phys. B, 40:241–247, 1980. W. Horsthemke & R. Lefever. Noise-induced transitions. Springer-Verlag, Berlin, Heidelberg, New York, 1984. ¨tte. Phase transitions W. Huisinga, S. Meyn & C. Schu and metastability in Markovian and molecular systems. Ann. Appl. Probab., 14(1):419–458, 2004. C.-R. Hwang & S.-J. Sheu. Large-time behavior of perturbed diffusion Markov Processes with applications to the Second Eigenvalue Problem for Fokker-Planck Operators and Simulated Annealing. Acta Appl. Math., 19:253–295, 1990. K. Ichihara & H. Kunita. A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30:235–254, 1974 and 39:81–84, 1977 (supplements and corrections). A.M. Ilin. On a class of ultraparabolic equations. Soviet Math. Dokl., 5:1673–1676, 1964. A.M. Ilin & R.Z. Khasminskii. Asymptotic behavior of solutions of parabolic equations and an ergodic property of nonhomogeneous diffusion processes. Amer. Math. Soc. Transl. Ser. 2, 49:241–268, 1965.

246 [Im-Ld 99]

[Im-Ld 01]

[Im-Mo 02] [In 68] [Ito 88] [Ito-Mik 96]

[Ja 92]

[Je 89] [Kt 80] [Ka-Kv 60]

[Ka-My 02]

[Kh 60]

[Kh 67]

[Kh 80]

[Kk-Hh-Lf 79]

[Kk-Hh-Lf-Ia 80]

[Kl 79]

[Kl 80]

Bibliography P. Imkeller & C. Lederer. An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator. Dyn. Stab. Syst., 14(4):385–405, 1999. P. Imkeller & C. Lederer. Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum. Dyn. Syst., 16(1):29–61, 2001. P. Imkeller & A. Monahan. Conceptual stochastic climate models. Stoch. and Dyn., 2:311–326, 2002. E. Infante. On the stability of some linear nonautonomous random systems. J. Appl. Mech., 35:7–12, 1968. H. Ito. Stochastic Disk Dynamo as a model of reversals of the earth’s magnetic field. J. Stat. Phys., 53(1/2):19–39, 1988. H. Ito & T. Mikami. Poissonian asymptotics of a randomly perturbed dynamical system: Flip-Flop of the Stochastic Disk Dynamo. J. Stat. Phys., 85(1/2):41–53, 1996. S. Jacquot. Asymptotic behavior of the second eigenvalue of Kolmogorov’s process (in French). J. Multivariate Analysis, 40:335–347, 1992. G. Jetschke. Mathematics of self-organization (in German). Deutscher Verlag der Wissenschaften, Berlin, 1989. T. Kato. Perturbation Theory for Linear Operators. 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1980. I. Kats & N. Krasovskii. On the stability of systems with random parameters. J. Appl. Math. Mech., 24:1225–1246, 1960. I. Kats & A. Martynyuk. Stability and Stabilization of nonlinear systems with random structure. Stability and Control: Theory, Methods and Applications Vol. 18, Taylor and Francis, London, New York, 2002. R.Z. Khasminskii. Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Th. of Probability and its Appl., 5(2):179–196, 1960. R.Z. Khasminskii. Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Th. of Probability and its Appl., 12(1):144–147, 1967. R.Z. Khasminskii. Stochastic stability of differential equations. Sijthoff and Noordhoff, Alphen, 1980. Translation of the Russian edition, Nauka, Moscow, 1969. K. Kitahara, W. Horsthemke & R. Lefever. Colourednoise-induced transitions: Exact results for external dichotomous Markovian noise. Phys. Lett. A, 70(5-6):377–380, 1979. K. Kitahara, W. Horsthemke, R. Lefever & Y. Inaba. Phase diagrams of noise induced transitions. Progr. Theoret. Phys., 64(4):1233–1247, 1980. W. Kliemann. Some exact results on stability and growth of linear parameter excited stochastic systems. In Stochastic control theory and stochastic differential systems. Proceedings, Bad Honnef 1979 (Eds: M. Kohlmann and W. Vogel), pages 456–471. Springer-Verlag (Lect. Notes Control Inf. Sci. 16), Berlin, New York, 1979. W. Kliemann. Qualitative theory of nonlinear stochastic systems (in German). PhD thesis, Universit¨ at Bremen, Report 32, Institut f¨ ur Dynamische Systeme, Bremen, 1980.

Bibliography [Kl 83a]

[Kl 83b]

[Kl 87] [Kl 88]

[Kl-Rm 81]

[Ko 34]

[Kr 40]

[Ku 90] [Kz 02]

[Ks 82a]

[Ks 82b]

[Lan 00] [Le-You 91] [Lz 89]

[Lg-SN-Tw 92]

[LC 11]

[Lv 48]

[Li-Qi 98]

247 W. Kliemann. Transience, recurrence and invariant measures for diffusions. In Nonlinear stochastic problems. Proceedings, NATO Adv. Study Inst., Arma¸cao de Pera 1982 (Eds: R. Bucy, & J. Moura), pages 437–454. Reidel (NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 104), Dordrecht, 1983. W. Kliemann. Qualitative theory of stochastic dynamical systems—applications to life sciences. Bull. Math. Biol., 45(4):483–506, 1983. W. Kliemann. Recurrence and invariant measures for degenerate diffusions. Ann. Probab., 15(2):690–707, 1987. W. Kliemann. Analysis of nonlinear stochastic systems. In Analysis and estimation of stochastic mechanical systems, Courses and lectures of the International Centre for Mechanical Sciences, 303 (Eds: W. Schiehlen, & W. Wedig), pages 43–102. Springer-Verlag, Wien, New York, 1988. ¨melin. On the growth of linear sysW. Kliemann & W. Ru tems parametrically disturbed by a diffusion process. Report 27, Institut f¨ ur Dynamische Systeme, Universit¨ at Bremen, 1981. A. Kolmogorov. Random motions - On the theory of Brownian motion (in German). Annals of Math. (2), 35(1):116–117, 1934. H. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4):284–304, 1940. H. Kunita. Stochastic flows and stochastic differential equations. Cambridge Univ. Press, Cambridge, 1990. A. Kunz. Extremes of Multivariate Stationary Diffusions and Applications in Finance. PhD thesis, Technische Universit¨ at M¨ unchen, Munich, 2002. H. Kushner. A cautionary note on the use of singular perturbation methods for “small noise” models. Stochastics, 6:117–120, 1982. H. Kushner. Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: Approximate invariant measures. Stochastics, 6:259–277, 1982. C. Landen. Bond pricing in a hidden Markov model of the short rate. Finance Stochast., 4:371–389, 2000. F. Ledrappier & L.-S. Young. Stability of Lyapunov exponents. Ergod. Th. and Dynam. Syst., 11:469–484, 1991. A. Leizarowitz. On the Lyapunov exponent of a harmonic oscillator driven by a finite-state Markov process. SIAM J. Appl. Math., 49(2):404–419, 1989. G. Leng, N. Sri Namachchivaya & S. Talwar. Robustness of nonlinear systems perturbed by external random excitation. Trans. ASME J. Appl. Mech., 59(4):1015–1022, 1992. T. Levi-Civita. On the linear equations with periodic coefficients and on the average motion of the lunar node (in French). ´ Norm. Sup. 3e S´ Ann. Scient. de l’Ec. erie, 28:325–376, 1911. N. Levinson. The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J., 15(1):111– 126, 1948. Li Yulin & Qian Minping. Hierarchical structure of attractors of dynamical systems. Sci. China, Ser. A, 41(11):1128– 1134, 1998.

