Probability and Partial Differential Equations in Modern Applied Mathematics (The IMA Volumes in Mathematics and its Applications)

The IMA Volumes in Mathematics and its Applications Volume 140

Series Editors Douglas N. Arnold Fadil Santosa

Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The primary mission of the IMA is to foster research of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems from other disciplines and industry. To this end, the IMA organizes a wide variety of programs, ranging from short intense workshops in areas of exceptional interest and opportunity to extensive thematic programs lasting a year . IMA Volumes are used to communicate results of these programs that we believe are of particular value to the broader scientific community. The full list of IMA books can be found at the Web site of the Institute for Mathematics and its Applications: http:j jwww.ima.umn.edujspringerjvolumes.html. Douglas N. Arnold, Director of the IMA

********** IMA ANNUAL PROGRAMS 1982-1983 1983-1984

Statistical and Continuum Approaches to Phase Transition Mathematical Models for the Economics of Decentralized Resource Allocation 1984-1985 Continuum Physics and Partial Differential Equations 1985-1986 Stochastic Differential Equations and Their Applications 1986-1987 Scientific Computation 1987-1988 Applied Combinatorics 1988-1989 Nonlinear Waves 1989-1990 Dynamical Systems and Their Applications 1990-1991 Phase Transitions and Free Boundaries 1991-1992 Applied Linear Algebra 1992-1993 Control Theory and its Applications 1993-1994 Emerging Applications of Probability 1994-1995 Waves and Scattering 1995-1996 Mathematical Methods in Material Science 1996-1997 Mathematics of High Performance Computing 1997-1998 Emerging Applications of Dynamical Systems 1998-1999 Mathematics in Biology Continued at the back

Edward C. Waymire

Jinqiao Duan

Editors

Probability and Partial Differential Equations in Modem Applied Mathematics

With 22 Illustrations

~ Springer

Edward C. Waymire

Jinqiao Duan

Department of Mathematics Oregon State University Covallis, OR 97331 [email protected]

Department of Applied Mathematics Illinois Institute of Technology Chicago. IL 60616 [email protected]

Series Editors: Douglas N. Amold Fadil Santosa Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455

Mathematics Subject Classification (2000) : 35Q30 , 35Q35. 37H05 . 60Hl5, 6OG60 Library of Congress Control Number: 2005926339 ISBN-IO: 0-387-25879-5 ISBN-13 : 978-0387-25879-9

Printed on acid-free paper.

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(MVY)

FOREWORD

This IMA Volume in Mathematics and its Applications

PROBABILITY AND PARTIAL DIFFERENTIAL EQUATIONS IN MODERN APPLIED MATHEMATICS contains a selection of articles presented at 2003 IMA Summer Program with the same title. We would like to th ank Jinqiao Duan (Department of Applied Mathematics, Illinois Institute of Technology) and Edward C. Waymire (Department of Mathematics, Oregon State University) for their excellent work as organizers of the two-week summer workshop and for editing the volume. We also take thi s opportunity to th ank th e National Science Foundation for their support of th e IMA.

Series Editors

Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA

PREFACE

The IMA Summer Program on Probability and Partial Differential Equations in Modern Applied Mathematics took place July 21-August 1, 2003, a fitting segue to the IMA annual program on Probability and Statistics in Complex Systems : Genomics, Networks, and Financial Engineering which was to begin September, 2003. In addition to the outstanding resources and staff at IMA, the summer program was developed with the assistance of the following members of the organizing committee: Rabi N. Bhattacharya, Larry Chen, Jinqiao Duan, Ronald B. Guenther, Peter E. Kloeden, Salah Mohammed, Sri Namachchivaya, Mina Ossiander, Bjorn Schmalfuss, Enrique Thomann, and Ed Waymire . The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. Researchers involved in contemporary problems concerning dispersion and flow , e.g. fluid flow, cash flow, genetic migration, flow of internet data packets , etc., were selected as speakers and to lead discussion groups. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. In organizing this program an explicit effort was made to bring together researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. A probabilistic representation of solutions to partial differential equations that arise as deterministic models, e.g. variations on Black-Scholes options equations, contaminant transport, reaction-diffusion, non-linear equations of fluid flow , Schrodinger equation etc . allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. In addition such approaches can be effective in sorting out multiple scale structure and in the development of both non-Monte Carlo as well as Monte Carlo type numerical methods. There is also a growing recognition of a role for the inclusion of stochastic terms in the modeling of complex flows. The addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations . During the last decade, significant progress has been made towards building a comprehensive theory of random dynamical systems, statistical cascades, stochastic flows, and stochastic pde's. A few core problems in the modeling, analysis and simulation of complex flows under uncertainty are : Find appropriate ways to incorporate stochastic effects into models; Analyze and express the impact of randomness on the evolution of complex vii

viii

PREFACE

systems in ways useful to the advancement of science and engineering; Design efficient numerical algorithms to simulate random phenomena. There is also a need for new ways in which to incorporate the impact of probability, statistics, pde's and numerical analysis in the training of present and future PhD students in the mathematical sciences. The engagement of graduate students was an important feature of this summer program. Stimulating poster sessions were also included as a significant part of the program. The editors thank the IMA leadership and staff, especially Doug Arnold and Fadil Santosa, for their tremendous help in the organization of this workshop and in the subsequent editing of this volume. The editors hope this volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and science. Jinqiao Duan Department of Applied Mathematics Illinois Institute of Technology Edward C. Waymire Department of Mathematics Oregon State University

CONTENTS Foreword

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Nonnegative Markov chains with applications. K.Bo Athreya Phase changes with time and multi-scale homogenizations of a class of anom alous diffusions Rabi Bhattacharya 0

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Semi-Markov cascad e representations of local solut ions to 3-D incompressible Navier-Stokes Rabi Bhattacharya, Larru Chen, Ronald B . Guenther, Chris Drum, Mina Ossuuuier, En rique Thomann, and Edward C. Waymire

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Amplitude equations for SPDEs: Approxim ate cent re manifolds and invariant measures Dirk Blomker and Martin Hairer

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Enstrophy and ergodicity of gravity currents Vena Pearl Bongolan- Walsh, Jinqiao Duan, Hongjun Gao, Tamay Ozgokm en, Paul Fischer, and Traian Iliescu Stochastic heat and Burgers equations and their singularities fan M. Davies, Aubrey Truman , and Huaizhong Zhao

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A gentle introduction to cluster expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 William G. Faris

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CONTENTS

Continuity of the Ito-rnap for Holder rough paths with applications to the Support Theorem in Holder norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 117 Peter K. Friz Data-driven stochastic processes in fully developed turbulence Martin Greiner, Jochen Cleve, Jiirgen Schmiegel, and K atepalli R . Sreenivasan Stochastic flows on the circle Yves Le Jan and Olivier Raimond Path integration: connecting pure jump and Wiener processes Vassili N. Kolokoltsov Random dynamical systems in economics Mukul Majumdar A geometric cascade for the spectral approximation of the Navier-Stokes equations M. Romito

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Inertial manifolds for random differential equations. . . . . . . . . . . . . . . . .. 213 Bjorn Schmalfuss Existence and uniqueness of classical , nonnegative, smooth solutions of a class of semi-linear SPDEs Hao Wang

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Nonlinear PDE's driven by Levy diffusions and related statistical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Wojbor A . Woyczynski List of workshop participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS· K.S . ATHREYAt Abstract. For a class of Markov chains that arise in ecology and economics conditions are provided for the existence, uniqueness (and convergence to) of stationary probability distributions. Their Feller property and Harris irreducibility are also explored. Key words. Population mod els, stationary measures, random iteration, Harris irreducibility, Feller property.

AMS(MOS) subject classifications. 60J05 , 92D25 , 60F05 .

1. Introduction. The evolution of many populations in ecology and that of some economies exhibit the following characteristics: a) It is random but the stochastic transition mechanism displays a time st ationary behavior, b) for small population size (and in small and fledgling economies) the growth rate is proportional to the current size with a random proportionality constant, c) for large populations the above growth rate is curt ailed by competition for resources (diminishing return in economies) . This leads to considering the following class of stochastically recursive time series model

(1) [0,00) ---+ [0,1] is continuous and decreasing , g(O) = 1, and are LLd. and independent of the initial value X o. These are called density dependent models (Vellekoop and Hognas (1997), Hassel (1974)). It is clear that {Cn}n >o defined by the above random iteration scheme is a Markov chain with stated space S = [0,00) and transition function where 9

{Cn}n~l

(2)

P(x, A)

= P(Cx g(x)

EA).

The goals of this paper are to describe some recent results on the existence of nontrivial stationary distributions, convergence to them, their uniqueness , etc .

2. Examples. a) Random logistic maps. The logistic model has been quite popular in the ecology literature to capture the density dependence as will as preypredator interaction (May (1976)). In the present context the parameter "Supported in part by Grant AFOSR IISI F49620-01-1-0076. This paper is based on the talk presented by the author at the IMA conference on Probability and P.D.E . in July-August, 2003. tSchool of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853 ([email protected]) ; and Iowa State University.

2

K.B . ATHREYA

C is allowed to vary in an LLd. fashion over time. Thus the model (1) becomes

(3)

n~ O

wit h X n E [0, 1]' Cn E [0,4] . T hus, the state space S = [0, 1] and g(x) == 1 - x has compact support . b) Random Ricker maps. Ricker (1954) proposed t he following model for th e evolution of fish population in Canada:

(4)

°

with X n E [0,00) , C n E [0, 00), < d < 00. Thus, th e st at e space S = [0,00) and g(x) == e- dx has exponential decay. c) Random Hassel maps. Hassel (1974) propos ed a model with a polynomial decay for large values. Here

(5)

°

with X n E [0, 00), C; E [0, 00), < d < 00. Here S = [0, 00), g(x) = (l+ x )-d . d) Yellekoop-Hiiqnos maps. A model that includes all th e previous cases was proposed by Vellekoop and Hognas (1997)

(6)

b>O

h : [0, 00) --+ [1,00), h(O ) = 1, h(·) is continuously differenti able and h(x ) = x~~W is nondecreasing. This family of maps exhibits behavior similar to th at of t he logistic fmaily such as pitchfork bifurcation of periodic behavior , chaotic behaivor as the parameter value is increased etc . The random logistic case was first introduced by R.N. Bhattachar ya and B.V. Rao (1993). Contribution s to it include Bhattacharya and Majumdar (2004), Bhattacharya and Waymire (1999), Athreya and Dai (2000, 2002), Athreya and Schuh (2002), Dai (2002), Athreya (2003), Athreya (2004a, b) . Deterministic interval maps have been studied a great deal in the dynamical systems literature. Random perturbations of such system have been investigated in th e book of Y. Kifer. Useful references for the deterministic case are the books by Devaney (1989), de Melo and van Strien (1993).

3. Random dynamical systems. The sto chastic recursive time series defined by (1) is an example of a random dynamical system obtained by iteration of random jointly measurable maps. This set up will be describ ed now. Let (S, s) and (K , K,) be two measur able spaces and f : K x s --+ S be jointly measurable, Le. (s x K" s) measurable. Let {Bi (W )} i ~l be a sequence of K

3

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS

valued random variables on a probability space (D, B , P). Let X o : D -. S be an S-valued r.v. Let

(7)

n ~

o.

Then for each n , X n : D -. S is a random variable and hence {X n+!(w)}n2:0 is a well defined S-valued stochastic process on (D,B,P). When {Bi };2:1 are LLd. LV. independent of X o then {X n }n2:0 is an Svalued Markov chain on (D, B, P) with transition function

P(x,A) = P{w : f(B(w), x)

(8)

f-

A} .

It turns out that if S is a polish space then for every probability tr ansition kernel P(·, .), i.e., a map from S x s -. [0,1] such that for each x, P(x , ·) is a probability measur e on (S, s) and for each A in s, P(" A) : S -. [0,1] is s measurable, there exists a random dynamical system of LLd. random maps {Ji(X,W)};2:1 from S x n -. S that is jointly measurable for each i and {Ji(·,W)}i2:1 are LLd. stoch astic processes such that the Markov chain generated by the recursive equation

(9) has transition function P(" .), i.e.

P(x , A)

=

P{w : f( x ,w) EA}.

See Kifer (1986) and Athreya and Stenflo (2000). As simple examples of this consider the following. 1. The vacillating probabilist . S=[O ,l], X _ Xn n +! 2 +

En+!

2

are LLd. Bernouilli (!) LV . Athreya (1996). 2. Sierpinski Gasket. Let S be an equilateral triangle with vertices Vl,V2,V3 and {Xn}n2:0 be define by {€n}n2:1

X

_ Xn n+! -

where

{€ n}n2:1

+ €n+! 2

are LLd. with distribution P(El

1

= Vi) = -3

i

= 1,2 ,3 .

3. Let {An , bn} n2:1 be LLd r.v. such that for each n, An is K matrix and b« is a K x 1 vector . Let

X

K real

4

K.B . ATHREYA

Suppose Elog IIAIII < 0 and E(log Ilblll)+ < 00 where "Alii is the matrix norm and Ilblll is the Euclidean norm. Then it can be shown that X n converges in distribution and the limit 1r is nonatomic (provided the distribution of (AI, bI) is not degenerate). Note that this example includes the previous two. Further, it can be shown that w.p.1 the limit point set of {Xn}n>O coincides with the support k of the limit distribution tt , This result has been used to solve the inverse problem of generating k. by running an appropriate Markov chain {Xn}n~O and looking at the limit point set of its sample path. For this the book by Barnsley (1993) may be consulted. When S is Polish and the {Jih>l are LLd. Lifschitz maps several sufficient conditions are known for the existence of a stationary distribution, its uniqueness and convergence to it . Two are given below. THEOREM 3.1. Let (S ,d) be Polish and (n,B,p) be a probability space. Let {Ji(x , w h~l be i.i.d. maps form S x n ~ S such that for each i fi is jointly measurable. Let Xn+l(w) = fn+I(Xn(w),w), n 2: 0 aJ Let Ji(·,w) be Lifschitz w.p.l and let

s(fl)

==

sup d(fl(x,w) ,fl(y,w) x #y d(x ,y)

Assume E(logs(fl)) < 0 and E(logd(fl(xo ,w),xo))+ < 00 for some Xo in S . Then, for any initial distribution, the sequence {X n} converges in distribution to a limit 1r that is unique and stationary for the Markou chain {X n } . bJ Let for some p > 0 sup E(d(fl(x,w),fl(y,w)))P < 1 x#y d(x ,y)

and for some Xo E(logd(fl(xo,w),xo))+
0, M < 00 such that i) "Ix ~ k, E(v(Xd IX o = x) - V( x) S -a. ii) "Ix E S, E(V(X l ) IX o = x) - V(x) S M . Then limf n,xo(k) 2 ",';M > O. In ecological and economic applications when S = [0, (0) , the above condition is verified for a compact set k c (0,00) so that I' is different from the delta measure at O. For proofs the above two results see Athreya (2004a, b) .

Is zr

I

Is

4. Stationary distributions for Markov chains satisfying (1) . Let {Xn}n >O be a Markov chain defined by (1). A necessary condition for the existence of a st ationary distribution 1r such that 1r(0 , (0) > 0 is provided below. THEOREM 4 .1. Let E(ln cl)+ < 00. Suppose there exists a probability distribution 1r on [0, (0) that is stationary for {Xn}n~O and 1r(0, (0) > O. Then, i) E(lncl)- < 00, ii) I Ilng(x)I1r(dx) < 00, iii) E In Cl = - I In g(x) 1r(dx) and hence strictly positive. CORO LLARY 4 .1. If E In Cl S 0 then 1r == 80 , the delta measure at 0 is the only stationary distribution for {Xn} n~O ' Furthe r, X n converges to 0 w.p.1 if E In Cl < 0 and in probability if E In Cl = O. A sufficient condition is given below. THEOREM 4.2. Let D == sup xg(x) < 00. Let i) EllnCll O, ii) Ellng(C l , D)I < 00. Then, there exists a stationary distribution 1r for { X n} such that 1r(0, (0) = 1. For the logistic case this reduces to ElnC l > 0 and Elln(4 - Cdl < 00 and for the Ricker case to E In Cl > 0 and EC l < 00. For proofs of these and more results see Athreya (2004) . The stationary distribution is not unique, in gener al. For an example in the logistic case see Athreya and Dai (2002). Under some smoothness hypothesis on the distribution of Cl uniqueness does hold as will be shown in the next section. For some convergence results see Athreya (2004a,b).

6

K.B. ATHREYA

5. Harris irreducibility. DEFINITION 5.1. A Markov chain {Xn}n>O with state space (S, s) and transition function P(·, .) is Hams irreducible with reference measure cp on

(S, s) if i) sp in a-finite and ii) cp(A) > 0 ===} P(Xn E A

for some

n ~ 11 X o = x)

is

>0

for every x in S.

(Equivalently if there exists a er-finite measure sp on (S, s) such that for each x in S, the Green's measure G(x, A) == I:~=o P(Xn E AI X o = x) dominates cp.) If S = N, the set of natural numbers and P == ((Pij)) is a transition probability matrix and if Vi, j :3 nij E Pi;ij > 0 then {Xn } is Harris irreducible with the counting measure on N as the reference measure. An important consequence of Harris irreducibility is the following THEOREM 5.1. Let {Xn}n;:::O be Hams irreducible with state space (S, s), transition function P(·,·) and reference measure cp. Suppose there exists a probability measure 1r on (S, s) that is stationary for P. Then i) n is unique. ii) For any x in S , the occupation measures r n,x(A) = ~ I:~-l P(Xj E AI X o = x) converge to 1r(') in total variation. iii) For any x in S, the empirical distribution Ln(A) ~ I:~-l lA (Xj) -+ 1r(A) w.p.l (Px ) (when X o = x) for each A in s. iv) {Xn}n>O is Hams recurrent i.e. cp(A) > 0 ::::} P(Xn) E A for some";~ 11 Xo = x) = 1 for all x in S.

The Markov chain vacillating probabilist (Example 3.1) is not Harris irreducible but will be if Ei has a distribution that has an absolutely continuous compnent. It is also known that if s is countably generated then every Harris recurrent Markov chain with state space (S, s) is regenerative in the sense its sample paths could be broken up into a sequence of LLd. cycles as in the discrete state space case. For a proof of this and Theorem 5.1 see Athreya and Ney (1978), Nummelin (1984), Meyn and Tweedie (1993). In the rest of this section conditions will be found for Harris irreducibility of {Xn}n;:::O defined by (1). Assume that {Cn}n;:::l are LLd. with values in (0, L), L :s 00 and for each c E (O,L), fc(x) == cxg(x) maps S = (O,k), k:S 00 to itself. For any function f : S -+ S the iterates of f are defined by

The first step is a local irreducibility result. THEOREM 5.2. Suppose i) :30 < a < 00, 8> 0, a Borel function lIt : J == (a - 8,a + 8) (0,00) -+ P(C1 E B) ::::: fBn] lIt(B)dB for all Borel sets B .

-+

NONNEGATIVE MARKOV CHAI NS WITH APPLICATIONS

ii) ::10 < P < 00, m f~m)(p) = p .

7

2: 1 su ch that for the fun ction f O;( x) == ax g(x ),

°-;

°

Then, ::11] > "Ix E I == (p -1] ,p + 1]) , Px(X m EA) > fo r all Bo rel sets A such that iP(A) == .\(A n 1) > where .\ is Lebesgn e m easure. COROLLARY 5. 1. Suppo se in additi on t o the hypotheses of Th eorem 5.1 , Px(Xn E I fo r some n 2: 1) is> for all x in (0, k) . Th en {X n }n2: 0 is Hams irreducible with st ate space S = (0, k). Using a deep result of Gu ckenheim er (1979) on S-unimodal m aps a sufficient condition for the hyp otheses of Corollar y 5.1 ca n be found. DEFINITION 5 .2. A ma p h : [0,1] -; [0,1] is S-unimodal if i) h( .) E C3, i. e. 3 times con ti nuous ly differentiable, ii) h(O) = h(l) = 0, iii) ::I < c < 1 ::1 h"(c) < 0, h is increasing in (0, c) an d decreasing in (c,l) an d

° °

°

' ) (S f )(X ) = - h"l(x) h' (X ) > Od zv h"(x) - '3( 2 hll(X h'(x))) 2'f Z an - 00 Z'f h' (x ) -- O

°

°

is < fo r all < x < 1. EXAMPLES. h( x) === cx (l - x ), 0 < c::; 4, h( x ) = x 2 sin JrX . DEFINITION 5 .3 . A number p in (0,1) is a stable periodic point fo r h if for som e m 2: 1: h(m )(p) = p and Ih(m)(p)1 < 1. DEFINITION 5.4 . For x in (0,1) th e orbit Ox is th e set {h (m)(x)}m 2:0 and w( x) is the limit point set of Ox. THEOREM 5.3 (Guckenheim er (1979)) . Let h be S-unimodal with a stable periodic point p . Let K = {x : < x < 1, w(x ) = w(p )} . Th en , .\(K) = 1 where .\( .) is the Lebesgu e m easure. Combining Theorem 5.2, 5.3 and Corollar y 5.1 leads to THEOREM 5.4 . Let S = [0,1] . A ssume i) "1 O < c < k , hc(x) == cx g(x ) is S -uni m odal. ii) ::I < P < 1, < a < L ::1 P is a st able peri odic point fo r hO;(x) == axg(x ). iii) ::I 8 > 0, a Bo rel fun ction Ill : J == (a - 8,a + 8) -; (0,00) ::1 P(C 1 E B) 2: f Bn J III(B)dB fo r all Borel sets B . Th en , th e Markov chain {X n} n2:0 defin ed by

°

°

°

n=0 ,1 ,2 , . .. where {Cn} n2:1 are i.i.d . is Ham s irreducibl e with state space (0,1) referenc e m easure cP(·) = .\(. n 1) where I = (p -1], p + 1]) for some approp riate 1] > 0. As a special cas e applied to random logistic map s one gets THEOREM 5 .5. Let S = [0,1]' let {Cn} n2:1 i.i.d. (0,4] valued r.v . and {X n}n2:0 be th e th e Markov chain defin ed by

n

2: 0.

Suppose ::I an open interval J C (0,4) and a function Ill : J -; (0,00) -; P(C1 E B) 2: fBnJ III(B)dB fo r all Borel sets B .

8 If J

K.B . ATHREYA

n (1,4)

Cl 3 f{3(x)

= ip , assume in addition, that :3 j3 > 1 in the support of == j3x(l - x) admits a stable periodic point p in (0,1). Then

{Xn}n>O is Hams irreducible . COROLLARY 5.2 . Suppose, in addition to the hypotheses of Theorem 5.5, that:3 InCI > 0 and Elln(4-CI )1< 00 . Then,:3 a unique stationary

measure 1r for {Xn } such that i) 1r(0 , 1) = 1, ii) 1r is absolutely continuous, iii) V 0 < x < 1, Px (X n E .)

--+

1r(') in total variation.

For proofs of all the results in this section except Theorem 5.1 see the Athreya (2003) . It has been pointed out by one of the referees that the above Corollary has been obtained independently by R.N. Bhattacharya and M. Majumdar in a paper entitled "Stability in distribution of randomly perturbed quadratic maps as Markov processes" , CAE working paper 0203, Department of economics, Cornell University. REFERENCES [1] ATHREYA K.B . (1996). The vacillating mathematician, Resonance. J . Science and Education, Indian Academy of Sciences , Vo!. 1, No. 1. [2] ATHREYA KB . (2003). Harris irreducibility of iterates of LLd. random maps on R+. Tech . Report, School of ORIE, Cornell University. [3] ATHREYA K .B . (2004a) . Stationary measures for some Markov chain models in ecology and economics. Economic Theory, 23 : 107-122. [4] ATHREYA K.B . (2004b) . Markov chains on Polish spaces via LLd. random maps . Tech. Report, School of ORIE, Cornell University. [5] ATHREYA K.B. AND DAI J.J. (2000). Random logistic maps I. J . Th. Probability, 13(2): 595-608. [6] ATHREYA KB . AND DAI J .J . (2002). On the nonuniqueness of the invariant probability for LLd. random logistic maps. Ann . Prob., 30: 437-442. [7] ATHREYA K.B . AND NEYP . (1978). A new approach to the limit theory of recurrent Markov chain . Trans. Am . Math. Soc., 245: 493-501. [8] ATHREYA K.B . AND SCHUH H.J. (2003). Random logistic maps Il, the critical case . J . Th. Prob., 16(4): 813-830. [9] ATHREYA KB . AND STENFLO O . (2000). Perfect sampling for Doeblin chains . Tech . Report, School of ORIE, Cornell University. (To appear in Sankhya, 2004). [10] BHATTACHARYA R.N . AND RAo B.V . (1993). Random iteration of two quadratic maps. In Stochastic processes: A. Fetschrift in honor of G. Kallianpur, pp . 1321, Springer. [11] BHATTACHARYA R .N . AND MAJUMDAR M. (2004). Random dynamical systems: A review . Economic Theory, 23(1) : 13-38. [12J BHATTACHARYA R.N. AND WAYMIRE E .C . (1999). An approach to the existence of unique invariant probabilities for Markov processes, colloq ium for limit theorems in probability and statistics. J . Bolyai Soc. Budapest. [13] BARNSLEY M.F. (1993). Fractals everywhere. Second edition, Academic Press, New York. [14] CARLSSON N. (2004). Applications of a generalized metric in the analysis of iterated random funct ions. Economic Theory, 23(1) : 73-84. (1980) . Iterated random maps on the interval as [15] DEVANEY R.L . (1989). An introduction to chaotic dynamical systems. 2nd edition, Academic Press, New York.

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS

9

[16] DE MELO W . AND VAN STRIEN S. (1993) . One dimensional dynamics. Springer. [17] DIACONIS P . AND FREEDMAN D.A . (1999) . Iterated random function . SIAM Review, 41 : 45-76. [18] GUCKENHEIMER J . (1987) . Limit sets of S-unimodal maps with zero entropy. Comm. Math. Physics, 110: 655-659. [19] HASSEL M.P. (1974) . Density dependence in single species populations. J . Animal Ecology, 44: 283-296. [20] KIFER Y. (1986) . Ergodic theory of random transformations. Brikhauser, Boston. [21] MAY R.M . (1976). Simple mathematical models with very complicated dynamics. Nature, 261: 459-467. [22] MEYN S. AND TWEEDIE R.L . (1993) . Markov chains and stochastic stability, Springer. [23] NUMMELIN E. (1984) . General irreducible Markov chains and nonegative operators. Cambridge University Press. [24] RICKER W .E . (1954) . Stock and recruitment. Journal of Fisheries Research Board of Canada, 11:559-623. [25] VELLEKOVP M.H . AND HOGNAS G . (1997) . Stability of stochastic population model. Studia Scientiarum Hungarica, 13 : 459-476. [26J WEI BIAO Wu (2002). Iterated random functions : Stationary and central limit theorems. Tech . Report , Dept. of Statistics, University of Chicago.

PHASE CHANGES WITH TIME AND MULTI-SCALE HOMOGENIZATIONS OF A CLASS OF ANOMALOUS DIFFUSIONS* RABI BHATTACHARYA t Abstract . Composite media often exhibit multiple spatial scales of heterogeneity. When the spatial scales are widely separated, t ransport through such medi a go through distinct phase changes as time progresses. In the pres ence of two such widely separated scales, one local and one large scale , the time scale for t he appearance of the effects due to the large scale fluctuations is det ermined. In the case of t ransport in period ic media with such slowly evolving heterogeneity and divergence-fr ee velocity fields , there is a first Gaussian phase which breaks down at the above t ime scale, and a second Gaussian phase occurs at a later time scale which is also precisely determined. In between there may be non-Gaussian phases, as shown by examples. Dep ending on the structure of the large scale fluctuations , the diffusion is either super-diffusive, with the effective diffusivity increasing to infinity, or it exhibit s normal diffus ivity which increas es to a finite limit as time increases. Sub-diffusivity, with the effect ive diffusion coefficient tend ing to zero in time, is shown to arise in a cert ain class of velocity fields which are not divergence-free.

1. Introduction. Electric and thermal conduction in composite media as well as diffusion of matter through them are problems of much significance in applications (see, [5-7, 16, 21]) . Ex amples of such composite media are natural heterogeneous material such as soils, polycrystals, wood , animal and plant tissue, cell aggregates and tumors , and synthetic products such as fiber composites, cellular solids , gels, foams, colloids, concrete, etc . The evolution equation that arises in such conte xt s is gener ally a Fokker-Planck equation of the form

(1.1)

ac(t, y) 1 at = 2" \7 . (D(y)\7c) - \7 . (v(y) c),

c(O, .) = Ox

where D( ·) is a k x k positive definite matrix-valued function depending on local properties of the medium, and its eigenvalues are assumed bounded away from zero and infinity; v(·) is a vector field which arises from other sources. To fix ideas one may think of v(·) as the velocity of a fluid (say, water) in a porous medium (such as a saturated aquifer) in which c(t , y) is the concentration of a solute (e.g., a chemical pollutant) injected at a point in the medium ([12 , 16,21 ,25,31 ,36,38]) . One may also think of (1.1) as the equation of transport, or diffusion, of a substance in a turbulent fluid ([1,3,35]). One of the main aims of the study of t ransport in disordered media is to derive from the local , or microscopic, Equation (1.1) a ma croscopic equation with const ant coefficients governing c over much larger space/time *Research supported by NSF Grant DMS-OO-73865. tDepartment of Mathematics, Univ ers ity of Arizona, ([email protected]) . 11

Tu cson ,

AZ

85721

12

RABI BHATTACHARYA

scales , under appropriate assumptions. Such a derivation is known as hom ogenization in par tial differential equations. The macroscopic equation is then of the form ac(t, y) = ~ ~ D . . a _ ~ e. ac at 2 L...J t ,) ay . ay . L...J t ay. ' 2c

(1.2)

i,j=l

i= l

t)

t

where D = (Di,j) is the effective dispersion or , diffusivity. This program has been carried out in complete generality for periodic D (·), v( ·) in Bensoussan et al. (1978) (also see [1, 2, 8, 23, 30, 38]). Another popular model assumes D (·), v (·) are stationary ergodic random fields ([1, 2, 7, 23,38]). P apanicolaou and Varadhan (1980) and Kozlov (1979) independently derived homogenizations when (1.1) is in divergence form (i.e., v(·) = 0 in (1.1)) . For a class of two-dimensional problems in such random media wit h D (·) = D constant and v(.) a (divergence free) shearing motion, a der ivation of homogenization and analysis of asymptotics is carried out in Avellaneda and Majda (1990), (1992) (also see [1]). From a probabilistic point of view, homogenization of (1.1) in the form (1.2) means t hat a diffusion (Markov process ) X( ·) generated by A = ~\7. (D (.) \7) + v( ·) . \7 converges in law, under a scaling of time and space with properly large units , to a Brownian motion WO with (constant) diffusion matrix D and (constant) drift velocity vector v: (1.3)

cX(!-) - ! v --. W(t) , c;2

E

(t

~

0),

as s ]

o.

It is known t hat if the coefficients are periodic, or stationary ergodic random fields, and v(·) is divergence free, the effective diffusivity is larger than the average of the local diffusivity D(· ). We have so far considered homogenization und er a single scale of het erogeneity. Natural composite media generally exhibit multiple scales of het erogen eity, i.e., heterogeneity th at evolves with distance. It has been observed in many instances, and sometimes verified theoretic ally, that this often leads to increase in the effective dispersivity D with the spatial scale, say, L. For t he case of solute dispersion in porous media, such as saturated aquifers, one may see this by int roducing a scale parameter in v(·), or by relating D to the correlation length, and still using a single large scale ([13, 23, 38]). Our objective in the present survey is to introduce different widely separated spatial scales of heterogeneity explicitly in the model and study (i) the effective diffusivity as a funct ion of the spatial scale, and (ii) the time scales for the different (Gaussian and non-G aussian ) ph ases t he diffusion pass es through as time progresses. In the next sect ion we give a fairly complete description of this for the case of periodic coefficients and a divergence free velocity field v(·) with two widely separ at ed scales- a local scale and a large scale . The case of additional appropriately widely separated

PHASE CHANGES WITH OF A CLASS OF ANOMALOUS DIFFUSIONS

13

scales may be understood from this. Examples in Section 4 illustrate the emergence of non-Gaussian phases in between Gaussian ones. Before concluding this introduction, let us mention the classical work of Richardson (1926) who looked at already existing data on diffusion in air over 12 or so different orders of spatial scale, and conjectured that the diffusivity DL at the spatial scale L satisfies

(1.4) This was related later by Batchelor (1952) to the turbulence spectrum v ex L 1/ 3 derived by Kolmogorov (1941). The length scale L(t) and the diffusion coefficient DL(t), as functions of time t, are now related using L(t) as the root mean squared distance from the mean flow (see Ben Arous and Owhadi (2002)): L2(t) ex DL(t)t ex L4/3(t)t, leading to L(t) ex t 3/ 2 and DL(t) ex t 2. This was also derived by Obhukov (1941) by a dimensional argument similar to that of Kolmogorov (1941). In particular, DL(t) ---+ 00 as t ---+ 00 , that is, this is a case of super-diffusivity. For a precise analysis of a two-dimensional model with constant D( .) = D and a stationary ergodic v( ·), we refer to Avellaneda and Majda (1990), (1992). 2. A general model with two spatial scales: The first phase of asymptotics and the time scale for its breakdown. Consider the general model (1.1) with v(·) of the form

v(y) = b(y) + I'(~) ,

(2.1)

where a is a large parameter, b(.), and 1'( -!a) represent the local and large scale velocities, respectively. The solution to (1.1) is the fundamental solution p(t;x, y) . Consider a diffusion X(t) , t 2 0, on R k with transition probability density p, starting at x = X (0). To avoid the artificial importance of the origin, take the initial point x to be

x = axo

(2.2)

where Xo is a given point in Rk, so that the initial value of I'Ua) is I'(xo). One may represent such a diffusion as the solution to the stochastic integral equation

X(t)

axo + it {b(X(S)) + d(X(s)) + 1'( X~s)) }dS

(2.3)

+ it a(X(s))dB(s) , where a(x) JD(x), d(x) (d1(x) , .. . ,d k(x))', dj(x) L.i(fJ/fJXi) Dij(x), and BC) is a standard k-dimensional Brownian motion. Since I'(-!a) changes slowly, at the rate of s.]«, one expects that for

14

RABIBHATTACHARYA

an initial period of time the process X (.) will behave like the diffusion Y(.) governed by

Y(t) =

axo + it {b(Y(s))

+ d(Y(s)) }ds + t,8(xo)

(2.4)

+ it (1(Y(s))dB(s) . Indeed, the £i-distance between p(t;x, y) and the tr ansition density q(t;x, y) of Y (t) is negligible for t he times t « a2 / 3 . Actually, the total variation distance l!Poot - QO,tllv between the distributions Poot of the process {X (s) : 0 'S s 'S t} and the distribution QO,t of the process {Y(s) : 0 'S s 'S t} goes to zero in this range . More precisely, one has the following result obtained in [12] (also see [9]). THEOREM 2.1. Assume b(.) and its first order derivatives are bounded, as are D( ·), ,8(.) and their first and second order derivatives. Assume also that the eigenvalues of D( .) are bounded away from zero and infinity. Then

l!Poot -

(2.5)

QO,t Ilv-----. 0

as

t

a2 / 3

-----.