248 [Li-Qi 99]

[Lm-Sn 99]

[Lt-Pn-Vu 96]

[Lu 75] [Mar 56]

[Mt-Sc 88]

[My-Zh 85]

[Mo 02]

[M-Bh 87]

[Nev 64]

[Nm-Co-Ki 85] [NgVP 74a]

[NgVP 74b] [Nc-Ni 81] [Nk 76]

[Ol-Va 05] [Os 68]

[Pa 00] [Pd 86]

Bibliography Li Yulin & Qian Minping. The sublimit distribution of small random perturbed dynamic systems. Acta Sci. Nat. Univ. Pekin., 35(1):1–12, 1999. G. Lohmann & J. Schneider. Dynamics and predictability of Stommel’s box model. A phase-space perspective with implications for decadal climate variability. Tellus A, 51:326–336, 1999. V. Loreto, G. Paladin & A. Vulpiani. Concept of complexity in random dynamical systems. Phys. Rev. E, 53(3):2087– 2098, 1996. D. Ludwig. Persistence of dynamical systems under random perturbations. SIAM Rev., 17(4):605–640, 1975. L. Markus. Asymptotically autonomous differential systems. In Contributions to the theory of nonlinear oscillations III. Annals of Mathematical Studies 36 (Ed: S. Lefschetz), pages 17–29. Princeton University Press, Princeton, 1956. F. Martinelli & E. Scoppola. Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition. Commun. Math. Phys., 120(1):25–69, 1988. P.-A. Meyer & W.A. Zheng. Construction of reversible Nelson processes (in French). In S´ emin. de probabilit´ es XIX. Proceedings, Universit´ e de Strasbourg 1983/1984 (Eds: J. Az´ ema and M. Yor), pages 12–26. Springer-Verlag (Lect. Notes Math. 1123), Berlin, Heidelberg, 1985. A. Monahan. Lyapunov exponents of a simple stochastic model of the thermally and wind-driven ocean circulation. Dyn. Atmos. Oceans, 35:363–388, 2002. ¨ller & U. Behn. Electrohydrodynamic instabilities R. Mu in nematic liquid crystals under dichotomous parametric modulation. Z. Phys. B Condensed Matter, 69:185–192, 1987. M. Nevel’son. On the behavior of the invariant measure of a diffusion process with small diffusion on a circle. Th. of Probability and its Appl., 9(1):125–131, 1964. C. Newman, J. Cohen & C. Kipnis. Neo-darwinian evolution implies punctuated equilibria. Nature, 315:400–401, 1985. Nguyen Viet Phu. On a problem of stability under small stochastic perturbations. Moscow Univ. Math. Bull., 29(56):5–10, 1974. Nguyen Viet Phu. Small Gaussian perturbations and Euler’s equation. Moscow Univ. Math. Bull., 29(5-6):52–57, 1974. C. Nicolis & G. Nicolis. Stochastic aspects of climatic transitions - Additive fluctuations. Tellus, 33:225–234, 1981. K. Nishioka. On the stability of two-dimensional linear stochastic systems. Kodai Math. Semin. Rep., 27:211–230, 1976. E. Olivieri & M. Vares. Large deviations and metastability. Cambridge University Press, Cambridge, 2005. V. Oseledets. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197–231, 1968. T. Palmer. Predicting uncertainty in forecasts of weather and climate. Rep. Prog. Phys., 63:71–116, 2000. E. Pardoux. Wide band limit of Lyapunov exponents. In Stochastic differential systems, Proc. 3rd Bad Honnef Conf.

Bibliography

[Pd-Wh 88]

[Pd-Wh 92]

[Pd-Ver 01]

[Pv 02] [Pl 96] [Pl-Sa 95a]

[Pl-Sa 95b]

[Per-Leb 71]

[Pe 29]

[Pk 93] [Pk-Fe 95] [Pi 86] [Pi-Wh 88] [Pi-Wh 91]

[Pu 99] [Ri-Na 55]

[Roc 97]

[Roy 89] [Ry 75]

[Ru 82]

249 1985 (Eds: N. Christopeit, K.Helmes and M. Kohlmann), pages 305–315. Springer-Verlag (Lect. Notes Control Inf. Sci. 78), Berlin, Heidelberg, New York, 1986. E. Pardoux & V. Wihstutz. Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion. SIAM J. Appl. Math., 48(2):442–457, 1988. E. Pardoux & V. Wihstutz. Lyapunov exponent of linear stochastic systems with large diffusion term. Stoch. Processes Appl., 40(2):289–308, 1992. E. Pardoux & A. Veretennikov. On the Poisson equation and diffusion approximation I. Ann. Probab., 29(3):1061–1085, 2001. I. Pavlyukevich. Stochastic Resonance. PhD thesis, Humboldt Universit¨ at zu Berlin, Logos-Verlag, Berlin, 2002. C. Penland. A stochastic model of IndioPacific sea surface temperature anomalies. Physica D, 98:534–558, 1996. C. Penland & P. Sardeshmukh. Error and sensitivity analysis of geophysical eigensystems. J. Climate, 8(8):1988–1998, 1995. C. Penland & P. Sardeshmukh. The optimal growth of tropical sea surface temperature anomalies. J. Climate, 8(8):1999–2024, 1995. O. Penrose & J. Lebowitz. Rigorous treatment of metastable states in the van der Waals-Maxwell theory. J. Stat. Phys., 3(2):211–236, 1971. O. Perron. On stability and asymptotic growth of the integrals of systems of differential equations (in German). Math. Zeitschr., 29:129–160, 1929. A. Pikovsky. Local Lyapunov exponents for spatiotemporal chaos. Chaos, 3(2):225–232, 1993. A. Pikovsky & U. Feudel. Characterizing strange nonchaotic attractors. Chaos, 5(1):253–260, 1995. M. Pinsky. Instability of the harmonic oscillator with small noise. SIAM J. Appl. Math., 46(3):451–463, 1986. M. Pinsky & V. Wihstutz. Lyapunov exponents of nilpotent Itˆ o systems. Stochastics, 25(1):43–57, 1988. M. Pinsky & V. Wihstutz. Lyapunov exponents of realnoise-driven nilpotent systems and harmonic oscillators. Stochastics Stochastics Rep., 35(2):93–110, 1991. M. Pituk. The Hartman-Wintner theorem for functional differential equations. J. Differ. Equations, 155(1):1–16, 1999. ¨ kefalvi-Nagy. Functional Analysis. F. Riesz & B. Szo Transl. from the 2nd French ed. by Leo F. Boron, Dover Publications, New York, 1955. R.T. Rockafellar. Convex analysis. 2nd edition, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1997. G. Royer. A remark on simulated annealing of diffusion processes. SIAM J. Control Optim., 27(6):1403–1408, 1989. B. Roynette. On the diffusion process in dimension 2 (in French). Z Wahrscheinlichkeitstheorie verw. Gebiete, 32(12):95–110, 1975. D. Ruelle. Characteristic exponents and invariant manifolds in Hilbert space. Annals of Mathematics 2nd Ser., 115(2):243– 290, 1982.