O.

Proof By the Cameron-Martin-Girsanov Theorem (Ikeda and Watanabe (1981), pp. 176-181) ,

Z(t) (2.6)

:=

it (1-1(Y(S)){ ,8(Y~S)) ,8(Y~O)) _~ it 1(1- 1(Y(S)){ ,8(Y~s)) ,8 ( Y~O) ) }1 -

}dB(S)

_

2

ds.

Since Eexp{Z(t)} = 1, Ell - exp{Z(t)}1 = 2E(1 - exp{Z(t)})+ 'S 2[EIZ(t)1 /\ 1]. Now the expected value of the second integral in (2.6) can be shown, using Ito 's Lemma ([26]), to be bounded by [C1t2j a2 + C2t3j a2+ c3t3ja4]jA where A is the infimum of all eigenvalues of D(.), and C1, C2,C3, depend only on the upper bounds of the components of b( ·), ,8(.), D( ·) and of their first order derivatives, and also of the second order derivatives of ,8(.). Since the expected value of the square of the norm of the stochastic integral equals the expected value of the Reimann integral of the squared norm of the integrand, one has 1 1/2 l!Poot - QOotllv 'S (} + 2(}

where () = [C1t2 ja 2 + C2t3j a2 + C3t3 j a4]jA. 0 One may show by examples (see Section 4) that the large scale fluctuations (namely, fluctuations of ,8Ua)) can not be ignored in general for times t of the order a 2 / 3 or larger, i.e., the time scale in (2.5) is precise.

PHASE CHANG ES WITH OF A CLASS OF ANOMALO US DIFFUSIONS

15

Theorem 2.1 implies that a first homogenization occurs for times 1 « t « a 2/3, provided y( .) defined by (2.4) is asymptotically Gaussian. This is the case, e.g., if b(·), D( ·) are period ic, or are ergodic random fields satisfying some additional condit ions ([1- 3, 14, 34,38]). No assumption is needed on f3 (.), except the smoothness and boundedn ess conditions impos ed in Theorem 2.1. To illustrat e thi s, let b(.) and D( ·) be periodic with the same period lattice, say, and assume for simplicity that

zr.

(2.7)

divb(·)

= o.

Then, by Bensoussan et al. (1978) (or, Bhattachar ya (1985) ), and Theorem 2.1, one has

(2.8)

lim

O.

Thus the transition density qa(t;x ,y), say, of Y(.) satisfies Doeblin's condition with the same lower bound J > 0 (at t = 1) and, therefore, one has (see, e.g., [15, pp. 214, 215]) (3.9)

sup a E Z\{O}, x E

T iTr Iqa(t;x,y)-lldx~c'e-fJt , 1

t

2 0,

1

for some positive constants c', J. For the transition density Pa of the diffusion XC) on the big torus Ta, (3.9) implies

18

RABIBHATTACHARYA

yielding the precise order of the relaxation time t » a 2 • For the analysis of final phase dispersion as a function of the scale parameter, it is convenient to again look at the related process Y(t) = X(a 2t)/a. Then Y(t) := Y (t) mod 1 is a diffusion on the unit torus 7i with generator (3.11) A a := 1) + a(b(a·) + (3( .)) . \1,

1

with

1)

LD 2 0 0' ·

=-

fj2

jj,

a a . Xj

J ,J

»r

We will sketch briefly th e main ideas of the derivation, the details of which may be found in [9, 10, 12]. Since b(a·) is rapidly oscillating, one may approximate Aa by (3.12)

A := 1) + a(h + (3(.)) . \1,

where b (hI," " hk ) is the mean of b(·) w.r.t. the uniform distribution on 7i. According to th e central limit theo rem for X (t) , with a fixed, the asymptotic dispersion (or variance) per unit tim e of Yj(t) is given by Djj - 2l1gjlli , where gj is the mean-zero solution of Aagj(x) = bj(ax) + (3j( x) - hj - ilj. Here Ilgjlll is the norm in t he complex Hilbert space HI = {h mean-zero, periodic: Ihl2 and l\1hl 2 integrable w.r.t. uniform distr ibution on 7i} endowed with the inner product ts, fh = J[O, l)k(\1g(x) )'D\1f(x)- dx, being the complex conjugate of f . One may replace gj by the solution hj to Ah j = {3j - ilj . The last equation may be expre ssed as

r:

(I +a1)-l(b+ {3(.)) . \1)hj

D - 1 ({3j - ilj) ,

+ as)hj

1)-l ({3j - ilj),

(I (3.13)

or

S .- 1)-l (b + (3(.)) . \1, I is the identity operat or.

Since S is a skew-symmetric compact operator on HI , one may now use the spectral decomposition of S to express h j in an eigenfunction expansion, arriving at Ilhj Ili ~ Ilgj Ili· This gives an asymptotic relat ion between a and the dispersion of X)(t) as th at of a2 times that ofYj(t) . The dominant term in this expansion of th e dispersion is 2a211(1)-1({3j -ilj ))Nlli where fN is the pro jection of f in HI ont o th e null space N of S. Thus, if (V- 1( {3j - ilj))N i:0, the disp ersion of Xj (t) per unit tim e grows with a quadratically and is asymptotically bounded away form 0 and 00 if (V- 1({3j - ilj ))N = O. We then have the following result s. First, we assume a technical condition which is most probably redundant : A4: (V- 1({3j( ') - ilj))N is twice continuously (3.14)

differenti able for 1 :$ j :$ p.

PHASE CHANGES WITH OF A CLASS OF ANOMALOUS DIFFUSIONS 19

Assume A1 - A4, with C V- I ({3j (·) - !Jj ))N , 1 ::; j ::; p, linearly independent, for some index p, 1 < p ::; k . Th en the follo wing hold: a) Th e eigenvalues of r := ~(( Kij)h~ i,j~P are bound ed away from zero and infinity as a ranges over the set of all positive integers. b) For the tim e scale t » a 2 , i. e., as t ----> oo, a ----> 00 , and THEOREM 3 .1 ([9, 10 , 12]) .

t

(3.15)

2 a

----> 00 ,

one has

where [... ]i denotes th e first p coordinates of the vector inside [...], and I p is the p x p iden tity m atrix. REMARK 1. Suppose t sat isfies (3.15) and t = 0(a 2 +8 ) for some 8 > O. 1

Then a = O( t (2+, < 11 >, .. .},

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

35

v = (2 , 1 ,2, . .. ) E 8V

( 11 2)

( 2 12)

( 11 )

( 2 1)

( 1)

( 2 2 2)

(22 )

( 2)

B FIG. 1. Full binary tree with index set V and boundary 8V . The path v = (2,1,2, . . .) E 8V is indicated in bold, with vlO = B, viI = (2), vl2 = (21), and vl3 = (212).

where {I, 2}O = {O} . Also let aV = n~o{1, 2} = {I , 2}N. A stochastic model consistent with (2.1) is obtained by consideration of a multitype branching random walk of nonzero Fourier wavenumbers ~ , thought of as particle types , as follows: A particle of type ~ =1= initially at the root 0 holds for a random length of time SIJ distributed according to p(~ , dt). When this clock rings, a coin /'i,1J is tossed and either with probability ~ the event [/'i,1J = 0] occurs and the particle is terminated, or with probability ~ one has [/'i,1J = 1], the clocks are re-set and the particle is replaced by two offspring particles of types TJ , ~ - TJ selected according to the probability kernel q(~ , dTJ) . This process is repeated independently for the particle types TJ and ~ - TJ rooted at the vertices < 1 >, < 2 >, respectively. A more precise description of the stochastic model requires more notation. For v = (Vl ,V2 , ,, . ,Vk) E V, let Ivl = k, 101 = 0. For v = (Vl,V2, .,,) E aV, and j = 0,1,2 , . " let vlj = (Vl ,,,.Vj) , vlo = e. That is, for v E aV, vlO, vll , vI2, ... may be viewed as a path through vertices of the tree starting from the root vlo = e. For u , v E aV, let [u /\ vi = inf {m 2': 1 : ulm =1= vim}. The following requirements provide the defining properties of the underlying stochastic model. The model depends on the initial frequency (wave number) ~ . For fixed ~ =1= let {(~v , /'i,v) : v E V} be the tree-indexed (discrete parameter) Markov pro cess starting at (~IJ , /'i,1J) with ~IJ = ~, /'i,1J E {a, I}, taking values in the state space (R3 \{0}) x {a, I}, and defined on a probability space (0, F , Pe) by the following properties:

°

°

36

RABI BHATTACHARYA ET AL .

1. Pe (~o E B , KO= K) = !8dB ), BE

e;

K E {O, I}. 2. For any fixed v E av, (~vlo , Kvlo) , (~Vl l' Kvp), (~vI2 ' KVI2) " .. is a Markov chain wit h transit ion probabilities

(2.17)

Pdevln+l E B , Kvln+l

1

= Kla( {(eu ,Ku) : lul ::; n})) = "2 q(e, B )

for B E B(R3\{0} ), K E {0,1} . 3. For any u , v , E av , {(eulm, Ku lm )}~=o and {(~v l m , Kvlm)}~=o are conditionally independent given a( {(~w , Kw) : [w] ::; [u 1\ v i}). 4. For v E V, ~v1 + ~v2 = ~v Pe - a.s., where vj = (V1.. . vn)j := (V1.. .Vn, j), j = 1,2 , . .. is the concatenation operat ion. 5. Let {Sv : v E V} be a collection of non-negative rand om variables such that for each m 2=: 1, conditionally given eo = and et := {ev : v E v ,lvl 2=: I} , the random variables Sv,v E V are independent with respective (conditional) margin al distributions p(ev, ds). Our objective now is to use the stochastic branching model to recursively define a random funct ional related to (2.1) through its expected value. By a backward recursion one may define a (non-random) function !(z , z+, s, s+,K,K+,t ) where z E Rk\{O} , z+ E (Rk\ {O})v+,s E [0, 00), s+ E [O,oo)v+,'and K E {1,2} , K+ E {1,2} v+, where

e

V+ := {v E V : Ivl 2=: I} ,

(2.18) such t hat

! (Z, Z+, S, S+,K, K+, t ) = a(z ,t )l [t ,oo)(s) + l [o,t)(s)l{1} (K)b(z, t - s) . ! (Zl, zt , Sl , st, K1,Kt, t - s) 0 z ! (Z2, zt, S2 , st, K2, Kt ,t - s) + l [o ,t)(s )l{o}(K)c(z , S,t - s), where for x E SV we define (2.19)

(2.20)

In the event that

K v = 1 for all v the recursion may not terminate. In this case one simply defines ! (z , z", s, s+, K, K+,t ) == 0). Otherwise the backward recursion is sure to termin ate and! is well-defined. For this given functio n ! let us now define a random functional of the random fields {~v : v E V }, {Kv : v E V} , and {Sv : v E V} defined on (0, F , P ) for wE 0 by t he composition

(2.21)

X(B, t)(w)

:= ! (~o (w ) , ~t (w ) , So(w), s t

(w), KO(W), Kt(W), t ).

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

37

A careful formulation and details of a proof is given in a PhD thesis by Orum (2004) which yields the following: THEOREM 2.1. If EIX (0,t)\ < 00 then a solution of (2.1) is given by x(~ , t) :=

Ef.o=f.X (0, t) .

We will conclude this section with an application to local existence theory for Navier-Stokes via a semi-Markov cascade representation. Taking a constant failure rate >.(~, t) = 6 > 0, this includes a special case obtained by Orum (2004) of local existence having exponentially distributed holding times. THEOREM 2.2. Assume that there is a 0 < T* :::; 00 such that for all ~ E Rk\{O}, max{la(~,t)l, Ib(~,t)l,suPo< s 0,

one obtains a local solution to (FNS) in

F h o,r ,O,T •.

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

39

3. Time-asymptotic steady state solutions. In this section we compute time-asymptotics under supplemental conditions for the existence of a unique global solution to 3-d incompressible Navier-Stokes equations. For this we begin with the following theorem quoted from Bhattacharya et al. (2003) for ease of reference. THEOREM 3 .1. Let h (~ ) be a standard maj orizing kern el with exponent B = 1. Fix 0 < T ~ +00. Suppos e that luol h .o . T ~ (/2ii) 311 / 2 and I(- ~) -1 gl.1"h .O . T ~ (/2ii)3 112 /4. Th en there is a un ique solution u

in the ball Bo(O ,R) cen tered at 0 of radius R = (/2ii )311/2 in the space Moreover the Fouri er transform of the solution is given by u(~ , t) =

Fh ,O ,T .

h(~)Ef.X (T(~ , t)) , ~ E W~3) and t ~ T . As an immediate consequ ence one readily obtains st eady-st at es as follows. COROLLARY 3 .1. Under the conditions of Th eorem 3.1 with T = 00, suppose further that limt->oo 9(~ , t) = 900(~) exists for each ~ -1= O. Then Uoo(~) :=

lim u(~ , t)

t-s- co

exists and satisfies the st eady state Na vier-Stokes (FNS)oo defin ed by

Uoo(O

= t OO e-VIf.12S { I~I (27r) -~ ( uoo (1])

lo

lR3

® f.

uoo (~ -1] )d1] + 900(0 }ds.

Proof. Note that the und erlying discrete par am et er binary branching is critical. Thus limt->oo X(B, t) exists a.s. as a finit e random product. Moreover, under the conditions of Theorem 3.1 one has for each t ~ 0, with probability one IX(B, t)1 ~ 1. Thus, by Lebesgue's Dominated Convergence Theorem Xoo (~ ) := lim X(~ , t)

t-+oo

exists for each Z. Now again apply Dominated Convergence to (FNS)h to obtain

Multiplication through by h(~) proves the assertion for uoo(~) 2g oo (f.) ( C) -- vlf.12h(f.)· an d 0, (:F(v + c/J), v) ::; KIIc/J11 4

(2.11)

-

811vl1 4 -

,L(Lv, v) ,

for any c/J,v E n' , Concerning the stochastic perturbation we will always assume that the following is true. (For a detailed discussion of Q-Wiener processes and stochastic convolutions see [18] .) ASSUMPTION 4. The noise process is formally given by ~ = QOtW , where W is a standard cylindrical Wiener process in 'It with the identity as a covariance operator and Q E £('It, 'It) is symmetric. Furthermore, there exists a constant a < such that

!

(2.12) where" . to 1-£ .

IIHS(1i)

denotes the Hilbert-Schmidt norm of an operator from 'It

REMARK 2.1. Straightforward computations, combined with the properties of analytic semigroups allow to check that Assumption 4 implies the following (see [18, Section 5.4) for the first assertion}: • The stochastic convolution Wdt) = f~ eL(t-s)Q dW(s) is an 1-£valued process with Holder continuous sample paths . • There exist positive constants C and 'Y such that

(2.13) holds for every t > O. 2.2. Note that we do not assume that Q and L commute. Hence , it is in general not true that Q and Pc commute. Therefore , the noise processes PcQW and PsQW will not necessarily be independent, which implies that the amplitude equation (2.4) and equation (2.5) for the second order correction are coupled through the noise. It is straightforward to verify that REMARK

46

DIRK BLOMKER AND MARTIN HAIRER

Therefore, the stochastic convolution is a Wiener process on N and it is a stable Ornstein-Uhlenbeck process on S . This means that the noise acts in two completely different ways on Peu and Piu for some mild solution u. To give a meaning to (2.1) we will always consider mild solutions, which are given by the following proposition. PROPOSITION 2.1. Under Assumption 1, 2, and 4, for any (stocha stic) initial condition Uo E 'H equation (2.1) has a unique local mild solution u. This means we have a stopping time t* > 0 and a stochastic process u such that u : [0, t*] --t 'H is a solution of (2.14) for t

~

u(t) = etLuo

+

it

e(t-T)L[€2Au + F(u)](r)dr

+ €2Wdt)

r.

Suppose additionally that Assumption 3 is true, then the solutions are global, which means t* = 00. For the proof note that the existence and uniqueness of local solutions is standard since we consider locally Lipschitz-continuous nonlinearities. See for example [18, Section 7] we can also apply the deterministic approach of [26, Thm. 3.3.3] path-wise. For LP-theory with application to NavierStokes eq. see for example [10, 11]. The global existence follows from standard a-priori estimates for v = u - WL , as v is a weak solution of the following PDE with random coefficients: (2.15)

The formal idea is to take the scalar product with v, in order to derive estimates for IIvl12 and hence Ilu112. 2.2. Examples of equations. In the literature there are many examples of equations of the type given by Assumptions 1, 2 or 3, and 4. For instance, the well-known Ginzburg-Landau equation (see [20] for a standard proof of existence)

and the Swift-Hohenberg equation OtU = -(~

+ 1)2u + vu -

u 3 + a~ ,

which was first used as a toy model for the convective instability in a Rayleigh-Benard problem (see [27] or [15]), fall into the scope of our work when the parameters v and a are small and of comparable order of magnitude. Both equations are considered on bounded domains with suitable boundary conditions (e.g. periodic, Dirichlet, Neumann, etc.). Other equations could involve nonlinearities of the type o;(u 3 ) or ox((oxu)3). The first nonlinearity is considered with the Sobolev space

AMPLITUDE EQUATIONS FOR SPD E

47

1t = H - 1 , while the second one has to be considered in £2 , provided we have the following Poincar e typ e inequality Il ull :S C lloxull for u E D (L ). Another example arising in t he t heory of surface growth is (2.16)

subject t o periodic bo undary condit ions an d which will ensure a Poincar e ty pe inequality. was rigorously treated in [28]. Here we can 2 2u (J = 0 (E ), whe re /la is such t hat L = - t1

moving fram e fe u dx = 0, The deterministic equat ion consider /l = /la + E2 and /lat1u fulfils Assumption 1.

3. Amplitude equations, main results. We review t he two main approaches to ver ify the approximation via amplitude equa t ions . One relies on a purely local picture and uses Assumption 2, while t he other t akes into acco unt the global nonlinear stability of the equation given by Assumpti on 3. 3.1. Attractivity. The at t ract ivity justifies the ansat z for the form al computation. It shows t ha t afte r a comparably short time the solution is of the form of the an satz (2.3) . THEOREM 3.1 (At t ractivity-local) . Under A ssum pt ion s 1, 2, and 4 there are cons tants e, > 0 and a tim e t e = 0(ln (c 1 ) ) suc h th at fo r all mild soluti ons u of (2.14) we can wri te u( t e ) = m e + E2Re with ae E N and Re E S , where (3.1)

1P' ( llaell:S s, 11 Re11 :S E- K ) 2: 1P'( llua ll :S

C3bE)

- cle-c2e- 2K ,

forallb >O and e c. (0,1). The proof of this result is a st raight forward modi fication of Theorem 3.3 of [4]. It relies on the fact t hat small solut ions of ord er O(e) are on sm all time-scales given by the lineari sed picture, which is dominat ed by the semigroup estimat es (2.7) and (2.8) . THEOREM 3.2 (Attractivity-global) . Let A ssumptions 1, 2, and 4 be satisfied. Th en f or all Ta > 0 an d p 2: 1 there are cons tants c p > 0 such that

(3.2) for all1t- valued m ild solutions u of equation 2.1 independent of the initial conditi on. Furthermore, th ere is a tim e t e = 0(ln(c 1 ) ) such that given a family of posit ive constants {bp} p~l there are positive constants { Cp} p~l , such that for all1t-valu ed mild solutions u of equation (2.1) with IEllu(O)IIP :S bpEP we have

(3.3)

IEllu(t)IIP :S CpEP and IE IIPs u (t + te)IIP :S CpE 2p

f or all t 2: 0 and E E (0,1 ).

48

DIRK BLOMKER AND MARTIN HAIRER

The proof is given by a-priori estimates. This was not directly proved in [7], but under our somewhat stronger assumptions this is similar to Lemma 4.3 of [7] . It relies on a-priori estimates for V o = u - WL - o with = O(€2) , which fulfils a random PDE similar to (2.15). We omit the proof, as it is technical but straightforward.

o

3.2. Approximation. For a solution a of (2.4) and 'l/J of (2.5) we define the approximations €Wk of order k by

The residual of

(3.4)

€W

is given by

Res(€w(t)) = -€w(t)

+

it

+ etL€w(O) + €2Wdt)

e(t- r)L[€3 Aw + F(€w)](r)dr.

In order to show that €W is a good approximation of a solution u of (2.14), the main step is to control the residual. The main idea is to obtain bounds on PeRes(€w) via the amplitude equation and to bound PsRes(€w) by using the stability of the equation which is ensured by our spectral gap (cf. (2.6) or (2.7)). These estimates require good a-priori bounds on the approx imation €Wk, but do not require any a-priori knowledge on the solut ion u of the original equation. Bounds on the residual easily imply approximation results, as we can establish bounds on the difference of euu; and u using (3.4) and (2.14). THEOREM 3 .3 (Approximation-local) . Suppose Assumptions 1, 2, and 4 are true . Fix the time To > 0 and som e sm all r;, E (0,1) . Then there are constants Gatt> 0 and c, > 0 such that for € E (0,1) we obtain for all solutions u of (2.14) and all solutions a of (2.4) (with I; instead of Fe) JP> (

sup tE[O,TQE-2j

lIu(t) - €wl(t)lIx ::::: Gatt€2-K)

2: 1 - JP> (lIuo - m(O)llx 2:

C1 €2-

K) - JP>(lluollx 2:

C2

€) - c3e-e41n(g-1)2 .

The proof of this result is a straightforward modification of Theorems 4.1 and 4.3 of [4] . We use ideas of [9] to allow for weaker bounds on SUPTE[O ,To]la(T)! by c* In(c 1 ) , which were not present in [4] or [5] . There the probability was bounded by terms of order 0(1) in e without any further information on the smallness. Nevertheless, we can easily improve these proofs. In that situation, we can use the following large devi ation bound JP> (

sup TE[O ,To]

la(T)I2: Gln(€-l)) ::::: Ge- cln(g-1)2 ,

AMPLITUDE EQUATIONS FOR SPDE

49

which is not exponentially small, but smaller than any power of c. This relies on the fact that the amplitude equation is nonlinear stable, which follows from Assumption 3. This stability allows to carry over large deviation results for the Brownian motion (3 in N ~ ~n to results for a. Under the stronger Assumption 3 we can prove a much better result. THEOREM 3.4 (Approximation-global) . Let Assumptions 1, 3, and 4 hold and let u be the mild solution of (2.1) with (random) initial value Uo satisfying (3.3). Then for all p > 0, 1 » /'\, > 0 and To > 0 there is a constant Cap p explicitly depending on p and To such that the estimate

holds for e E (0,1) . The proof is Corollary 3.9 of [7] .

4. Applications. We give results for approximate centre manifolds and the dynamics of random fixed points. 4.1. Approximate centre manifold. This section uses the approximation results of the previous section . We rely especially on Theorem 3.4 to extend the results, which were briefly sketched in [4] or [5], by using second order corrections introduced in [7] . That is why we restrict ourselves to nonlinear stable equations given by Assumption 3. Our main result will show th at th e flow along N is well approximated by the solution a of the amplitude equation on a slow time scale. The distance from N is given by a fast oscillation 'IjJ, which is a stationary Ornstein-Uhlenbeck pro cess. And everyt hing is valid only up to errors of order O(c3 - K ) and with high probability. The flow given by (2.1) is approximated with high prob ability as sketched in Figure 1. There the typical behaviour of solutions is given. THEOREM 4 .1. Suppose A ssumptions 1, 3, and 4 are true, then there is a logarithm ic tim e t e = O(ln(c l ) ) such that the following is tru e. For an arbitrary mild solution u(t) of (2.1) with (random) initial condition uo, such that IElluollP ::; JpcP for som e fixed family of constants { O. Then there exists a constant C > 0 such that

holds for e E (0,1) .

50

DIRK BLOMKER AND MARTIN HAIRER

N in X £a(O)

e: 2 'l/J (0)

~ U (O)

( J

2

a ( E: t

g2'l/J(t)

u(t )

FIG . 1. Typical trajectory on the approximate centre manifold .

Proof. First we use global nonlinear attractivity in logarithmic time t~l) for arbitrary initial conditions (cf. Theorem 3.2). Then we approximate with solution aCt) of (2.4) and ;Pet) of (2.5) for times t E [t~l), Toc 2 J. Define a version of the stationary Ornstein-Uhlenbeck process by (4.1)

'l/J*(t)

=

{too e-L(t-s) dPsQW(s) ,

where W(s) = W(s) for s > 0 and it is an independent Wiener process for s < O. For (3 in the amplitude equation, we need only th e rescaling (3(T) = gPc QW (T c 2 ) . The difference between ;p and 'l/J* is trivially small in any p-th moment, if we wait another sufficiently large logarithmic time tF). Define now tE: := t~l)

+ tF).

The difference between aCt) and aCt) is small by the approximation result, because first lIa(tE:) - a(to)11 = O(g3-K) by Theorem 3.4. Then, by the same theorem Ila(t) - a(t)11 ~ lIa(t) - Pcu(t)11 + IlPcu(t) - a(t)11 = O(g3-K). 0 4.2. Dynamics of the random attractor. We can determine the dynamics of random fixed points by the approximation result over a very long time-scale with high probability. It suffices indeed to apply the results of the previous section by starting the equation in the random fixed point.

AMPLITUDE EQUATIONS FOR SPDE

51

Let us first fix some notation. If we consider a two sided Wiener process W = {W(t)}tEIR, then it is well known that solutions of (2.14) define a random dynamical system (e.g. via transformation to (2.15)) . Here cp(t,Uo, W) is the solution u(t) given initial condition Uo and twosided noise path W. A random fixed point ao(W) is a random variable such that cp(t,ao(W) , W) = ao('19 tW) , where t9 t W (s ) = W(t + s) - W(t) . For a detailed discussion of random dynamical systems see [1] . For the existence of random (set) attractors see for example [14, 41, 39]. COROLLARY 4.1. Under the assumptions of Theorem 4.1 let a., be a random fixed point of the random dynamical system generated by (2.14}. Then

where a(O) = Pea o , and a is a solution of (2.4). The proof is basically just a simpler case of Theorem 4.1. We start the system in the random fixed point ao. In this case , we do not need time for attractivity, as due to the stationarity of ao and Theorem 3.2 u(O) := ao already fulfils the assumptions of Theorem 3.4 . REMARK 4.1. We do not use uniformity in the initial condition. Hence, we can only prove results for random fixed points, and not for random set attractors but it would be an interesting result , whether we have

on time intervals of order O(c- 2 ) with high probability. REMARK 4 .2 . The restriction to random fixed points still covers several cases. For example for dissipative nonlinearities c 2 A + F (e.g. -c 2 u - u 3 ) it is well known, that the random attractor is just a single random fixed point. For non-dissipative nonlinearities (e.g. c 2 u - u 3 ) it is in most cases completely open what the topology of the random aitracior is. But, nevertheless, in many examples of non-trivial random attractors for SPDEs these attractors contain random fixed points. If the attractor for the amplitude equation is a single stable fixed point a*, which is exponentially attracting, then we can proof a much stronger result. We suppose the following . ASSUMPTION 5 . Suppose that the random dynamical system generated by the amplitude equation (2.4) has a unique random fixed point a* that is exponentially attracting in p-th mean. This means, for any p > 0 there are constants 6 > 0 and M';» 0 such that

for any solution a of (2.4) .

52

DIRK BLOMKER AND MARTIN HAIRER

For simplicity, we will rescale the equation to the slow time-scale T = €2t. Consider the rescaling v(T) = €-l u(Tc 2), which is a solution of (4.2)

aTV =

€-2 Lv

+ Av + F(v) + aTQW ,

where W is just a rescaling of W. Let a be a solution of the amplitude equation with j3 = PcQW and let 'l/Je be the rescaled Ornstein-Uhlenbeck process

Consider now the random dynamical system generated by the triple (v, a, €'l/Je). It is obvious that random fixed points of this equation are just rescaled versions of random fixed points for the original system of equation. We can prove the following theorem. THEOREM 4 .2. Suppose Assumptions 1, 3, 4, and 5 with random fixed point a* are true. Let v* be any fixed point of the rescaled equation (4.2) and denote by 'I/J; the rescaled stationary 0 U-process. Then for any small K, E (0,1) and any p > 0 there is a constant C such that lE ( Ilv* - a* - €'I/J; II~ )

I/ P

::; C€2-t 0 such that

(15) for any q E Wi (D) which follows st raight forwardly by regularity properties of a linear elliptic boundary problem. Note th at wt is a Sobolev space with respect to the third derivatives. Hence we get :

IIJ(S,1/')11£2::; sup (lox1/'(x,z)1 + loz1/'(x,z)l) x (x ,z )ED

X

(l'OxS(X, z)1

+ lozS(x,z)ldD) .

68

VENA PEARL BONGOLAN-WALSH ET AL.

The second factor on the right hand side is bounded by

On account of the Sobolev embedding Lemma, we have some positive constants C2, C3 such that sup (Iax 'l/J(x, z)/ + laz'l/J(x, z)l) :Sc211'V'l/J 1 1wi (D ) :Sc31Iqllwi(D) :Sc31Iullv. (x ,z)ED Hence we have a positive constant

C4

such that

for u E V . We now show that

(J(S, 'l/J ),S) =

o.

We obtain via integration by parts

LaxSaz'l/J SdD - LazSax'l/J SdD = - La;zS'l/J SdD + La;xS'l/JS dD -LaxS'l/JazS dD

+

razS'l/JaxSdD + r

}D

J(O ,I)

axS'l/JSI~~5dx -

r

J(O,I)

azS'l/JSI~~5dz = 0

because 'l/J is zero on the boundary Bl), This relation iso true for a set of sufficiently smooth functions 'l/J , S which are dense in W~(D) x Wi(D) . By the continuity of F I , as just shown in Lemma 3.2, we can extend this o . property to W~(D) x Wi(D). 0 LEMMA 3.3 . The following estimate holds

for some positive constant cs . Proof By simple calculation, the proof is obtained.

0 We have obtained a differential equation without white noise but with random coefficients. Such a differential equation can be treated samplewise for any sample w . We are looking for solutions in v E C([O, T]; H) n L 2(0, T; V) ,

for all T > O. If we can solve this equation then u := v + 7J defines a solution version of (6). For the well posedness of the problem we now have the following result.

ENSTROPHY AND ERGODICITY OF GRAVI TY CURRENT S

69

3.4 (Well-Posedness) . For any time r > 0, there exists a uni que solution of (13) in C( [O, r ]; H ) n £ 2(0, r ; V). In particular, the solution mapping THEOREM

IR+ x

nx

H 3 (t ,w , vo) ---+ v(t) E H

is measurable in its arguments and the solution mapping H 3 Vo ---+ v(t ) E H is continu ous. Proof By the properties of A and Ft (see Lemm a 3.2), t he random differenti al equation (13) is essentially similar to t he 2 dim ensional Navier Stokes equation. Note th at F2 is only an affine mapping. Hence we have 0 existence and uniqueness and the above regulari ty assertions. On account of t he transform ation (12), we find t hat (6) also ha s a uniqu e solution. Since the solution mapping IR+ x n x H 3 (t ,w, vo)

---+

v(t ,w , vo) = : 4?(t,w,vo ) E H

is well defined, we can introdu ce a random dyn amical system . On n we can define a shift operator ()t on th e paths of the Wiener process t hat pushes our noise:

w( ·,{hw) = w( · + t ,w) - w(t,w)

for t E IR

which is called the Wiener shift. Then {()t} tE IR forms a flow which is ergodic for the probability meas ure JP>. The properties of t he solution mapp ing cause the following rela tions

4?(t + r ,w ,u) = 4?(t , ().,w, 4?(r ,w ,u))

for

t, r 2: 0

4?(O,w, u) = u for any wE n and u E H. This prop erty is called t he eocycle proper ty of 4? which is import ant to study the dynamics of random syste ms. It is a generalization of t he semigroup property. The eocycle 4? toget her with the flow () forms a random dynamical system. 4. Dissipativity. In t his section we show t hat the random dynamical syste m (13) for gravity curre nts is dissip ative, in the sense t hat it has an absorbing (random) set . This means that the solut ion v is cont ained in a particular region of t he phas e space H after a sufficiently long tim e. This dissip ativity will help us to obtain asymptoti c est ima tes of the enst rophy and salinity evolution. Dynamical properties that follow from this dissipat ivity will be considered in the next sect ion. In par ticular , we will show that the system has a ra ndom at t rac tor , and is ergodic und er suit able conditions. We introduce the spaces

fI = £2(D ) V = Wi(D).

70

VENA PEARL BONGOLAN-WALSH ET AL.

We also choose a subset of dynamical variables of our system (1)

v=

(16)

S-

TJl .

To calculate the energy inequality for V, we apply the chain rule to We obtain by Lemma 3.2

Ilvllk.

~ Il vlll + 211V' v11L

(17)

=2(J(TJl, 1jJ), v). The expression V'v is defined by (V' x,zv). We now can estimate the term on the right hand side. By the Cauchy inequality, integration by parts and Poincare ine9uality AlllqllL~ "V'qIIL~ for q E W~(D) and A211iillL2 < IIV'iiIIL2 for v E Wi(D), we have

s

(18) For q, we have the following estimate

(19) From (18) and (19), we have

(20)

:t (2 11v111 + Ra~A~ Ilqll2) + IIV'vIIL +

(Ra;A~ -

2 2,xi11TJl1 }1V'qI12

$

~~ IITJlI1 2.

DEFINITION 4 .1. A random set B = {B(W)}wEO consisting of closed bounded sets B(w) is called absorbing for a random dynamical system ep if we have for any random set D = {D(W)}WEO, D(w) E H bounded, such that t ---+ SUPyED(O,w) IlyllH has a subexponential growth for t ---+ ±oo

(21)

ep(t,w,D(w)) C B(Btw) ep(t, B_tw, D(B_tw)) C B(w)

for for

t2to(D,w) t 2 to(D, w).

B is called forward invariant if ep(t,w,uo)EB(Btw)

ifuoEB(w)

fort20.

Although v is not a random dynamical system in the strong sense we can also show dissipativity in the sense of the above definition. LEMMA 4.2. Let ep(t , w, vo) E H for Vo E H be defined in (6), and 1

Ra2A~ -

2

2

2A 1lEIITJlI > O.

ENSTROPHY AND ERGODICITY OF GRAVITY CU RRENTS

71

Then the closed ball B(O,RI(w)) with radius RI(w) =

2[°00 e"'T:~ 1I1']1112dT

is forward invariant and absorbing. The proof of this lemma follows by int egration of (20) . For the appli cations in th e next section we need that the elements which are contained in th e absorbing set satisfy a particular regularity. To this end we introduce th e functi on space

:= {u EH : Ilull; := IIA~ ull~ < oo}

'W

where s E R The operator As is th e s-th power of th e positive and symmetric operator A. Note that these spaces are embedded in the Slobodeckij spaces HS, s > 0. The norm of thes e spaces is denoted by I1 . 1I n-. This norm can be found in Egorov and Shubin [7]' P age 118. But we do not need this norm explicitly. We only mention that on re th e norm I . lisof HS is equivalent to the norm of re for < s, see [8] . Our goal is it to show that v(l, w , D) is a bounded set in 'H" for some s > 0. This property causes th e complete continuity of th e mapping v(l, w, .). We now derive a differential inequ alit y for tllv(t )II;. By the chain rule we have d d

°

dt (t [[ v(t)II; ) =

Il v(t)ll; + t dt Il v(t)II ;·

Note that for the emb edding const ant

Cs

between H' and V

1Ilvll;ds::; 1Il vll~ds t

t

c;

for s ::; 1

such th at the left hand side is bound ed if the initial condit ions Vo are contained in a bounded set in H . The second term in the above formula can be expressed as followed:

d ..