250 [Ru 88]

[Rm 78]

[Rm 79]

[Sn-SM 80]

[So 81]

[Ste 89] [Sto 61] [Stra-Yor 67] [Str-Vdh 72]

[Str-Vdh 97]

[Su 81] [Tg-Mo-Fy 04] [Ti-Lm 00]

[Tr 82]

[Va 96]

[Vc 77] [Wa 00] [Wg-Bc-Fg 99]

Bibliography D. Ruelle. Can nonlinear dynamics help economists? In The Economy as an Evolving Complex System, Proceedings of the Evolutionary Paths of the Global Economy workshop, September 8-18, 1987 in Santa Fe, New Mexico (Eds: P.W. Anderson, K.J. Arrow & D. Pines), pages 195–204. Addison-Weseley Publishing Company, Redwood City (CA), 1988. ¨melin. Stability and growth of the solution of y¨ +ft y = W. Ru 0 where ft is a positive stationary Markov process. In Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes. Proceedings, Prague 1978, Vol B, pages 149–161. Reidel, Dordrecht, Boston, 1978. ¨melin. Stability and growth of the solution of y¨ + W. Ru ft y = 0 where ft is a positive stationary Markov process (in German). PhD thesis, Universit¨ at Bremen, Bremen, 1979. J. Sancho & M. San Miguel. External non-white noise and nonequilibrium phase transitions. Z. Phys. B, 36:357–364, 1980. U. Sommer. The growth numbers of three-dimensional linear, parameter-excited systems (in German). Z. Angew. Math. Mech., 61:T353–T355, 1981. L . Stettner. Large deviations of invariant measures for degenerate diffusions. Probab. Math. Stat., 10(1):93–105, 1989. H. Stommel. Thermohaline convection with two stable regimes of flow. Tellus, 13(2):224–230, 1961. A. Strauss & J. Yorke. On asymptotically autonomous differential equations. Math. Syst. Theory, 1:175–182, 1967. D. Stroock & S. Varadhan. On degenerate ellipticparabolic operators of second order and their associated diffusions. Commun. Pure Appl. Math., 25:651–713, 1972. D. Stroock & S. Varadhan. Multidimensional diffusion processes. 2nd corrected printing, Springer-Verlag, Berlin, 1997. A. Sutera. On stochastic perturbation and long-term climate behavior. Quart. J. R. Met. Soc, 107:137–151, 1981. Q. Teng, A. Monahan & J. Fyfe. Effects of time averaging on Climate Regimes. Geophys. Res. Lett., 31(22), 2004. A. Timmermann & G. Lohmann. Noise-induced transitions in a simplified model of the Thermohaline Circulation. J. Phys. Oceanogr., 30:1891–1900, 2000. V. Tran. On the asymptotic behavior of solutions of ordinary asymptotically periodic differential equations with random parameters (in German). Acta Math. Acad. Sci. Hungar., 39(4):353–360, 1982. M. Vares. Large deviations and metastability. In Disordered Systems. Travaux en Cours, Vol. 53 (Eds: R. Bam´ on, S. Martinez) pages 1–62, Hermann, Paris, 1996. O. Vasicek. An Equilibrium Characterization of the Term Structure. J. Financial Econ., 5:177–188, 1977. W. Walter. Ordinary differential equations (in German). 7th edition, Springer-Verlag, Berlin, Heidelberg, 2000. B. Wang, A. Barcilon & Z. Fang. Stochastic Dynamics of El Ni˜ no-Southern Oscillation. J. Atmosph. Sci., 56:5–23, 1999.

Bibliography [Web 51]

[Wd 71]

[Wt 70] [We-Fr 69]

[We-Fr 70]

[We 79]

[We-Fr 72]

[Wh 75]

[Wh 01]

[Wi 50] [Wi 57] [Wt-Ne-Kt 97]

[Wo 92] [Ze 88]

251 M. Weber. The fundamental solution of a degenerate partial differential equation of parabolic type. Trans. Amer. Math. Soc., 71(1):24–37, 1951. W. Weidlich. The statistical description of polarization phenomena in society. Brit. J. math. statist. Psychol., 24:251–266, 1971. E. Weintraub. Stochastic stability of short-run market equilibrium: Comment. Quart. J. Econom., 84(1):161–167, 1970. A. Wentzell & M. Freidlin. On the limiting behavior of an invariant measure under small random perturbations of a dynamical system. Soviet Math. Dokl., 10(5):1047–1051, 1969. A. Wentzell & M. Freidlin. On small random perturbations of dynamical systems. Russ. Math. Surveys, 25(1):1–55, 1970. A. Wentzell. Theory of random processes (in German). Akademie-Verlag, Berlin, 1979. German translation of the Russian edition, Nauka, Moscow, 1975. A. Wentzell & M. Freidlin. Some Problems concerning stability under small random perturbations. Th. of Probability and its Appl., 17(2):269–283, 1972. V. Wihstutz. On stability and growth of solutions of linear differential equations with stationary random parameters (in German). PhD thesis, Universit¨ at Bremen, Bremen, 1975. V. Wihstutz. Communication structure of discretized degenerate diffusion processes and approximation of Lyapunov exponents. Monte Carlo Methods and Appl., 7(3–4):411–419, 2001. A. Wintner. On free vibrations with amplitudinal limits. Quart. Appl. Math., 8(1):102–104, 1950. A. Wintner. On Riccati’s resolvent. Quart. Appl. Math., 14(4):436–439, 1957. A. Witt, A. Neiman & J. Kurths. Characterizing the dynamics of stochastic bistable systems by measures of complexity. Phys. Rev. E, 55(5):5050–5059, 1997. R. Wolff. Local Lyapunov exponents: Looking closely at chaos. J. R. Statist. Soc. B, 54(2):353–371, 1992. E. Zeeman. Stability of dynamical systems. Nonlinearity, 1:115–155, 1988.