(d

)

t dt(A 2V,A 2V)H =2t dt v,Asv H

= - 2t(Au, ASV)H + 2t(FI( v + l'](B t w)), ASV)H

+ 2t(F2(v + rJ(Btw)) ,ASV)H. We have

(Av , ASV)H

= IIA~+ ~vIIH = Ilvlli+s'

Similar to the argument of [21] and th e estim ate for the existence of absorbing, and applying some embedding theorems, see Temam [18] Page 12 we have got

Ilvll; ::; C(t , IlvollH , sup tElO,I]

111']1IlD(A')) ' for t

E [0,1] .

72

VENA PEARL BONGOLAN-WALSH ET AL .

By the results of [12] and [21], we know sup 11711 IlD(AO) ::; C(trL2Q)
0 depends only on physical data. By the estimates of Corollary 5.7, we are able to use the well known Krylov-Bogolyubov procedure to conclude the existence of invariant measures of the Markov semigroup T(t). COROLLARY 5.9 . The semigroup of Markov operators {T(t)h2:o possesses an invariant distribution J1i in M 2 : T(t)J1i = J1i

for t 2: O.

In fact, the limit points of

{ ~t lot T(T) J1

0dT}

t2:0

for t --+ 00 are invariant distributions. The existence of such limit points follows from the estimate in Corollary 5.7. In some situations, the invariant measure may be unique. For example, the unique random fixed point in Theorem 5.6 is defined by a random

76

VENA PEARL BONGOLAN-WALSH ET AL.

variable u*(w) = v*(w) + 1](w) . This random variable corresponds to a unique invariant measure of the Markov semigroup T (t). More specifically, this unique invariant measure is the expectation of the Dirac measure with the random variable as the random mass point

Because the uniqueness of invariant measure implies ergodicity [13], we conclude that the gravity currents model (1) is ergodic under the suitable conditions in Theorem 5.6 for physical data and random noise. We reformulate it as the following ergodicity principle. THEOREM 5.10 (Ergodicity) . Assume that the salinity boundary flux data 11F11L2' the Prandtl number Pr, and the trace of the covariance for the noise trL 2 Q are sufficiently small. Then the gravity currents system (1) is ergodic, namely, for any observable of the gravity currents, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long. In the regime of ergodicity, the gravity currents system can be numerically simulated with (almost surely) one random sample. This is what we call ergodicity-based numerical simulation of stochastic systems, in contrast to sample-wise ("Monte Carlo") simulations. Acknowledgements. This work was partly supported by the NSF Grants DMS-0209326 and DMS-0139073, and a Grant of the NNSF of China. H. Gao would like to thank the Illinois Institute of Technology,

Chicago, for hospitality.

REFERENCES [lJ L. ARNOLD. Random Dynamical Systems. Springer, New York, 1998. [2J I.D. CHUESHOV, J . DUAN, AND B. SCHMALFUSS . Probabilistic dynamics of two-layer geophysical flows. Stochastics and Dynamics, pp . 451-476, 2001. [3J H. CRAUEL AND F . FLAN DOLI. Hausdorff dimension of random attractors. J. Dyn. Diff. Eq., 1999. [4] A. DEBUSSCHE. Hausdorff dimension of a random invariant set. J. Math . Pure Appl., 9 :967-988, 1998. [5] H . A . DIJKSTRA. Nonlinear Physical Oceanography. Kluwer Academic Publishers, Boston, 2000. [6] J . DUAN, H . GAO, AND B . SCHMALFUSS . Stochastic Dynamics of a Coupled Atmosphere-Ocean Model, Stochastics and Dynamics, 2(3) : 357-380, 2002. [7] Yu. V. EGOROV AND M.A .SHUBIN. Partial Differential Equations, Vol. I of Encyclopedia of Mathematical Sciences. Springer, New York, 1991. [8] J.L. LIONS AND E . MAGENES. Problemes aux Limites Non Homoqenes et Applications, Vol. 1, Dunhod, 1968. [9] F . FLANDOLl AND B . SCHMALFUSS . Random attractors for the stochastic 3D Navier-Stokes equation with multiplicative white noise. Stochastics and Stochastics Reports, 59 : 21-45, 1996. [10] H.J . GAO AND J . DUAN . Dynamics of a coupled atmosphere-ocean model. Nonlinear Analysis B, in press, 2004.

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[11] T . OZGOKMEN, W .F. JOHNS , H . PETERS , AND S. MATT . Turbulent mixing in the Red Sea outflow plume from a high-resolution nonhydrostat ic model. 1. Physical Oceanogr ., 33 : 1846-1869, 2003. [12] G . DA PRATO AND J . ZABCZYK. Evolution equations with white noise boundary conditions, Stochastics and sto chastics Reports, Vol. 42 : 167-182, 1993. [13] G . DA PRATO AND J . ZABCZYK . Ergodicity for Infinite Dimensional Systems , Cambridge University Press, Cambridge, 1996. [14] B. SCHMALFUSS . A random fixed point theorem and th e random graph transformation. Journal of Mathematical Analysis and Applications, 225(1): 91-113, 1998. [15] B. SCHMALFUSS . Invariant attracting sets of nonlinear stochastic differential equat ions. In H. Langer and V. Nollau, editors, ISAM Seminar - GaufJig, Vol. 54, pp . 217-228. Mathematical Research, Akademie--Verl ag, 1989. [16] B. SCHMALFUSS. Long-time behaviour of the stochastic Navier-Stokes equation. Mathematis che Na chri chten, 152:7-20, 1991. [17J G. SIEDLER, J . CHURCH , AND J. GOULD . Ocean Cir culation and Climate: Observing and Modeling the Global Ocean. Academic Press, San Diego , USA, 2001. [18J R. TEMAM . Navier-Stokes equation and Nonlinear Functional Analysis. CBMS NSF Regional Conference Series in Applied Mathematics. SIAM , Philadelphia, 1983. [19] O . THUAL AND J. C . MCWILLlAMS . The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box model. Geophys . Astrophys. Fluid Dynamics, 64: 67-95, 1992. [20] J . ZABCZYK . A mini course on stochastic partial differential equations. In P. Imkeller and J .S. von Storch, editors, Sto chastic climate models, Progress in Probability, 49 : 257-284, Boston , 2001. [21] JINQIAO DUAN AND 13 . SCHMALF USS. The 3D quasigeostrophic fluid dynamics und er random forcin g on boundary, Com m. Math . S ci., 1: 133-151 , 2003.

STOCHASTIC HEAT AND BURGERS EQUATIONS AND THEIR SINGULARITIES IAN M. DAVIES· , AUBREY T RUMAW AND HUAIZHO NG ZHAOt Abstract. T he Arno l'd- T horn classificati on of caustics for t he Burgers equation suggests t hat th er e should be an analogous one for t he wavefronts of the correspo ndi ng heat equation. We present a general th eorem for Hamiltoni an systems char acte rizing how the level surfaces of Hamilton 's principal funct ion meet th e ca ust ic surfa ce in both the det erminist ic and stochastic cases . Such a char acte rizat ion allows one to give a fairly detail ed description of th e behaviour of th e solution of th e heat equa ti on in the vicinity of th e wavefront and caustic. It allows one to propose some reasons for the "blow-up" of th e Burgers velocity field on t he ca ust ic. In t he case of sm all noise th e sha pes of the rand om wavefront and rand om caust ic may easily be obtained , and to first order the caustic is merely displaced . In th e st ochastic case we have the possibility of "rapid" cha nges in the caustic-wavefront intersecti on . This will engender stoc has t ic turbulence in th e Burgers velocity field and, du e to its stochasticity, may be of an intermittent nature. There is no an alogue of this in the deterministic case. Th roughout our studies much use has been made of computer algebra packages in building an und erstanding of the ar chetypal cases . Numerical simulat ions and num eri cal solut ions of th e partial differen ti al equat ions involved have been imm ensely useful in clarifying conjec t ures and det erm ini ng apt char ac terizat ions. AMS(MOS) subject classifications. P rimary 60H15, 60H30; Seconda ry 76M35, 35R60.

1. Introduction. Stochast ic Bur gers equations have attracted a considerable amount of attent ion in recent years, e.g. [1,3, 5, 9, 10, 18, 19,20, 21, 22, 28, 29, 30, 31]. See also [2, 4, 16, 24] for related works. They have been used to give models of t urbulence (see especially [9]) and to model th e large scale st ruct ure of th e universe [27]. Here we shall be int erested in what has come to be called Bur gulence. Primarily we show how a knowledge of the geometry of the level surfaces of a Hamilton - Jacobi function and the associated causti c surface can be used in det ermining the behaviour of th e velocit y field of th e viscous Burgers fluid. Th e presence of viscosit y provides access to a range of powerful analytical methods . Consider the stochastic viscous Burgers equation for th e velocity field vJl. = vJl. (x , t), x E ]Rd, t > 0, € E ]R small,

with initial velocit y vJl.(x, 0) = Y'So(x) + O(J..L2 ) where J..L 2 is th e coefficient of viscosity. Here c and k are C 2 functions and Wt is a real valued Wiener • Department of Mathematics , University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK. t Department of Mat hematical Sciences, Loughborough Univers ity, Loughb orough , LEll 3T U, UK.

79

80

IAN M. DAVIES, AUBREY TRUMAN, AND HUAIZHONG ZHAO

process on the probability space {O, F , P} . We shall be interested in the 'blow-up' of, i.e. the advent of discontinuities in, vO(x , t) where

The corresponding heat equation for uJ.l = uJ.l(x, t) is the Stratonovich equation

auJ.l J.l2 = -f),uJ.l at 2

-

1

€

•

+ -c(x)uJ.l + -k(x t)uJ.l 0 Wt J.l2 J.l2 ' ,

uJ.l(x,O) = To(x) exp (-SO(x)/J.l2) , the connection between uJ.l and vJ.l being the Hopf-Cole logarithmic transformation vJ.l = _J.l2\7 In ur . It is this correspondence with the stochastic heat equation that enables us to appeal to asymptotic methods in our study of the viscous Burgers fluid. Moreover , these methods highlight the importance of the stochastic dynamical flow and the stochastic Hamilton - J acobi function in determining the behaviour of solutions. With this in mind we would expect, from the work of Donsker, Freidlin et al. [15, 17, 29], that we have as J.l -> 0, -J.l 2 In uJ.l(x , t)

(1.1)

->

inf [A(xo,x, t) I Q

+ So(xo)] = S(x , t) ,

with

A(xo, x, t) =

inf

X (s) X(Q) =XQ X( t ) = x

A[X],

where A[X] is the st ochastic action

A[X] =

rIX(s)1

21 lo

2

r

ds - lo c(X(s))ds -

€

r

lo k(X(s), s)dWs .

We require absolute continuity of X (d. Davies and Truman [7] and references therein) and note that we will have X continuous almost surely. S(x, t), as defined in Equation (1.1), is the minimising solution of the stochastic Hamilton-Jacobi equation

2

d.S;

I\7SI + -2dt + c(x) dt + €k(x, t) dWt

= 0,

S(x ,O) = So(x),

and as such is Hamilton's principal function for a stochastic mechanical path. From Equation (1.1), as J.l -> 0, we expect that uJ.l switches from being exponentially large to exponentially small across the level surface

S(x,t) =

°

(zero level surface),

STOCHASTIC HEAT AND BURGERS EQ UATI ONS

81

since ul-' rv e- S (x ,t )/ 1-'2. It is also possible for jump discontinuities to appear and our methods allow for th e precise nature of such discontinui ti es to be studied. Our use of pre-level surfaces and t he analogous precausti c greatly simplifies their considerati on. Such behaviour will be reflected in that of the Burgers fluid. If we demand that X (t ) = x , for fixed t and x, X (s) appea ring in th e stoc hastic action above may not be unique. Hence, we expect that shockwaves for v arise from precausti cs (in (xo, t) variables) when infinitely man y of these classical mechanical paths from Xo and a neighbourhood focus, in a set of zero volume, on X (t ). The condit ion for paths starting from Xo focusing at a point X (t) at tim e t is Det

(aaxo X (t) )

= 0

(precaus tic) .

2. Stochastic heat and Burgers equations. We st udy the inviscid limit of the stochastic viscous Burgers equation, for the velocity field

vl-'(x, t), avl-'

-at + (vI-' · \l x )vl-' = -\le -

. E\lkW:t

~2

+ -2 !:lvll '

where vl-'(x, 0) = \lSo(x ) + O(~2) for some given So, Noise, by using t he Hopf-Cole t ra nsformation,

w, repr esenting White

with ul-' satisfying the stoc hastic heat equation of Stratonovich ty pe

du'(:

=

[~2 !:lu'(: + ~-2cu'(:]

dt + E~-2ku'(: 0 dW t ,

with u~ (x ) = To (x ) exp (- So(x) / ~2) , So as before and To a smoot h positive function . In general c and k are functions of (x , t ). We event ually t ake To equal to 1 in our study. REMARK 2.1. For sm all E and gen eral potentials, we are able to deriv e the shape of the random shockwave for the stochastic Burgers velocity field and are also able to give th e equ ation determining the random wavefront for the stochastic heat equ ation correct to first order in E. The solution of the viscous st ochasti c Bur gers equation can be surprisingly written in closed form [29], for each m 2: 0,

82

IAN M. DAVIES , AUBREY TRUMAN, AND HUAIZHONG ZHAO

where Vj(x,t) = \lSj(x,t), and the Sj satisfy

-c for j = 0,1,2, . .. , with the convention !~S-1 white noise. The Nelson diffusion process y~ satisfies ,

EkWt , Wt

being

m

dy~ = j1.dB s

-

\l

L j1.2 j Sj(Y~, t -

s) ds,

j=O y~ =

x,

B; is Brownian Motion.

The above is extremely brief and it is important to note that SI, and hence all Sj, are To dependent [29] . We now begin to introduce the familiar terminology of dynamics starting with the flow for the stochastic case. The stochastic action will be introduced shortly. Define the random map 0 is the inverse temperature parameter, and v(p, q) is the potential energy. The situation considered here corresponds to 0 ::; v(p, q) ::; +00. The number of interactions for pairs of particles at two locations p, q is L{p,q} (N) = N (p)N (q) for p =1= q. The number of interactions for pairs of particles at a single location p = q is L{p,q} (N) = (N~p)) . The interaction for all pairs of particles is defined to be

(3.12)

K(N)

IT (1 - t(p,

=

q))L{p,q} (N) .

{p,q} This says that the weight of a configuration N is decreased by a factor (1 - t(p, q)) for each pair of particles at locations p, q. In the following we write op for the multi-index defined by op(q) = Opq. This represents a single particle at the location p. Similarly, let t p be the function tp(q) = t(p , q). This represents the interaction of a single particle at p with another particle at an arbitrary point. LEMMA 3.1. Suppose that the interaction coefficients K are given by a two-location interaction t . Then the effect of adding one more particle at location pis K(N + op) = K(N)(l- tp)N . Proof. Add one particle at p. This changes the particle configuration to N +8p. Then the number of pairs of particles at p, q increases to L{p,q} (N + op) = L{p,q}(N) + N(q) . Thus one more particle at p decreases the weight of the configuration N by a factor of 1 - t(p, q) for each single particle at q. The total decrease is I1 q(l - t(p,q))N(q) = (1- tp)N. 0 LEMMA 3.2 (Partition function equation) . The derivative of the partition junction is given in terms of modified activities by

(3.13)

Proof. The effect of differentiating is to add one additional external particle. That is, if one differentiates the sum over N of 1jN!K(N)w N and makes the change of variable N (p) - 1 f-- N (p), then the result is the sum over N of 1 jN!K (N + )w N . Therefore according to the lemma the derivative is

s,

(3.14)

However, this is just the rescaled partition function.

o

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

105

THEOREM 3.2 (Second Mayer equation) . For the discrete gas system with two location interactions the expected number of particles is expressed in terms of modified activities by

(3.15)

This may also be written (3.16)

where Tp(M) = 1 - (1 - tp)M. Proof Start with the partition function equation. Multiply by w(p) and divide by ZA(W). The left hand side is the expected number of particles. The right hand side involves a quotient of partition functions and hence an exponential of a difference of pressures . 0 It is now possible to understand the behavior of the Mayer equations, at least on a non-rigorous level. Suppose that the interactions or the activities are quite small. Then one may approximate the difference of pressures by a derivative : (3.17)

FA(w - tpw) - FA(w):::::; - Lt(p,q)w(q)aFA(w)/aw(q) . q

The first Mayer equation gives

(3.18)

FA(w - tpw) - FA(w) :::::; L t(p , q)EA(N(q)). q

The second Mayer equation then gives

(3.19)

EA(N(p)) :::::; w(p) exp( - "[)(p, q)EA(N(q))) . q

This is an approximate equation for the mean number of particles given at each point. It says that the mean number of particles at p is given by the weight w(p) times an exponential factor involving the interaction of a particle at p with the mean number of particles at all the other sites . The interaction between locations p and q is t(p, q) = 1 - exp( -(3v(p, q)) :::::; (3v(p, q). 3.2. Cluster estimates. The second Mayer equation is a complicated equation, since the Tp(M) factor is nonlinear in t p and in M . However there is a remarkable upper estimate in terms of a quantity that is linear in these variables . Set (tM)(p) = L: q t(p, q)M(q) . LEMMA 3.3 (Interaction estimate) .