Index

action functional 55 arrow 92 asymptotic integration (Hartman and Wintner) 29 asymptotically autonomous ODE 44 asymptotically constant, linear ODE 28 attracting switching surface 20 backward operator 128f. backward (parabolic) equation 61 boundedness in probability 103f. cocycle 22 commodity prices 114 comparison 163 contraction principle 68 cycle 93ff. diagonal matrix plus small skewsymmetric perturbation 47, 223f. duality correspondence 135 dynamic programming PDE 136 El Ni˜ no-Southern Oscillation 119 ellipticity 56, 129 entrance point of a cycle 96 ergodic dynamical system 22 evolution model 121 exit point of a cycle 96 exit position 79 exit probability in the degenerate case 125ff., 130ff., 134ff., 140f. in the non-degenerate case 72ff. exit rate of a cycle 93ff., 97

exit time in the degenerate case 127ff. in the non-degenerate case 72ff., 79ff. first exit position see exit position first exit time see exit time Fokker-Planck equation see Kolmogorov forward equation forward (parabolic) equation 61 Freidlin-Wentzell theory 53ff. Furstenberg-Khasminskii formula 27, 209 graph 92 harmonic oscillator 50 Hartman-Wintner-Perron theorem 33 Hern´ andez-Lerma theorems 130ff., 134ff. hierarchy of cycles 93ff. hypoellipticity 26, 128, 186 i-graph 92 inclusion probability 130 invariant (with respect to Markov semigroup) probability measure 62 Jacobi equation 11, 22f., 156 Jacquot condition 64 Kolmogorov backward equation 62f. forward equation 62f. large deviation principle (LDP) 68ff. law of large numbers for the top Lyapunov exponent 27 Legendre transformation 135

253

254 linearized SDE 2, 44ff., 222 Liouville equation see Jacobi equation local growth rate of the determinant 156 local hypoellipticity 186 local Lyapunov exponent 4ff., 45, 143ff. − upper and lower bound 145ff. − limit in probability 151 in the diagonal case 157ff., 175 in the general case 206ff. locality 53 logarithmic equivalence 100 logistic ODE see Verhulst ODE Lyapunov exponent in the deterministic case 30, 33ff. in the stochastic case 2, 23ff. main state of a cycle 93ff. Markovian noise 9 mean-reversion 114 metastability 1, 53, 91, 107, 144 metastable state 3, 100f. metric (measure preserving) dynamical system 21 multiplicative ergodic theorem 23 noisy north-south-flow 122 occupation time of the real noise system 179ff., 188ff., 198ff. of the underlying diffusion 102 Ornstein-Uhlenbeck process 45, 65, 109 Oseledets see multiplicative ergodic theorem parabolic equations see backward equation, forward equation parametrically excited system 9 projection method 12ff., 20 quasi-deterministic approximation 53, 102 quasi-deterministic behavior of the non-degenerate diffusion 102 of the real noise system 179ff., 188ff., 198ff. quasipotential 56, 76ff. random dynamical system 22 random Riccati-type differential inequality 153 random Riccati-/Verhulst-type differential inequality 164

Index rate function 68ff. real noise system 9 recurrence on time scales 228 repelling switching surface 20 rheolinear system 9 Riccati − Riccati-type differential inequality 31 − random Riccati-type differential inequality 153 − random Riccati-/Verhulst-type differential inequality 164 rotation rate of a cycle 93ff. Saksaul see tree population model Schilder’s Theorem 70 separated switching surfaces (curves) 185 simulated annealing 108, 228 stability on time scales 225 stationary distribution rate of a cycle 93ff. stationary probability distribution 21, 62f. stochastic disk dynamo model 121 strictly separated switching surfaces (curves) 185, 209f. strong hypoellipticity 186 sublimiting distribution 3, 101f. sublimiting Furstenberg-Khasminskii formula 209 switching surfaces (curves) 18, 185 thermohaline circulation 115ff. trace formula for the sum of the Lyapunov exponents 24, 156 transience on time scales 227 tree population model 120 two well potential 110, 175 ultraparabolic operator 129 value function 127, 187 variational equation see linearized SDE Vasicek model for the interest rates 113 Verhulst − Verhulst-ODE 163 − random Riccati-/Verhulst-type differential inequality 164 W -graph 92 white noise system 10, 226 wide band noise limit 226

Lecture Notes in Mathematics For information about earlier volumes please contact your bookseller or Springer LNM Online archive: springerlink.com

Vol. 1780: J. H. Bruinier, Borcherds Products on O(2,1) and Chern Classes of Heegner Divisors (2002) Vol. 1781: E. Bolthausen, E. Perkins, A. van der Vaart, Lectures on Probability Theory and Statistics. Ecole d’ Eté de Probabilités de Saint-Flour XXIX-1999. Editor: P. Bernard (2002) Vol. 1782: C.-H. Chu, A. T.-M. Lau, Harmonic Functions on Groups and Fourier Algebras (2002) Vol. 1783: L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization (2002) Vol. 1784: L. H. Eliasson, S. B. Kuksin, S. Marmi, J.-C. Yoccoz, Dynamical Systems and Small Divisors. Cetraro, Italy 1998. Editors: S. Marmi, J.-C. Yoccoz (2002) Vol. 1785: J. Arias de Reyna, Pointwise Convergence of Fourier Series (2002) Vol. 1786: S. D. Cutkosky, Monomialization of Morphisms from 3-Folds to Surfaces (2002) Vol. 1787: S. Caenepeel, G. Militaru, S. Zhu, Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations (2002) Vol. 1788: A. Vasil’ev, Moduli of Families of Curves for Conformal and Quasiconformal Mappings (2002) Vol. 1789: Y. Sommerhäuser, Yetter-Drinfel’d Hopf algebras over groups of prime order (2002) Vol. 1790: X. Zhan, Matrix Inequalities (2002) Vol. 1791: M. Knebusch, D. Zhang, Manis Valuations and Prüfer Extensions I: A new Chapter in Commutative Algebra (2002) Vol. 1792: D. D. Ang, R. Gorenflo, V. K. Le, D. D. Trong, Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction (2002) Vol. 1793: J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems (2002) Vol. 1794: N. Pytheas Fogg, Substitution in Dynamics, Arithmetics and Combinatorics. Editors: V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel (2002) Vol. 1795: H. Li, Filtered-Graded Transfer in Using Noncommutative Gröbner Bases (2002) Vol. 1796: J.M. Melenk, hp-Finite Element Methods for Singular Perturbations (2002) Vol. 1797: B. Schmidt, Characters and Cyclotomic Fields in Finite Geometry (2002) Vol. 1798: W.M. Oliva, Geometric Mechanics (2002) Vol. 1799: H. Pajot, Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral (2002) Vol. 1800: O. Gabber, L. Ramero, Almost Ring Theory (2003) Vol. 1801: J. Azéma, M. Émery, M. Ledoux, M. Yor (Eds.), Séminaire de Probabilités XXXVI (2003) Vol. 1802: V. Capasso, E. Merzbach, B. G. Ivanoff, M. Dozzi, R. Dalang, T. Mountford, Topics in Spatial Stochastic Processes. Martina Franca, Italy 2001. Editor: E. Merzbach (2003)

Vol. 1803: G. Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation (2003) Vol. 1804: I. Cherednik, Ya. Markov, R. Howe, G. Lusztig, Iwahori-Hecke Algebras and their Representation Theory. Martina Franca, Italy 1999. Editors: V. Baldoni, D. Barbasch (2003) Vol. 1805: F. Cao, Geometric Curve Evolution and Image Processing (2003) Vol. 1806: H. Broer, I. Hoveijn. G. Lunther, G. Vegter, Bifurcations in Hamiltonian Systems. Computing Singularities by Gröbner Bases (2003) Vol. 1807: V. D. Milman, G. Schechtman (Eds.), Geometric Aspects of Functional Analysis. Israel Seminar 20002002 (2003) Vol. 1808: W. Schindler, Measures with Symmetry Properties (2003) Vol. 1809: O. Steinbach, Stability Estimates for Hybrid Coupled Domain Decomposition Methods (2003) Vol. 1810: J. Wengenroth, Derived Functors in Functional Analysis (2003) Vol. 1811: J. Stevens, Deformations of Singularities (2003) Vol. 1812: L. Ambrosio, K. Deckelnick, G. Dziuk, M. Mimura, V. A. Solonnikov, H. M. Soner, Mathematical Aspects of Evolving Interfaces. Madeira, Funchal, Portugal 2000. Editors: P. Colli, J. F. Rodrigues (2003) Vol. 1813: L. Ambrosio, L. A. Caffarelli, Y. Brenier, G. Buttazzo, C. Villani, Optimal Transportation and its Applications. Martina Franca, Italy 2001. Editors: L. A. Caffarelli, S. Salsa (2003) Vol. 1814: P. Bank, F. Baudoin, H. Föllmer, L.C.G. Rogers, M. Soner, N. Touzi, Paris-Princeton Lectures on Mathematical Finance 2002 (2003) Vol. 1815: A. M. Vershik (Ed.), Asymptotic Combinatorics with Applications to Mathematical Physics. St. Petersburg, Russia 2001 (2003) Vol. 1816: S. Albeverio, W. Schachermayer, M. Talagrand, Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilités de Saint-Flour XXX-2000. Editor: P. Bernard (2003) Vol. 1817: E. Koelink, W. Van Assche (Eds.), Orthogonal Polynomials and Special Functions. Leuven 2002 (2003) Vol. 1818: M. Bildhauer, Convex Variational Problems with Linear, nearly Linear and/or Anisotropic Growth Conditions (2003) Vol. 1819: D. Masser, Yu. V. Nesterenko, H. P. Schlickewei, W. M. Schmidt, M. Waldschmidt, Diophantine Approximation. Cetraro, Italy 2000. Editors: F. Amoroso, U. Zannier (2003) Vol. 1820: F. Hiai, H. Kosaki, Means of Hilbert Space Operators (2003) Vol. 1821: S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics (2003)

Vol. 1822: S.-N. Chow, R. Conti, R. Johnson, J. MalletParet, R. Nussbaum, Dynamical Systems. Cetraro, Italy 2000. Editors: J. W. Macki, P. Zecca (2003) Vol. 1823: A. M. Anile, W. Allegretto, C. Ringhofer, Mathematical Problems in Semiconductor Physics. Cetraro, Italy 1998. Editor: A. M. Anile (2003) Vol. 1824: J. A. Navarro González, J. B. Sancho de Salas, C ∞ – Differentiable Spaces (2003) Vol. 1825: J. H. Bramble, A. Cohen, W. Dahmen, Multiscale Problems and Methods in Numerical Simulations, Martina Franca, Italy 2001. Editor: C. Canuto (2003) Vol. 1826: K. Dohmen, Improved Bonferroni Inequalities via Abstract Tubes. Inequalities and Identities of Inclusion-Exclusion Type. VIII, 113 p, 2003. Vol. 1827: K. M. Pilgrim, Combinations of Complex Dynamical Systems. IX, 118 p, 2003. Vol. 1828: D. J. Green, Gröbner Bases and the Computation of Group Cohomology. XII, 138 p, 2003. Vol. 1829: E. Altman, B. Gaujal, A. Hordijk, DiscreteEvent Control of Stochastic Networks: Multimodularity and Regularity. XIV, 313 p, 2003. Vol. 1830: M. I. Gil’, Operator Functions and Localization of Spectra. XIV, 256 p, 2003. Vol. 1831: A. Connes, J. Cuntz, E. Guentner, N. Higson, J. E. Kaminker, Noncommutative Geometry, Martina Franca, Italy 2002. Editors: S. Doplicher, L. Longo (2004) Vol. 1832: J. Azéma, M. Émery, M. Ledoux, M. Yor (Eds.), Séminaire de Probabilités XXXVII (2003) Vol. 1833: D.-Q. Jiang, M. Qian, M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States. On the Frontier of Probability and Dynamical Systems. IX, 280 p, 2004. Vol. 1834: Yo. Yomdin, G. Comte, Tame Geometry with Application in Smooth Analysis. VIII, 186 p, 2004. Vol. 1835: O.T. Izhboldin, B. Kahn, N.A. Karpenko, A. Vishik, Geometric Methods in the Algebraic Theory of Quadratic Forms. Summer School, Lens, 2000. Editor: J.-P. Tignol (2004) Vol. 1836: C. Nˇastˇasescu, F. Van Oystaeyen, Methods of Graded Rings. XIII, 304 p, 2004. Vol. 1837: S. Tavaré, O. Zeitouni, Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilités de Saint-Flour XXXI-2001. Editor: J. Picard (2004) Vol. 1838: A.J. Ganesh, N.W. O’Connell, D.J. Wischik, Big Queues. XII, 254 p, 2004. Vol. 1839: R. Gohm, Noncommutative Stationary Processes. VIII, 170 p, 2004. Vol. 1840: B. Tsirelson, W. Werner, Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilités de Saint-Flour XXXII-2002. Editor: J. Picard (2004) Vol. 1841: W. Reichel, Uniqueness Theorems for Variational Problems by the Method of Transformation Groups (2004) Vol. 1842: T. Johnsen, A. L. Knutsen, K3 Projective Models in Scrolls (2004) Vol. 1843: B. Jefferies, Spectral Properties of Noncommuting Operators (2004) Vol. 1844: K.F. Siburg, The Principle of Least Action in Geometry and Dynamics (2004) Vol. 1845: Min Ho Lee, Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms (2004) Vol. 1846: H. Ammari, H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (2004) Vol. 1847: T.R. Bielecki, T. Björk, M. Jeanblanc, M. Rutkowski, J.A. Scheinkman, W. Xiong, Paris-Princeton Lectures on Mathematical Finance 2003 (2004)