(3.20)

Tp(M)

~

(tM)(p) .

106

WILLIAM FARIS

The interaction estimate is elementary [8], but very useful. Write (3.21 )

for the bound on the power series expansion of the expected number of particles at location p. The inter action estimate leads to the following proposition. PROPOSITION 3.1. The bound on the expected number of particles satisfies the inequality (3.22)

n(p) ~ Iw(p)1 exp(L t(p, q)n(q)) . q

The next task is to estimate these quantities. This is done in terms of an energy bound A(q) 2:: O. This is to be chosen in a convenient manner for the problem at hand. Associated with this is an expected particle number bound Iw(q)1 exp(A(q)). The simplest choice is to take A(q) = 1 for all locations q; this works for some problems . The Kotecky-Preiss cluster condition [5] is that for each point p there is an estimate (3.23)

L t(p , q)lw(q)1 exp(A(q)) ::; A(p) . q

This says that the interactions of a particle at p with particles at other points q satisfying the expected particle number bound satisfy the energy bound. THEOREM 3.3 (Stability bound for expected particle number) . Consider the discrete gas system with two location interactions. Suppose the cluster condition is satisfied. Then the bound on the particle density satisfies (3.24)

n(q ) ::; Iw(q)1 exp(A(q)) .

Thus the radius of convergence of the series for the expected number of particles does not depend on the size A of the system. Proof Here is a proof of th e stability bound, following Ueltschi [8] . Let

(3.25)

Consider the assertion that there is a bound (3.26)

nk(p) ::; Iw(p)1 exp(A(p)).

107

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

Since C(O) = 0, the value of no(p) = 0, and so this is true for k = O. Suppose it is true for some arbitrary k 2: 0. From the second Mayer equation and the interaction estimate (3.27)

nk+l (p) :::; Iw(p)1 exp(L t(p, q)nk(q)). q

By the supposition (3.28)

nk+I (p) :::; Iw(p)1 exp(L::: t(p, q)w(q) exp(A(q))) . q

Then from the cluster estimate

nk+l(p) :::; Iw(p)1 exp(A(p)) .

(3.29)

This says that it is true for k + 1. Thus the bound for k implies the bound for k + 1. It follows from induction that it is true for all k. Since the right hand side of the bound is independent of k, this also gives the bound for the infinite series. 0 COROLLARY 3.1 (Stability bound for interaction). Suppose the cluster condition is satisfied. Then

L t(p ,q)n(q) :::; A(p) .

(3.30)

q

The importance of these results is in the fact that th e bounds do not depend on the size A of the system. In fact the convergence of the series gives an expression for the expected particle number at p for an infinite system: (3.31) as A

-+

P . The sum on the right is over all multi-indices of finite support.

3.3. Abstract polymer systems. An abstract polymer system (or hard-core discrete gas system) is the special case when t(p, p) = 1 (no two particles at the same location) and when also t(p, q) is always either or 1 (no int eraction or total exclusion). This is th e framework for much of the recent work [5, 3, 1, 7]. For each location p define the corresponding set of incompatible locations to be I(p) = {q I t(p ,q) = I} . For an abstract polymer system the expected number of particles at a point is the same as the probability of a particle at a point, that is,

°

(3.32)

PA(N(p)

= 1) =

L N(p)=l

PA(N)

= w(p) ZAi:((~\w) .

108

WILLIAM FARIS

The interaction of a particle configuration M with a location p given by Tp(M) = 1 - (1 - tp)M is 1 or 0 according to whether M(I(p)) > 0 or = O. The interaction estimate says that the indicator function of the event M(I(p)) > 0 that a particle is at a location incompatible with p is bounded by the number M(I(p)) of particles that are at locations incompatible with p. For an abstract polymer system the first Mayer equation expresses the particle probability as (3.33)

The second Mayer equation says that it is also equal to (3.34)

PA(N(p) = 1) = w(p) exp( -

L

c~) w'j) .

M(I(p))>O

This may be viewed as a statement of the principle that a ratio of partition functions is the exponential of a difference of thermodynamic potentials. Consider an abstract polymer system. The set of particle locations is a graph, where two locations are connected if they are incompatible. It may be shown that c(M) =J 0 implies that the support of M is connected. This is one sense in which the coefficients of the expansion may be thought of as being associated with connected clusters of locations. For an abstract polymer system the cluster estimate is (3.35)

L

Iw(q)1 exp(A(q)) :S A(p).

qE[(p)

The stability bound for the probability of a particle at q is (3.36)

1 LM N(q) MI1c(M)/IwAIM :S w(q)exp(A(q)). .

The stability bound for the expected number of particles at locations that are incompatible with location p is

(3.37) 4. Polymer systems. A polymer system is a realization of an abstract polymer system. There is a given set T of sites. The set P of locations consists of all finite non-empty subsets of T. If Y is a finite nonempty subset of T, the presence or absence of a particle at location Y is identified with the presence or absence of a polymer occupying the sites in Y . The exclusion interaction between locations is such that if the subsets

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

109

Y and Y' overlap, then there cannot be a polymer occupying the set Y and also another polymer occupying the set Y'. The set combinatorial exponential may be expressed as a polymer system. The object of interest is (4.1 )

K(X)

L

=

C(Yd .. · C(Yk ) .

f={Y1 ,...,Ykl

The sum is over partitions I' = {Y1 , . . . , Yk } . Each}j has at least one point, the Yj are disjoint, and the union of the }j is X. Then K is the combinatorial exponential of C. The interpretation is that K (X) is the partition function, which is a function of the variables C(Y) . Each r is a polymer configuration. The interaction is the condition that no pair of sets Y in the partition can overlap. This gives rise to a problem: The union constraint is not a two-location interaction. However, this problem has a solution. Suppose Y having one point implies that C(Y) = 1. The sum is now over sets r = {Y1 , ... , Yr } . Each }j has at least two points, and the }j are disjoint. However there is no longer a requirement on the union of the }j. This is now precisely a polymer system, where the weights are the C(Y), and the two-body interaction is the disjointness condition. In this combinatorial setting the cluster estimate is expressed in terms of positive quantities A(Y) that depend on the set Y. A typical choice is A(Y) = a\YI, where a > 0 is a constant and IYI is the number of points in Y . The cluster estimate is a condition on the cluster coefficients of the form

L

(4.2)

IC(Z)I exp(A(Z)) ::; A(Y) .

ZnYji0

Typically there is some kind of graph structure on T, and C(Z) = 0 except subsets Z that are connected. This limits the number of terms in the sum for a given size IZ I. Furthermore, the analysis is restricted to a regime where C(Z) approaches zero very rapidly as the size IZI gets large. Thus there are situations where such a cluster estimate holds . THEOREM 4 .1. Suppose that the interaction coefficients K(X) are the set combinatorial exponential of the cluster coefficients C(Z). Suppose that IZI = 1 implies C(Z) = 1. Assume that the cluster estimate holds. Then the probability that a polymer occupies the set Y is

(4.3)

C(Y) K(X \ Y) = K(X)

L M(Y) c(~) IT C(Z)M(Z) . M

M.

ZcX

Here the c(Y) are the cluster coefficients for the polymer system. These satisfy the estimate (4.4)

L M(Y) 1c(~)1 IT IC(Z)IM(Z) ::; exp(A(Y)). M

M.

zs:x

110

WILLIAM FARIS

Thus the ratio is expanded in a series whose radius of convergence depends on Y but not on X. Proof The proof amounts to a translation from the polymer language to the combinatorial language . The set of locations A is the set of nonempty subsets of X . Each location p in A is a subset Ye X. The weight w(p) is the coefficient C(Y). Similarly, the partition function Z/\(w) is the coefficient K(X). The interaction is an exclusion interaction: t(p, q) = 1 translates to Z n Y =I- 0. The translation of the polymer system identity

(4.5)

Z/\(w) = exp(L

c~) IT w(p)M(p))

M

pE/\

into the combinatorial language is

(4.6)

K(X) = exp(L M

c(~)

IT C(Z)M(Z)).

M. z-:x

The ratio Z/\\I(p)(w)/Z/\(w) of partition functions becomes the combinatorial ratio K(X \ Y)/ K(X). Now use the exponential representation (4.7)

The cluster estimate (4.8)

L

Iw(q)1 exp(A(q)) ::; A(p)

qEI(p)

implies the stability bound and convergence.

o

5. Cluster expansions. This section is a brief sketch of the general problem of constructing a probability measure from a density on a high dimensional space using cluster expansions. The book of Malyshev and Minlos [6] gives a much more complete account. Let E be a countable infinite set. Consider a probability measure J-L defined on the Borel subsets of the space RC . In the following the same notation J-L will be used for the expectation associated with this probability measure. Thus, if f is a bounded Borel measurable function on RC, then the expected value J-L(J) = Jf dJ-L is the integral of f with respect to the probability measure J-L. More than one measure may be considered; typically the expectation will be identified with the corresponding measure. If A is a finite subset of E, and f is a function on the finite dimensional space R /\, then f defines a corresponding function fA on RC that depends only on the coordinates in A. The value of f/\ on w in RC is f/\(w) = f(w/\),

GE NTLE INTRODU CTIO N TO CLUSTER EXPA NSIONS

111

where WA in RA is the restriction of w to A. Every function that depends only on th e coordin at es in A arises in this way. Fur thermore, the prob ability measure J.l is determined by the corresponding expect at ions for bounded measurable functions th at depend on only finitely many coordinates, for all such choices of finite subsets A c 1:-. It is assumed th at there is an initial measure , denoted simply by u, and that one can already do calculations or at least estimates with this measur e. For instance, it could be a product measure or a Gaussian measure. For each finite subset A c I:- let PA > 0 be a positive function on R.c that depends only on the coordinates in A. The task is to calculate with a new measure J.l~ with expectation given by this density : (5.1)

I

J.lA

(1)

=

J.lUPA) () . J.l PA

The hope is that even if there is no limiting densit y as A approaches 1:-, there will be a limiting measure J.l' . To show this , the main task is to get estimates that are independent of A. The idea is to express the expectation with J.l~ in a series in terms of certain expectations with u. The problem is that one needs to exhibit a cancellation between the numerator and denominator. This will be difficult unless there is a factorization of th e density with some approxim at e independ ence prop erty. In t his case, it someti mes possible to express the denominator (partition function)

(5.2) as a combinatorial exponential

(5.3)

K(A) =

L IT C(Y) , r

YEr

where r ranges over partitions of A. The C(Y) are called cluster coefficients . The idea is that if the factorization has good independence properties, then the domin ant contribution will come from the partition into one point subsets. There will be a similar expression for the numerator. There is often a formula for the numerator expressing it in terms of computable quantities together with the K(A \ Z) , where Z ranges over subsets of A. So the problem reduces to an estimation of th e rations K(A \ Z) / K (A) that is uniform in A. If the cluster coefficients corresponding to subsets with more than one point are small, then there is some hope of obtaining an estimate of these ratios. This is done by an expansion in terms of the cluster coefficients. In each case, the analys is is in stages: find the cluster representation, estimate the cluster coefficients, and then carr y out the analysis of convergence. Here we only indicate the first stage.

112

WILLIAM FARIS

As a first example consider a perturbation of a Gaussian measure by a density PA . Suppose that PA factors as

(5.4)

PA =

IT A

p

pEA

with a factor for each point in A. The factor Ap is a function that only depends on the p coordinate. There are two procedures that are used in such a situation. The first procedure is the usual perturbation expansion in terms of potential energy. The second is an expansion in terms of the density factors . The first method is expansion in the potential. The idea is to write Ap = exp( -(3pUp) in terms of a potential Up. The parameter {3p > plays a role similar to that of an inverse temperature. It is convenient for the expansion to allow (3p to depend on p. Such a representation in terms of potential energy is natural in physics. Then the relation between the combinatorical exponential and the exponential gives an explicit cancellation. This leads to an elegant representation for the expectation in terms of cumulants of the Up . PROPOSITION 5.1. Let the measure /lA be expressed in terms of the reference measure /l and a density PA that is the product of the Ap = exp( -(3pUp) for p in A. Let fj be functions that each depend on only finitely many coordinates corresponding to some subset of A. Th en the /lA

°

cumulants of the and Up by (5.5)

Ii

are expressed in terms of the /l cumulants of the

C'(L) =

Ii

LM M1,C(L,M)(-{3)M . .

for L #- 0, where the sum is over all multi-indices M supported in A. Proof Write tf = LjE} tj/j and {3U = LpEA {3pUp. Then the /l'

moment generating function for the

(5.6)

Ii

has the representation

'( ( f)) - /l(exp(tf - (3U)) /l exp t - /l(exp(-{3U))

as a quotient of /l moment generating functions for the fj and the Up. This may be written as a relation for exponential generating functions :

(5.7) Write each of these exponential generating functions for moments in terms of a thermodynamic potential that is an exponential generating function for cumulants. The quotient then becomes a difference. This gives the relation for the potentials:

(5.8)

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

113

When we write this out, we see a spectacular cancellation: the troublesome division becomes a simple subtraction. Th e explicit expression is

(5.9)

",1,

",11

L

L

M

L.JL!C(L)t =L.JL!M!C(L,M)t (-{3) , L

L ,M

where the sum is over all L =I 0 and all M . Equating the coefficients of t L gives the result stated in the proposition. 0 The second method is expansion in the interaction factor. The idea is to use the combinatorial exponential with cumulants of the Ap , which express dependence. The price one pays is that the result involves an ugly quotient. However this problem is solved by use of the polymer expansion explained in the previous section . PROPOSITION 5.2 . Let the measure fLA be expressed in terms of the Gaussian reference measure fL and a density PA that is the product of the Ap for p in A. Let K(X) = fL(I1 PE x Ap ) be the moment corresponding to the subset X. Let Ji be function s that depend only on the j coordinate. Let fw = I1 j Ew fj· Then the expectation has a representation (5 .10)

, fLA(fW)

=

L

G(Z)

K(A \ Z) K(A) .

WCZCA

Furthermore, the moments

(5.11)

K(X)

=L r

IT C(Y) YEr

have a representation as the combinatorial exponential of the cumulants o] the Ap . Proof The representation given above of the denominator K(A) = fL(PA) = fL(I1 PE A Ap ) is just the representation of a moment as a sum of

products of cumulants. For the numerator the factors Ap for p in Ware replaced by factors JpA p. This changes the moments and cumulants to (5.12)

K1(X) =

L IT C1(Y) r

YEr

However for each Y that does not intersect W th e cumulant Cl (Y) C(Y) . This is because the combinatorial logarithm formula shows that it is determined by moments K 1 (Z) = K (Z) for Z c Y . For each partition

I', there are certain Y in I' that have a non-zero intersection with W . Let the union of these Y be denoted Z. Then the numerator is (5 .13)

K1(A) =

L WcZcA

G1(Z)K(A \ Z) .

114

WILLIAM FARIS

Here (5.14)

G1(Z) =

L IT C1(Y), ~

YE~

where t1 ranges over partitions of Z with the property that every element of the partition has a non-zero intersectiori'with W. 0 As a second example, consider the perturbation of a product measure by a density PII. . PROPOSITION 5.3. Let the measure tL~ be expressed in terms of the product reference measure tL and a density PII. . Let fw be a functions on R w, where W c A. The tL~ expectation of fw has a representation (5.15) where

(5.16)

L IT C(Y)

K(X) =

fCII. YEf

is the sum over partitions. Proof. The idea is to use the combinatorial exponential to write

(5.17)

LIT py .

PX =

f

YEf

By the combinatorial logarithm formula py depends only on the oz for Z c Y . Since az depends only on the coordinates in Z, it follows that py depends only on the coordinates in Y . Since tL is a product measure, if we set K(X) = JL(px) and C(Y) = tL(Py), then we get the representation (5.18)

tL(PII.)

IT C(Y)

= K(A) = L f

where

r

ranges over partitions of A. For the numerator we can write

Iw o«

(5.19)

L

=

gz

WCZCII.

where (5.20)

r

YEf

L IT py, f

YEr

is summed over partitions of A \ Z. The factor gz is given by

gz =

Lfw ~

IT py, YE~

where t1 is summed over partitions of Z with the property that each element Y of t1 has a non-empty intersection with W. Let G(Z) = JL(gz) . Then we get (5.21)

~(fwplI.)

=

L G(Z)K(A \ Z). z

o

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

115

Acknowledgements. The author is grateful for the hospitality of the Courant Institute of Mathematical Sciences, New York University, where this work was begun. He also thanks Daniel Ueltschi for helpful comments.

REFERENCES [1] A. BOVIER AND M. ZAHRADNiK , A simple inductive approach to the problem of convergence of cluster expansions of polymers, J . Stat. Phys. 100 (2000), pp. 765-778. [2] D.C . BRYDGES, A short course on cluster expansions, in K. Osterwalder and K. Stora, eds ., Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII, 1984, Elsevier, Amsterdam, 1986, pp . 129-183. [3] R.L. DOBRUSHIN, Perturbation methods of the theory of Gibbsian fields, in Lectures on Probability Theory and Statistics (Lectures Notes in Math. #1648), Springer-Veriag, Berlin, 1996, pp . 1-66 . [4] W .G . FARIS AND R.A . MINLOS, A quantum crystal with multidimensional harmonic oscillators, J. Stat. Physics 94 (1999), pp. 365-387. [5] R . KOTECKY AND D . PREISS, Cluster expansions for abstract polymer models, Commun. Math. Phys. 103 (1986), pp . 491-498. [6] V.A . MALYSHEV AND R .A. MINLOS, Gibbs Random Fields: Cluster Expansions, Kluwer, Dordrecht, 1991. [7] S. MIRACLE-SOLE, On the convergence of cluster expansions , Physica A 279 (2000) , pp . 244-249. [8] D. UELTSCHI, Cluster expansions and correlation functions , Moscow Math. J ., 4 (2004) , pp. 509-520.