Vol. 1848: M. Abate, J. E. Fornaess, X. Huang, J. P. Rosay, A. Tumanov, Real Methods in Complex and CR Geometry, Martina Franca, Italy 2002. Editors: D. Zaitsev, G. Zampieri (2004) Vol. 1849: Martin L. Brown, Heegner Modules and Elliptic Curves (2004) Vol. 1850: V. D. Milman, G. Schechtman (Eds.), Geometric Aspects of Functional Analysis. Israel Seminar 20022003 (2004) Vol. 1851: O. Catoni, Statistical Learning Theory and Stochastic Optimization (2004) Vol. 1852: A.S. Kechris, B.D. Miller, Topics in Orbit Equivalence (2004) Vol. 1853: Ch. Favre, M. Jonsson, The Valuative Tree (2004) Vol. 1854: O. Saeki, Topology of Singular Fibers of Differential Maps (2004) Vol. 1855: G. Da Prato, P.C. Kunstmann, I. Lasiecka, A. Lunardi, R. Schnaubelt, L. Weis, Functional Analytic Methods for Evolution Equations. Editors: M. Iannelli, R. Nagel, S. Piazzera (2004) Vol. 1856: K. Back, T.R. Bielecki, C. Hipp, S. Peng, W. Schachermayer, Stochastic Methods in Finance, Bressanone/Brixen, Italy, 2003. Editors: M. Fritelli, W. Runggaldier (2004) Vol. 1857: M. Émery, M. Ledoux, M. Yor (Eds.), Séminaire de Probabilités XXXVIII (2005) Vol. 1858: A.S. Cherny, H.-J. Engelbert, Singular Stochastic Differential Equations (2005) Vol. 1859: E. Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras (2005) Vol. 1860: A. Borisyuk, G.B. Ermentrout, A. Friedman, D. Terman, Tutorials in Mathematical Biosciences I. Mathematical Neurosciences (2005) Vol. 1861: G. Benettin, J. Henrard, S. Kuksin, Hamiltonian Dynamics – Theory and Applications, Cetraro, Italy, 1999. Editor: A. Giorgilli (2005) Vol. 1862: B. Helffer, F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians (2005) Vol. 1863: H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms (2005) Vol. 1864: K. Efstathiou, Metamorphoses of Hamiltonian Systems with Symmetries (2005) Vol. 1865: D. Applebaum, B.V. R. Bhat, J. Kustermans, J. M. Lindsay, Quantum Independent Increment Processes I. From Classical Probability to Quantum Stochastic Calculus. Editors: M. Schürmann, U. Franz (2005) Vol. 1866: O.E. Barndorff-Nielsen, U. Franz, R. Gohm, B. Kümmerer, S. Thorbjønsen, Quantum Independent Increment Processes II. Structure of Quantum Lévy Processes, Classical Probability, and Physics. Editors: M. Schürmann, U. Franz, (2005) Vol. 1867: J. Sneyd (Ed.), Tutorials in Mathematical Biosciences II. Mathematical Modeling of Calcium Dynamics and Signal Transduction. (2005) Vol. 1868: J. Jorgenson, S. Lang, Posn (R) and Eisenstein Series. (2005) Vol. 1869: A. Dembo, T. Funaki, Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilités de Saint-Flour XXXIII-2003. Editor: J. Picard (2005) Vol. 1870: V.I. Gurariy, W. Lusky, Geometry of Müntz Spaces and Related Questions. (2005) Vol. 1871: P. Constantin, G. Gallavotti, A.V. Kazhikhov, Y. Meyer, S. Ukai, Mathematical Foundation of Turbulent Viscous Flows, Martina Franca, Italy, 2003. Editors: M. Cannone, T. Miyakawa (2006)

Vol. 1872: A. Friedman (Ed.), Tutorials in Mathematical Biosciences III. Cell Cycle, Proliferation, and Cancer (2006) Vol. 1873: R. Mansuy, M. Yor, Random Times and Enlargements of Filtrations in a Brownian Setting (2006) Vol. 1874: M. Yor, M. Émery (Eds.), In Memoriam PaulAndré Meyer - Séminaire de Probabilités XXXIX (2006) Vol. 1875: J. Pitman, Combinatorial Stochastic Processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII-2002. Editor: J. Picard (2006) Vol. 1876: H. Herrlich, Axiom of Choice (2006) Vol. 1877: J. Steuding, Value Distributions of L-Functions (2007) Vol. 1878: R. Cerf, The Wulff Crystal in Ising and Percolation Models, Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004. Editor: Jean Picard (2006) Vol. 1879: G. Slade, The Lace Expansion and its Applications, Ecole d’Eté de Probabilités de Saint-Flour XXXIV2004. Editor: Jean Picard (2006) Vol. 1880: S. Attal, A. Joye, C.-A. Pillet, Open Quantum Systems I, The Hamiltonian Approach (2006) Vol. 1881: S. Attal, A. Joye, C.-A. Pillet, Open Quantum Systems II, The Markovian Approach (2006) Vol. 1882: S. Attal, A. Joye, C.-A. Pillet, Open Quantum Systems III, Recent Developments (2006) Vol. 1883: W. Van Assche, F. Marcellàn (Eds.), Orthogonal Polynomials and Special Functions, Computation and Application (2006) Vol. 1884: N. Hayashi, E.I. Kaikina, P.I. Naumkin, I.A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations (2006) Vol. 1885: A. Telcs, The Art of Random Walks (2006) Vol. 1886: S. Takamura, Splitting Deformations of Degenerations of Complex Curves (2006) Vol. 1887: K. Habermann, L. Habermann, Introduction to Symplectic Dirac Operators (2006) Vol. 1888: J. van der Hoeven, Transseries and Real Differential Algebra (2006) Vol. 1889: G. Osipenko, Dynamical Systems, Graphs, and Algorithms (2006) Vol. 1890: M. Bunge, J. Funk, Singular Coverings of Toposes (2006) Vol. 1891: J.B. Friedlander, D.R. Heath-Brown, H. Iwaniec, J. Kaczorowski, Analytic Number Theory, Cetraro, Italy, 2002. Editors: A. Perelli, C. Viola (2006) Vol. 1892: A. Baddeley, I. Bárány, R. Schneider, W. Weil, Stochastic Geometry, Martina Franca, Italy, 2004. Editor: W. Weil (2007) Vol. 1893: H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems, Results and Examples (2007) Vol. 1894: C.W. Groetsch, Stable Approximate Evaluation of Unbounded Operators (2007) Vol. 1895: L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces (2007) Vol. 1896: P. Massart, Concentration Inequalities and Model Selection, Ecole d’Été de Probabilités de SaintFlour XXXIII-2003. Editor: J. Picard (2007) Vol. 1897: R. Doney, Fluctuation Theory for Lévy Processes, Ecole d’Été de Probabilités de Saint-Flour XXXV-2005. Editor: J. Picard (2007) Vol. 1898: H.R. Beyer, Beyond Partial Differential Equations, On linear and Quasi-Linear Abstract Hyperbolic Evolution Equations (2007)