CONTINUITY OF THE ITO-MAP FOR HOLDER ROUGH PATHS WITH APPLICATIONS TO THE SUPPORT THEOREM IN HOLDER NORM PETER K. FRIZ' Abstract . Rough Path theory is currently formulated in p-variation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric p can be used . Based on fine-estimates in Lyons ' celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (between Rough Path spaces equipped with p). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t . p. In combination with our Holder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the Stroock-Varadhan Support Theorem in Holder topology as immediate corollary. Key words. Rough Path theory, Ito-rn ap, Universal Limit Theorem, p-variation vs. Holder regularity, Support Theorem. AMS(MOS) subject classifications. 60Gxx .

-

®etoiDmet oem 2!nDeneen an 13rof. Dtto 13regL -

1. Introduction. 1.1. Background in Rough Path theory. Over the last years T . Lyons and co-authors developed a deterministic theory of differential equations capable of dealing with "rough" driving signals such as typical realizations of Brownian Motion . To explain what is now known as Rough Path Theory we consider controlled ordinary differential equations. To fix ideas , set V = ]Rd and W = ]RN and assume (Xt)tE[O,lj is a V-valued path of (piecewise) C 1_ regularity,

(1.1) Let I = (h, ..., Id) be a collection of d vector-fields on W, identified with a map

(1.2)

I :W

--+

L (V, W) ,

'Courant Institute, NYU , New York, NY 10012. The author acknowledges financial support by the Austrian Academy of Science. 117

118

PETER K. FRIZ

and consider the controlled ODE d

dYt = f(Yt)dxt = f(YdXt dt =

(1.3)

L fi(Ydx~dt . i=l

Of course, under standard conditions on f and for fixed initial point Yo, there is a unique W-valued solution path on our chosen time-horizon

[0,1]. Assume that one can find a metric d on Cl ([0,1], V) resp. Cl ([0,1], W) such that the Ita-map C l ([0,1] , V)

(1.4)

{

X .......

Cl ([0,1], W)

Y

is uniformly continuous (at least on bounded sets). Then the meaning of (1:"3) can be extended to driving signals in the closure of Cl-paths, x E Cl ([0,1] , V),

(1.5)

the closure being taken in the topology induced by d. EXAMPLE 1. Let V = ]R2, W =]R3 and consider

with

h = (1,0, _y 2 / 2), 12 = (0,1, y 1 /2) . With Yo =

(1.6) (1.7)

°

integration yields y 3(t) =

t r (x 2 lo

~

ldx 2 _

x 2dx l )

== At(x) .

Note that At(x) has a simple geometric interpretation as area. particular,

leads to At(x(m)) = nt independent of m. On the other hand xt(m) ---uniformly as m tends to infinity.

In

°

The above example shows that the uniform topology is not suited to carry out the extension (1.5). It was realized by T. Lyons that stronger path-space topologies, such as p-variation topology, are suited. We recall

ROUGH PATHS AND SUPPORT THEOREM IN HOLDER NORM

119

DEFINITION 1.1. The p-variation semi-norm of a path x with values in a normed vector-space is defined as

''tJ' (~ lx"

(1.8)

- x" _'IP )

I Jp

where sup D runs over' all dissections D = {a = to < tl < ... < tlDI = I} of [0, 1]. For fixed starting point XQ this provides a genuine norm on pathspace. EXAMPLE 2 . Almost every Brownian path has finite p-variation for p > 2, an immediate consequence of its lip-Holder regularity. On the other hand, its p-variation with pE [1,2] is known to be almost surely infinity. Actually, in the case p E [1,2) one does not have to carry out the closing procedure (1.5) . The differential equation (1.3) may be interpreted directly as Young-integral equation. Recall from [Y] the classical THEOREM 1.1 (L .C . Young) . Letv ,w E C([O ,I],lR) offinitep- resp. q-variation such that

1 p

1 q

-+->1. Then the Riemann-sum below converges and defines the Young Integral :

:J

lim

LVti (Wti - Wti_l ) ==

m"h( D )-.O i

lot v dw,

where D are dissections of [0, t]. We cite from the recent survey paper [Le] the following THEOREM 1.2 . Let f : W -+ L (V, W) be C2 with bounded derivatives up to order 2. Then for any x E C ([0,1], V) of finite p-variation with p E [1,2) and fixed starting point Yo , there exists a unique y E C ([0, 1] , W) , also of finite p-variation, such that (1.3) makes sense as Young integral equation. Moreover, the Ito-mop x f--4 y is continuous in p-variation norm. It is clear from example 2 that this result does not cover differential equations driven by Brownian motion. A new idea is needed and we give some motivation: • From (1.6) we see that iterated integrals are related to the issue of Ita-map continuityIdiscontinuity. • Consider the case of a linear ODE, e.g . f = A, a constant coefficient tensor in W 0 ~V* 0 V * so that (1.3) reads N

dy = Aydx

{:=}

dy~

d

L A~,kyt dx~. j=lk=l

= L

At least when x E Cl ([0,1], V) we may expand u

Yt = Ys + i t A (YS

+i

AYsdx v

+ ...)

dx.,

120

PETER K. FRIZ

and see that, keeping A fixed, the evolution from Ys to Yt is fully determined by iterated integrals of form

X:,t :=

1

S

1 when 1 ~ p < 2. For p ?: 2 simply set n = tpj and define the distance between two paths x,x E Cl ([0,1] , V) as

In the ground-breaking paper [L98] Lyons showed uniform continuity (over bounded sets) of the Ita-map (1.4) w .r .t, generalized p-variation distance for any p ?: 1. Then, of course, the closing procedure (1.5) works. However, there is no reason to expect that this abstract closure can be identified with a subset of the actual path-space C ([0,1] , V ). On the other hand, it is plausible from (1.9), and shown in det ail in [LQ], that it can be identified with a closed subset of C (.6., V EB ... EB Vjm (t)'1J jm

jm

(x) ,

of the velocity field, once inserted into (1.1), motivates the dynamical toymodel amplitude equations

(5.3)

OtVl

= ik 1

L c(l; 2, 3)V2V

3 -

lJk;Vl + it

,

2 ,3

known as the hierarchical shell model [20]. The last two terms only act at the small and large scales. Here it would be very interesting to find out which, if any, structure of the coupling c(1=jlml ; 2=12m2, 3=J3m3) between wavelet modes is required to reproduce the spatial statistics of the random multiplicative branching processes. This then would serve as a first link between the empirical random multiplicative cascade processes and the Navier-Stokes equation. More dreams in this direction are always at work! REFERENCES (1) A .S . MONIN AND A .M . YAGLOM, Statistical Fluid Mechanics, Vols . 1 and 2, MIT Press, 1971. [2J U . FRISCH, Thrbulence, Cambridge University Press, 1995. [3J R. BENZI, S . CILIBERTO , R . TRIPICCIONE, C . BAUDET, F . MASSAIOLOI, AND S . SUCCI, Extended self-similarity in turbulent flows, Phys. Rev. E, 48 (1993) , pp . R29-R32. [4) B .R . PEARSON , P .A . KROGSTAD , AND W . VAN DE WATER, Mea.surements of the turbulent energy dissipation rate, Phys. Fluids, 14 (2002) , pp . 1288-1290 . [5) B . DHRUVA, An Experimental Study of High Reynolds Number Thrbulence in the Atmosphere, PhD thesis , Yale University, 2000 . [6] J. CLEVE, M. GREINER , AND K .R . SREENIVASAN , On the surrogacy of the energy dissipation field in fully developed turbulence, Europhys. Lett ., 61 (2003) , pp . 756-761. [7) M. GREINER, P . LIPA, AND P . CARRUTHERS, Wavelet-correlations in the p-model, Phys.Rev. E, 51 (1995) 1948-1960. [8) M. GREINER, J. GIESEMANN, P. LIPA, AND P. CARRUTHERS, Wavelet-correlations in hierarchical branching processes, Z. Phys. C, 69 (1996), pp. 305-321. [9) M . GREINER, H . EGGERS, AND P . LIPA, Analytic multivariate generating function for random multiplicative cascade processes, Phys. Rev . Lett., 80 (1998) , pp . 5333-5336.

150

MARTIN GREINER ET AL.

[101 M . GREINER, J . SCHMIEGEL, F . EICKEMEYER, P . LIPA, AND H. EGGERS, Spatial [11] [12]

[13] [14] [15] [16J [17]

[18] [19J [20]

correla tions of singularity strengths in multifractal branching processes , Phys. Rev. E, 58 (1998), pp. 554-564. H. EGGERS, T . DZIEKAN , AND M . GREINER, Translat ionally invariant cumulants in energy cascade models of turbulence, Phys. Lett. A, 281 (2001), pp. 249- 255. J . CLEVE, T . DZIEKAN , J . SCHMIEGEL, O .E . BARNDORFF-NIELSEN, B.R. PEARSON , K.R. SREENIVASAN, AND M. GREINER, Data-driven derivation of the turbulent energy cascade generator, arXiv:physics/0312113 . J . CLEVE, M . GREINER, B .R. PEARSON, AND K .R. SREENIVASAN, On the intermittency exponent of the turbulent energy cascade, arXiv:physics/0402015. B . JOUAULT , P . LIPA, AND M. GREINER, Multiplier phenomenology in random multiplicative cascade processes, Phys. Rev . E , 59 (1999) , pp. 2451-2454. B. JOUAULT, M . GREINER, AND P. LIPA, Fix-point multiplier distr ibutions in discrete turbulent cascade models , Physica D, 136 (2000) , pp. 125-144. M. GREINER AND B . JOUAULT , An experimentalists view of discrete and con ti n uous cascade models in fully developed turbulence, Fractals, 10 (2002) , pp . 321-327. J . SCHMIEGEL, J . CLEVE, H. EGGERS , B . PEARSON , AND M . GREINER, Sto chastic energy-cascade model for 1+1 dimensional fully developed turbulence, Phys. Lett. A, 320 (2004), pp. 247-253. J . GIESEMANN, M . GREINER, AND P . LIPA, Wavelet cascades, Physica A, 247 (1997), pp. 41-58. M . GREINER AND J . GIESEMANN, Correlation structure of wavelet cascades, in Wavelets in Physics, L .Z. Fang and R.L. Thews (ed.), (1998) , pp. 89-131. R . BENZI, L . BIFERALE, R. TRIPICCIONE, AND E. TROVATORE, (1+1)-dimensional turbulence, Phys. Fluids, 9 (1997) , pp. 2355-2363.

STOCHASTIC FLOWS ON THE CIRCLE YVES LE JAW AND OLIVIER RAIMOND* Abstract. Brown ian flows on th e circle associated with singular covariances are studied . Key words. Stochastic differential equations, Wiener chaos decomposition, coalescence. AMS(MOS) subject classifications. Primary. 60HlO.

Contents 1

2 3

4

5

Introduction . Flows of diffeomorphisms . The Krylov Veretennikov expansion 3.1 Lipschitz case . 3.2 Non-Lipschitz case . 3.3 A flow of infinite matrices . Flows of kernels and flow of maps . 4.1 n-point motions . 4.2 Flow of maps . 4.3 Diffusive flow of kernels 4.4 Diffusive or coalescing? Classification of the solutions of the SDE 5.1 Solutions of th e SDE . Extension of th e noise and weak solutions 5.2

151 152 153 153 154 156 157 157 158 158 158 160 160 160

1. Introduction. The purpose of th is note is to provid e a simple introduction to the papers [6, 7] by considering the example of stochastic flows on the circle which was not treated in these papers. We refer to these papers for the detail of th e proofs, but the ideas can be explained more clearly and more rapidly in this simpler context. We will finally st ate a conject ure on the classification of isotropi c flows of kernels.

Notation. In all the following, we will denote by § the unit circle

1R/21TZ, by m the Lebesgue measure on § and by P(§) the set of Borel probability measures on S. The set P(§) is equipped with a metric compatible with t he weak convergence. Th e er-field of Borel sets of § and of

P(§) are respectively denoted B(§) and B(P(§)). Let (W , F W , Pw) be the canonical probability space of a sequence of independent Wiener processes (Wtk , k ~ 0, t ~ 0). For all s < t, let F~ denote the er-field generated by the random variables W: - W:, s ~ u < v ~ t and k ~ o. Being given (akh~o a sequence of nonnegative *Mathematiques, Bat 425, Universite Par is-Sud , 91405 Ors ay cedex , FRANCE. 151

152

YVES LE JAN , AND OLIVIER RAIMOND

numbers such that Lk>O a~ < 00, we set C(z) = Lk>O a~ cos(kz). Note that all real positive definite functions on § can be written in this form and that C(O) = Ek~O a~. 2. Flows of diffeomorphisms. Assume that Ek>l k2a~ < 00. Then by a stochastic version of Gronwalllemma, it can be shown that for each Xo E § the stochastic differential equation (SDE) (2.1)

Xt = Xo

+ aoW? +

L ak (1 k~l

t sin(kxs)dW;k-l

+

0

it

COS(kxs)dW;k)

0

has a unique strong solution. These solutions can be considered jointly to form a stochastic flow of diffeomorphims (k

k-I

X

cl,k- l;

v, = L

l

2"

X

eu : »

1=1

and

lHere and in the following, ek,l denotes th e infinite matrix such that ek,t(i,j) = 8k ,i 81,j .

157

STOCHASTIC FLOWS ON THE CIRCLE

M

Set

ii;

2k

= ak

o, -

0 Ui. - U

(

k + Vk

o;0 - V

k )

.

= L:1~1 l X el,l and

MO = ao (~o

~o) .

Then, we have

(3.4)

Set now Ak,l = (o:k ,CI)L2(m), Bk,l and Dk ,l

= (o:k,CI)L2(m)'

Set M

= (o:k,SI)L2(m), Ck,l = ({3k,CI)L2(m) = (~ ~), with A = ((Ak ,l)), B =

((Bk,l)), C = ((Ck,l)) and D = ((Dk ,l)). Then, for all s < t

u., = u., + l:.: M k t u.; dW: + k~l

(3.5)

MO with Ms,s =

is

it Ms,udW~ itu.; - N

du

(~ ~), where I = L:l~o el,l·

4. Flows of kernels and flow of maps.

4.1. n-point motions. Let p~n) be the family of random operators acting on LOO(mi ,j::>n

The case (a) appears when the diagonal is absorbing for the two-point motion . If it is not the case, we are in case (b).

158

YVES LE JAN, AND OLIVIER RAIMOND

4.2. Flow of maps. In case (a), it can be shown (using the Feller property) that there exists 'P = ('Ps,t) a family of random mappings such that for all s ~ t and all f E L 2 (m), Ss,d = f 0 'Ps ,t in L 2 (m ® Pw ), 'Ps ,t : (S x W , B(§) ® J:W) ....... (S, B(§» is measurable and (i) Co cycle property: 'Ps ,u(x) = 'Pt ,u 0'Ps,t(x) a.s . for all s < t < u and all x. (H) Stationary increments: for all s ~ t, 'Ps ,t and 'PO ,t-s have the same law. ~ . . . ~ t n , 'Pto ,t), . .. ,'Ptn_) ,tn are independent. Such a family is called a measurable stochastic flow of measurable mappings. It can be shown that this flow solves the SDE (2.1).

(Hi) Independent increments: for to

4.3. Diffusive flow of kernels. In case (b), it can be shown (still using the Feller property) that there exists K = (Ks ,t) a family of random kernels such that for all s ~ t and all f E L2(m), Ss,d = Ks ,d in L 2(m ® Pw ), Ks ,t : (S x W, B(§) ® J::w) ....... (P(§),B(P(§») is measurable and (i) Co cycle property: Ks,u(x) = Ks ,tKt,u(x) a.s, for all s < t < u and all x. (H) Stationary increments: for all s ~ t , Ks ,t and KO,t-s have the same law. (Hi) Independent increments: for to ~ . . . ~ t n , Kt o,t), . . . , Ktn _),tn are independent. Such a family is called a measurable stochastic flow of kernels and it will be called diffusive when the kernels are not induced by maps, which clearly happens in case (b). This solves the SDE in the sense that for all f E C 2 (§), s ~ t and x E S,

Ks,d = f

+L

k>l -

(4.2)

ao

ak (it s

Ks,u(sk!')dW~k-l + it KS'U(Ckf')dW~k) +

it Ks,u(f')dW~

s

+

C;O)

it

Ks,uf"du.

We still say that K is a generalized solution of the original SDE (2.1), when K is not induced by a flow of maps. In fact the adjective generalized can be omitted like in [7]. In the following, the flow 'P (in case (a» or the flow K (in case (b) will be called the Wiener solution of the SDE (2.1). Since there is un iqu eness to a solution (Ss,d, such that Ss,t is F~-measurable, the Wi ener solution 'P (or K) is the unique solution of SDE (2.1) (or of (4.3» such that 'Ps,t (or Ks ,t) is F~-measurable.

4.4. Diffusive or coalescing? A stochastic flow of maps ('Ps,t) for which there exist x =1= Y E § such that P[3t > 0 : 'PO,t(x) = 'PO,t(Y)] > is called a coalescing flow. A diffusive flow for which the two-point motion hits the diagonal fj. = {(x,x) , x E S] is called diffusive with hitting.

°

159

STOCHASTIC FLOWS ON THE CIRCLE

The diffusion Zt E [0, 21r), such that Zt = X, - Yt modulo 21r , where (X t , Yt) is the two point motion, has a natural scale. The speed measure m of this diffusion is given by m(dz) = (C(O) - C(z))-1dz . Let I'\, be defined by I'\,(z) = C(O)::::~(x)dx. Note that 1'\,(0+) = 00 implies that m((O, 21r)) = 00 .

f:

THEOREM 4.1.