Vol. 1899: Séminaire de Probabilités XL. Editors: C. Donati-Martin, M. Émery, A. Rouault, C. Stricker (2007) Vol. 1900: E. Bolthausen, A. Bovier (Eds.), Spin Glasses (2007) Vol. 1901: O. Wittenberg, Intersections de deux quadriques et pinceaux de courbes de genre 1, Intersections of Two Quadrics and Pencils of Curves of Genus 1 (2007) Vol. 1902: A. Isaev, Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds (2007) Vol. 1903: G. Kresin, V. Maz’ya, Sharp Real-Part Theorems (2007) Vol. 1904: P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions (2007) Vol. 1905: C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations (2007) Vol. 1906: T. Schuster, The Method of Approximate Inverse: Theory and Applications (2007) Vol. 1907: M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems (2007) Vol. 1908: T.J. Lyons, M. Caruana, T. Lévy, Differential Equations Driven by Rough Paths, Ecole d’Été de Probabilités de Saint-Flour XXXIV-2004 (2007) Vol. 1909: H. Akiyoshi, M. Sakuma, M. Wada, Y. Yamashita, Punctured Torus Groups and 2-Bridge Knot Groups (I) (2007) Vol. 1910: V.D. Milman, G. Schechtman (Eds.), Geometric Aspects of Functional Analysis. Israel Seminar 2004-2005 (2007) Vol. 1911: A. Bressan, D. Serre, M. Williams, K. Zumbrun, Hyperbolic Systems of Balance Laws. Cetraro, Italy 2003. Editor: P. Marcati (2007) Vol. 1912: V. Berinde, Iterative Approximation of Fixed Points (2007) Vol. 1913: J.E. Marsden, G. Misiołek, J.-P. Ortega, M. Perlmutter, T.S. Ratiu, Hamiltonian Reduction by Stages (2007) Vol. 1914: G. Kutyniok, Affine Density in Wavelet Analysis (2007) Vol. 1915: T. Bıyıkoˇglu, J. Leydold, P.F. Stadler, Laplacian Eigenvectors of Graphs. Perron-Frobenius and Faber-Krahn Type Theorems (2007) Vol. 1916: C. Villani, F. Rezakhanlou, Entropy Methods for the Boltzmann Equation. Editors: F. Golse, S. Olla (2008) Vol. 1917: I. Veselić, Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger (2008) Vol. 1918: B. Roberts, R. Schmidt, Local Newforms for GSp(4) (2007) Vol. 1919: R.A. Carmona, I. Ekeland, A. KohatsuHiga, J.-M. Lasry, P.-L. Lions, H. Pham, E. Taflin, Paris-Princeton Lectures on Mathematical Finance 2004. Editors: R.A. Carmona, E. Çinlar, I. Ekeland, E. Jouini, J.A. Scheinkman, N. Touzi (2007) Vol. 1920: S.N. Evans, Probability and Real Trees. Ecole d’Été de Probabilités de Saint-Flour XXXV-2005 (2008) Vol. 1921: J.P. Tian, Evolution Algebras and their Applications (2008) Vol. 1922: A. Friedman (Ed.), Tutorials in Mathematical BioSciences IV. Evolution and Ecology (2008) Vol. 1923: J.P.N. Bishwal, Parameter Estimation in Stochastic Differential Equations (2008) Vol. 1924: M. Wilson, Littlewood-Paley Theory and Exponential-Square Integrability (2008)

Vol. 1925: M. du Sautoy, L. Woodward, Zeta Functions of Groups and Rings (2008) Vol. 1926: L. Barreira, V. Claudia, Stability of Nonautonomous Differential Equations (2008) Vol. 1927: L. Ambrosio, L. Caffarelli, M.G. Crandall, L.C. Evans, N. Fusco, Calculus of Variations and NonLinear Partial Differential Equations. Cetraro, Italy 2005. Editors: B. Dacorogna, P. Marcellini (2008) Vol. 1928: J. Jonsson, Simplicial Complexes of Graphs (2008) Vol. 1929: Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes (2008) Vol. 1930: J.M. Urbano, The Method of Intrinsic Scaling. A Systematic Approach to Regularity for Degenerate and Singular PDEs (2008) Vol. 1931: M. Cowling, E. Frenkel, M. Kashiwara, A. Valette, D.A. Vogan, Jr., N.R. Wallach, Representation Theory and Complex Analysis. Venice, Italy 2004. Editors: E.C. Tarabusi, A. D’Agnolo, M. Picardello (2008) Vol. 1932: A.A. Agrachev, A.S. Morse, E.D. Sontag, H.J. Sussmann, V.I. Utkin, Nonlinear and Optimal Control Theory. Cetraro, Italy 2004. Editors: P. Nistri, G. Stefani (2008) Vol. 1933: M. Petkovic, Point Estimation of Root Finding Methods (2008) Vol. 1934: C. Donati-Martin, M. Émery, A. Rouault, C. Stricker (Eds.), Séminaire de Probabilités XLI (2008) Vol. 1935: A. Unterberger, Alternative Pseudodifferential Analysis (2008) Vol. 1936: P. Magal, S. Ruan (Eds.), Structured Population Models in Biology and Epidemiology (2008) Vol. 1937: G. Capriz, P. Giovine, P.M. Mariano (Eds.), Mathematical Models of Granular Matter (2008) Vol. 1938: D. Auroux, F. Catanese, M. Manetti, P. Seidel, B. Siebert, I. Smith, G. Tian, Symplectic 4-Manifolds and Algebraic Surfaces. Cetraro, Italy 2003. Editors: F. Catanese, G. Tian (2008) Vol. 1939: D. Boffi, F. Brezzi, L. Demkowicz, R.G. Durán, R.S. Falk, M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications. Cetraro, Italy 2006. Editors: D. Boffi, L. Gastaldi (2008) Vol. 1940: J. Banasiak, V. Capasso, M.A.J. Chaplain, M. Lachowicz, J. Mie¸kisz, Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic. Be¸dlewo, Poland 2006. Editors: V. Capasso, M. Lachowicz (2008) Vol. 1941: S.M.J. Haran, Arithmetical Investigations. Representation Theory, Orthogonal Polynomials, and Quantum Interpolations (2008) Vol. 1942: S. Albeverio, F. Flandoli, Y.G. Sinai, SPDE in Hydrodynamic. Recent Progress and Prospects. Cetraro, Italy 2005. Editors: G. Da Prato, M. Röckner (2008) Vol. 1943: L.L. Bonilla (Ed.), Inverse Problems and Imaging. Martina Franca, Italy 2002 (2008) Vol. 1944: A. Di Bartolo, G. Falcone, P. Plaumann, K. Strambach, Algebraic Groups and Lie Groups with Few Factors (2008) Vol. 1945: F. Brauer, P. van den Driessche, J. Wu (Eds.), Mathematical Epidemiology (2008) Vol. 1946: G. Allaire, A. Arnold, P. Degond, T.Y. Hou, Quantum Transport. Modelling, Analysis and Asymptotics. Cetraro, Italy 2006. Editors: N.B. Abdallah, G. Frosali (2008) Vol. 1947: D. Abramovich, M. Mariño, M. Thaddeus, R. Vakil, Enumerative Invariants in Algebraic Geo-

metry and String Theory. Cetraro, Italy 2005. Editors: K. Behrend, M. Manetti (2008) Vol. 1948: F. Cao, J-L. Lisani, J-M. Morel, P. Musé, F. Sur, A Theory of Shape Identification (2008) Vol. 1949: H.G. Feichtinger, B. Helffer, M.P. Lamoureux, N. Lerner, J. Toft, Pseudo-Differential Operators. Quantization and Signals. Cetraro, Italy 2006. Editors: L. Rodino, M.W. Wong (2008) Vol. 1950: M. Bramson, Stability of Queueing Networks, Ecole d’Eté de Probabilités de Saint-Flour XXXVI-2006 (2008) Vol. 1951: A. Moltó, J. Orihuela, S. Troyanski, M. Valdivia, A Non Linear Transfer Technique for Renorming (2009) Vol. 1952: R. Mikhailov, I.B.S. Passi, Lower Central and Dimension Series of Groups (2009) Vol. 1953: K. Arwini, C.T.J. Dodson, Information Geometry (2008) Vol. 1954: P. Biane, L. Bouten, F. Cipriani, N. Konno, N. Privault, Q. Xu, Quantum Potential Theory. Editors: U. Franz, M. Schuermann (2008) Vol. 1955: M. Bernot, V. Caselles, J.-M. Morel, Optimal Transportation Networks (2008) Vol. 1956: C.H. Chu, Matrix Convolution Operators on Groups (2008) Vol. 1957: A. Guionnet, On Random Matrices: Macroscopic Asymptotics, Ecole d’Eté de Probabilités de SaintFlour XXXVI-2006 (2009) Vol. 1958: M.C. Olsson, Compactifying Moduli Spaces for Abelian Varieties (2008) Vol. 1959: Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties (2008) Vol. 1960: J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes (2009) Vol. 1961: G. Buttazzo, A. Pratelli, E. Stepanov, S. Solimini, Optimal Urban Networks via Mass Transportation (2009) Vol. 1962: R. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart, Y. Xiao, A Minicourse on Stochastic Partial Differential Equations (2009) Vol. 1963: W. Siegert, Local Lyapunov Exponents (2009)

Recent Reprints and New Editions Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999 – Corr. 3rd printing (2007) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 – 2nd corr. and augmented edition (2007) Vol. 1693: S. Simons, From Hahn-Banach to Monotonicity (Minimax and Monotonicity 1998) – 2nd exp. edition (2008) Vol. 470: R.E. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. With a preface by D. Ruelle. Edited by J.-R. Chazottes. 1975 – 2nd rev. edition (2008) Vol. 523: S.A. Albeverio, R.J. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integral. 1976 – 2nd corr. and enlarged edition (2008) Vol. 1764: A. Cannas da Silva, Lectures on Symplectic Geometry 2001 – Corr. 2nd printing (2008)

LECTURE NOTES IN MATHEMATICS

123

Edited by J.-M. Morel, F. Takens, B. Teissier, P.K. Maini Editorial Policy (for the publication of monographs) 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Monograph manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Manuscripts should be submitted either to Springer’s mathematics editorial in Heidelberg, or to one of the series editors. In general, manuscripts will be sent out to 2 external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted: The author will be informed of this. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Authors should be aware that incomplete or insufficiently close to final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. Authors should also be aware that parallel submission of their manuscript to another publisher while under consideration for LNM will in general lead to immediate rejection. 3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. For evaluation purposes, manuscripts may be submitted in print or electronic form, in the latter case preferably as pdf- or zipped ps-files. Lecture Notes volumes are, as a rule, printed digitally from the authors’ files. To ensure best results, authors are asked to use the LaTeX2e style files available from Springer’s web-server at: ftp://ftp.springer.de/pub/tex/latex/svmonot1/ (for monographs).

Additional technical instructions, if necessary, are available on request from: [email protected]. 4. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files (and also the corresponding dvi-, pdf- or zipped ps-file) together with the final printout made from these files. The LaTeX source files are essential for producing the full-text online version of the book (see www.springerlink.com/content/110312 for the existing online volumes of LNM). The actual production of a Lecture Notes volume takes approximately 12 weeks. 5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3% on the price of Springer books purchased for their personal use, if ordering directly from Springer. 6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor J.-M. Morel, CMLA, ´ Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan Cedex, France E-mail: [email protected] Professor F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands E-mail: [email protected] Professor B. Teissier, Institut Mathématique de Jussieu, ´ UMR 7586 du CNRS, Equipe “Géométrie et Dynamique”, 175 rue du Chevaleret 75013 Paris, France E-mail: [email protected] For the “Mathematical Biosciences Subseries” of LNM: Professor P.K. Maini, Center for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LP, UK E-mail: [email protected] Springer, Mathematics Editorial I, Tiergartenstr. 17 69121 Heidelberg, Germany, Tel.: +49 (6221) 487-8259 Fax: +49 (6221) 4876-8259 E-mail: [email protected]

Local Lyapunov exponents: Sublimiting growth rates of linear random differential equations

Lyapunov Exponents

Linear Ordinary Differential Equations

Linear ordinary differential equations

Half-linear differential equations

Half-Linear Differential Equations

Linear Ordinary Differential Equations

Basic Linear Partial Differential Equations

Differential equations with linear algebra

Non-linear partial differential equations

Basic linear partial differential equations

Basic Linear Partial Differential Equations

Differential Equations with Linear Algebra

Ordinary Non-Linear Differential Equations

Differential equations and linear algebra

Differential Equations an Linear Algebra

Differential Equations with Linear Algebra

Wildlife Population Growth Rates

Galois Theory of Linear Differential Equations

Linear systems of ordinary differential equations

Galois Theory of Linear Differential Equations

Loewy Decomposition of Linear Differential Equations

Lie Algebraic Solution of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Introduction to linear systems of differential equations

Loewy Decomposition of Linear Differential Equations

Galois Theory of Linear Differential Equations

Local Lyapunov exponents: Sublimiting growth rates of linear random differential equations

Lyapunov Exponents

Linear Ordinary Differential Equations

Linear ordinary differential equations

Half-linear differential equations

Half-Linear Differential Equations

Linear Ordinary Differential Equations

Basic Linear Partial Differential Equations

Differential equations with linear algebra

Non-linear partial differential equations

Basic linear partial differential equations

Basic Linear Partial Differential Equations

Differential Equations with Linear Algebra

Ordinary Non-Linear Differential Equations

Differential equations and linear algebra

Differential Equations an Linear Algebra

Differential Equations with Linear Algebra

Wildlife Population Growth Rates

Galois Theory of Linear Differential Equations

Linear systems of ordinary differential equations

Galois Theory of Linear Differential Equations

Loewy Decomposition of Linear Differential Equations

Lie Algebraic Solution of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations

Introduction to linear systems of differential equations

Loewy Decomposition of Linear Differential Equations

Galois Theory of Linear Differential Equations

Recommend Documents