(a) If 1'\,(0+) = 00, then the Wiener solution is a stochastic flow of maps, which is not a coalescing flow. (b) If m((O, 21r)) = 00 and 1'\,(0+) < 00, then the Wiener solution is a coalescing flow. (c) If m( (0,21l')) < 00, then the Wiener solution is a diffusive flow with hitting. Proof To prove this theorem, we study the diffusion Zt, and the (we use Breiman's terminology, see [2]). Since the boundary point scale function s is finite on [0, 21r), is an open boundary point (i.e., P z(3t > 0, Zt = 0) = 0) if and only if 1'\,(0+) = 00 (see theorem 3.2 in

°

[4]).

°

°

°

Suppose that 1'\,(0+) = 00, then is open and since m((O, 21r)) = 00, is not an entrance boundary point. This implies that the two point motion does not leave ~ , and cannot hit ~ when starting out of ~ . This property implies (a). Suppose m((0,21r)) = 00 and 1'\,(0+) < 00 . In this case is an exit boundary point. This implies (b) . Suppose m((O, 21r)) < 00. In this case, is a regular boundary point. Thus one has to decide if this diffusion is absorbed or reflected at 0. As in theorem 9.4-c in [6], it can be proved that is instantaneously reflecting. 0 This implies (c).

°

° °

Let a% = k-(1+o:), with 0: > 0. Then, If 0: > 2, the Wiener solution is a stochastic flow of G1_ diffeomorphisms. If 0: = 2, the Wiener solution is a stochastic flow of maps, which is not a coalescing flow. If 0: E [1,2) , the Wiener solution is a coalescing flow. If 0: E (0,1), the Wiener solution is a diffusive flow with hitting. Proof (a) follows from the fact that when 0: > 2, Lk k2a~ < 00 . COROLLARY 4.1.

(a) (b) (c) (d)

LEMMA 4 .2 .

°

• When 0: = 2, there exists C2 > such that as Z - . 0, C(O) -C(z) = -c2z2log Izl(l + 0(1)) . • When 0: E (0,2), there exists Co: > such that as Z - . 0, C(O) C(z) = co:zO:(l + 0(1)). Proof We prove this lemma only for 0: E (0,2). For Z E (0,21r), G'(z) is well defined and G'(z) = Lk::::1 k-O: sin(kz) . Using the fact that

°

160

YVES LE JAN, AND OLIVIER RAIMOND

k- o =

(00 e-ks s0-1 r(o)' ds

Jo

oo

L r e-kss o- I sin(kz)~ k~IJO r(a) = _2- L r e-kssO-I(eikz _ e-ikz)~

G'(z) = -

oo

2ik~l~

r(a)

1 I" (e-Se iZ e-Se- iZ) ds = - 2i Jo 1 - e-seiz - 1 _ e-se-iz so-I r(a)

r

oo

=- Jo

e- s sin z 0-1 ds 2s s 1-2e-scosz+er(a)

= _zo-I

100 I(t, z)dt

o 1 sin z t Note t h at t here exi h I( were t, ) z = 1-2e-e-tz ere exists a constant G t • cosz+e 2t. r(o)' o3 such that for t > 1, II(t,z)l::; Gxt and for t E (0,1]' II(t,z)l::; Gxt o - I . Thus, using Lebesgue dominated convergence theorem, we prove that there exists a constant Co such that, as z ~ 0, G'(z) = -aco x zo-I + o(zo-I) . This implies the lemma. 0

This implies that m((0 ,2rr)) = 00 for a E [1,21 and m((O,2rr)) < 00 for a E (0,11. When a = 2, 1'1:(0+) = 00 and when a < 2, 1'1:(0+ ) < 00. Th eorem 4.1 implies th e corollary. 0 REMARK 4.1. The case a = 2 has been studied in (1,3, 8J. In fact, it is shown that the maps of the flow are homeomorphisms. 5. Classification of the solutions of the SDE. 5.1. Solutions of the SDE. Let (0, A, P) be an extension of the probability space (W , F W , Pw) . We say that a measurable flow of measurable maps cP = (CPs ,t} is a weak solution of (2.1) if it satisfies (2.1) without being F~-measurable . Similarly, a measurable flow of kernels K = (Ks ,t) will be called weak (generalized) solution of the SDE (2.1) if it satisfies (4.3) without being F~-measurable . We have seen that uniqueness is verified if one assumes in addition is F~ for all s ::; t. Wiener measurability:

«.,

5.2. Extension of the noise and weak solutions. In case (b), a different consistent syst em of Feller semigroups p~n),c can be constructed by considering the coalescing n-point motion Xt( n ),c associated with Xt( n ), the n-point motion of the Wiener solution. A measurable flow of coalescing maps cP~ t, whose n-point motion is n ),c , can be defined on an extension (0 , A, P) of the probability space (W, F W , Pw). This coalescing flow also solves the SDE (2.1). It is a weak solution. For s ::; t, set F~ t = (j(cp~ v' s::; u ::; v::; t) . Then (F~ t)s O. This completes the proof of Proposition 3.3. 0 Remarks. 1. The assumption that the measure V is finite was made for technical simplifications, and seemingly can be removed. 2. The assumption dim(V) > d - 1 is essential for (3.9) to be true. However, one can regularize (3.9) appropriately to include more general V (see [K2]). 3. There is seemingly some overlap of ideas between our construction of the Wiener path integral for the Schrodinger equation and the theory of rough paths of T . Lyons [L]. 4. It is worth noting that the natural topology on the path space CPL is the one induced from the uniform topology of continuous paths (or from the Hilbert space topology of the Cameron-Martin space). This topology enjoys the following properties: (i) it is compatible with the measure, (ii) when reduced to any finite-dimensional simplex it yields its natural Euclidean topology, (iii) any simplex Simr is the boundary for the simplex Simr+l, (iv) the topology is not locally compact, but the whole space is a countable union of locally compact spaces . 4. Two remarks on parabolic equations in momentum representation. As was already mentioned, the first definition of the Feynman path integral representing solutions for the Schrodinger equation as a genuine Lebesgue integral arising from a pure jump Markov process was given in [M], [MCh]. This integral was defined for the Scgrodinger equation in momentum representation with potentials satisfying Ito's complex measure condition. This was an important breakthrough. As for the diffusion equation the familiar Feynman-Kac representation exists, the analogous result for the diffusion equation in momentum representation did not receive much attention. This is also due to the fact that unlike Schrodinger equation the momentum representation for the diffusion equation often does not seem very natural physically, though it does make sense in the study of tunnel effects in quantum mechanics. A recent paper [Ch] is devoted to an interesting detailed analysis of the underlying jump processes for diffusion equations in momentum representation under Ito's complex measure condition for sources (potentials) and drifts. In this section I like to point out two simple observations about this theory. Firstly, in some cases one can get meaningful path integral representation for diffusion equations in momentum representation even when the source does not satisfy Ito's condition (so that the underlying process is not of pure jump type) and when an unbounded source prevents the possibility of using the standard FeynmanKac formula with the Wiener measure. Secondly, a curious asymptotic formula can be obtained by passing to a central limit in a pure jump path integral representation for the diffusion equation. Moreover, instead of just diffusion equations one can directly consider more general parabolic differential and even pseudo-differential equations without any increase in the complexity.

178

VASSILI N. KOLOKOLTSOV

1. Stable laws for parabolic equations. Consider the pseudo-differential parabolic equation

(4.1) in R d , where a > 0, G > 0, (3 E (0,1) are given constants (in fact, instead of 1.6.la one can take even more general operators f(I.6.1) with non-negative continuous function I). Since

with some c depending on d and {3 (see any book discussing stable processes, e.g. [K2J), in momentum representation, Le. for u(p) = I e-ipx'IjJ(x) dx, the equation (4.1) takes the form

(4.2)

~~ (p) = _GlpI2au(p) + c

J

(u(p

+~) - u(p))I~I-d-{3 d~.

As the second operator in this equation generates the Feller semigroup of a (3-stable Levy process, the following result is straightforward. PROPOSITION 4 .1. Solution to (4.2) with an arbitrary bounded initial fun ction Uo is given by the formula u(t,p) =

s,

[exp{ -G

I

t

ly(s)1 2a ds}uo(y(t))] ,

where E p denotes the expectation with respect to the corresponding (3 -stable Levy motion starting at p. 2. A central limit for pure jump path integrals. Consider now the equation (4.3)

with a small parameter h > 0, a, G are again positive constants and V(x) = with some finite positive measure M, Le. V satisfies Ito 's condition. Suppose also that M is symmetric in the sense that I ~M (d~) = 0, and that it has finite moments at least up to the third order. Denote by II(M) = {lIij(M)} the matrix of the second moments I ~i~jM(d~) of M. Performing the h-Fourier transform (which is usual in the theory of semiclassical asymptotics, see e.g. [MFJ), Le. passing to the function

I eix e M (~)

Uh(t,p) =

J

e-ipx /h'IjJ(t,x)dx

{=:::}

'ljJh(t,X) = (21rh)-d

yields for Uh(t,p) the equation (4.4)

ou =

-

ot

(h 0)

- Glpl2a u+ -12 V -;- u h Z op

J

eipx/hu(t,p)dp

CONNECTING PURE JUMP AND WIENER PROCESSES

179

for Uh(t,p). PROPOSITION 4.2. For the solution of (4.4) with the initial condition Uo the following asymptotic formula holds:

lim exp{th- 21IMII}uh(t,p)

h......O

(4.5)

= E w exp { - G

I

t

Ip + W(sWO ds }UO(P + W(t)),

where Ew denotes the expectation with respect to the d-dimensional Wiener process W(s) with the covariance matrix v(M), and where IIMII is, of course, the full measure M(Rd). Proof Using the properties of pure jump processes, one can write the solution to (4.4) with the initial condition Uo as the path integral (4.6)

Uh(t,p) =

exp{th-21IMII}E~ exp { -

G

I

t

jy(sW ds },

where E~ denotes the expectation with respect to the process of jumps which are identically independently distributed according to the probability measure J.1.(dp) = M(d(p/h))/IIMII and which occur at times Sj from [0, t] that are distributed according to the Poisson process of intensity h- 21IMII. Observing that

h- 2V(hx)

= h- 2

J

(eihxe - l)M(de)

+ h- 211MII

d

= -

L

XiXjVij

+ O(h) + h- 21IMII,

i ,j=l

we conclude that the characteristic exponent of our compound Poisson process converges, as h ---+ 0, to the characteristic exponent -(v(M)x,x) of the Wiener process indicated above. This implies the convergence of the corresponding measures on trajectories, and (4.5) follows . 0 Acknowledgements. I am grateful to E. Waymire for inviting me on a very stimulating summer conference in Minnesota on probability and partial differential equations, and to O. Gulinskii and A. Fesenko for useful discussions on the Feynman integral.

REFERENCES [A]

[ABB]

S. ALBEVERIO ET AL. Schrodinger operators with potentials supported by null sets. In: S. Albeverio et a!. (Eds.) Ideas and Methods in Quantum and Statistical Physics, in Memory of R. Hoegh-Krohn, Vo!. 2, Cambridge Univ. Press, 1992, 63-95. S. ALBEVERIO, A. BOUTET DE MONVEL-BERTIER, AND ZD. BRZEZNIAK . Stationary Phase Method in Infinite Dimensions by Finite Approximations: Applications to the Schrodinger Equation. Paten. Anal. 4:5 (1995), 469502.

180 [AKS]

[Ch]

[ChQ]

[K1]

[K2] [K3]

[K4] [K5] [L] [M] [MCh]

[MF] [Me]

[Mey] [PQ] [SS]

VASSILI N. KOLOKOLTSOV S. ALBEVERIO, V.N . KOLOKOLTSOV, AND O.G. SMOLYANOV. Representation des solutions de l'equation de Belavkin pour la mesure quantique par une version rigoureuse de la formule d'integration fonctionnelle de Menski, C.R. Acad. Sci . Paris , Ser . 1, 323 (1996), 661-664 . L. CHEN ET AL. On Ito 's Complex Measure Condition. In: Probability, statistics and their applications: paper in honor of Rabi Bhattacharya, p. 65-80, IMS Lecture Notes Monogr . Ser. 41, Inst. Math. Statist., Beachwood, OH, 2003. A.M . CHEBOTAREV AND R.B . QUEZADA. Stochastic approach to time-dependent quantum tunnelling. Russian J. of Math . Phys . 4:3 (1998), 275-286 . V. KOLOKOLTSOV. Complex measures on path space: an introduction to the Feynman integral applied to the Schrodinger equation. Methodology and Computing in Applied Probability 1:3 (1999), 349-365. V. KOLOKOLTSOV. Semiclassical Analysis for Diffusion and Stochastic Processes. Monograph. Springer Lecture Notes Math. Series , Vol. 1724, Springer 2000. V. KOLOKOLTSOV. A new path integral representation for the solutions of the Schrodinger, heat and stochastic Schrodinger equations. Math. Proc. Cam . Phi!. Soc. 132 (2002), 353-375. V.N. KOLOKOLTSOV. Mathematics of the Feynmann path integral. Proc. of the International Mathematical conference FILOMAT 2001, University of Nis, FILOMAT 15 (2001), 293-312 . V.N . KOLOKOLTSOV. On the singular Schrodinger equations with magnetic fields. Matem. Zbomik 194:6 (2003), 105-126 . T.J . LYONS. Differential equations driven by rough signals. Revista Matematica lberoamericana 14:2, 1998. V .P. MASLOV. Complex Markov Chains and Functional Feynman Integral. Moscow, Nauka, 1976 (in Russian) . V.P . MASLOV AND A.M. CHEBOTAREV. Processus it. sauts et leur application dans la mecanique quantique. In: S. Albeverio et al. (Eds .). Feynm an path integrals. LNP 106, Springer 1979, pp. 58-72. V .P . MASLOV AND M.V . FEDORYUK. Semiclassical approximation in quantum mechanics. Reidel, Dordrecht, 1981. M.B. MENSKI. The difficulties in the mathematical definition of path int egrals are overcome in the theory of continuous quantum measurements. Teor. Mat. Fizika 93:2 (1992), 264-272. P . MEYER. Quantum Probability for Probabilists. Springer Lecture Notes Math. 1538, Springer-Verlag 1991. P . PEREYRA AND R. QUEZADA. Probabilistic representation formulae for the time evolution of quantum systems. J. Math . Phys. 34 (1993), 59-68. O.G . SMOLYANOV AND KT. SHAVGULIDZE. Kontinualniye Integrali. Moscow Univ . Press, Moscow, 1990 (in Russian) .

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS MUKUL MAJUMDAR" Abstract. Random dynamical systems are useful in modeling the evolution of economic processes subject to exogenous shocks . One obtains strong results on the existence, uniqueness, stability of the invariant distribution of such systems when an appropriate splitting condition is satisfied. Also of importance has been the study of random iterates of maps from the quadratic family. Applications to economi c growth models are reviewed . . Key words. Dynamical systems, Markov processes, iterated random maps, invariant distributions, splitting, quadratic family, estimation, economic growth.

1. Introduction. Consider a random dynamical system (B, I' , Q) where B is the state space (for example, a metric space) , r an appropriate family of maps on B into itself (interpreted as the set of all possible laws of motion) and Q is a probability measure on (some er-field of) r . The evolution of the system can be described as follows: initially, the system is in some state x ; an element a1 of r is chosen randomly according to the probability measure Q and the system moves to the state Xl = a1(x) in period one. Again, independently of aI, an element a2 of r is chosen ac, cording to the probability measure Q and the state of the system in period two is obtained as X 2 = a2(a1(x)). In general, starting from some x in B, one has

(1.1) where the maps (an) are indepenent with the common distribution Q. The initial point x can also be chosen (independently of (an)) as a random variable X o. The sequence X n of states obtained in this manner is a Markov process and has been of particular interest in dynamic economics. It may be noted that every Markov process (with an arbitrary given transition probability) may be constructed in this manner provided B is a Borel subset of a complete separable metric space , although such a construction is not unique [Bhattacharya and Waymire [1] , p. 228]. Hence, random iterates of affine, quadratic or monotone maps provide examples of Markov processes with specific structures that have engaged the attention of probability theorists. Random dynamical systems have been studied in many contexts in economics, particularly in modeling long run evolution of economic systems subject to exogenous random shocks. The framework (1.1) can be interpreted as a descriptive model; but one may also start with a discounted (stochastic) dynamic programming problem , and directly arrive at a stationary optimal policy function, which together with the given law of transition describes the optimal evolution of the states in the form (1.1). "Department of Economics, Cornell University, Ithaca, New York 14853. 181

182

MUKUL MAJUMDAR

Of particular significance are recent results on the "inverse optimal problem under uncertainty" due to Mitra [10] which assert that a very broad class of random systems (1.1) can be so interpreted. To begin with, in order to provide the motivation, I present two examples of deterministic dynamical systems arising in economics. The first is a descriptive growth model that leads to a dynamical system with an increasing law of motion. The second shows how laws of motion belonging to the quadratic family can be generated in dynamic optimization theory. In Section 3 we review some results on random dynamical systems that satisfy a splitting condition, first introduced by Dubins and Freedman [7] in their study of Markov processes. This condition has been recast in more general state spaces (see (3.10)) . The results deal with : (i) The existence, uniqueness and global stability of a steady state (an invariant distribution): a general theorem proved in Bhattacharya and Majumdar [3] is first recalled (Theorem 3.1). The proof relies on a contraction mapp ing argument that yields an estimate of the speed of convergence [see (3.11) and (3.13)]. Corollary 3.1 deals with "split" dynamical systems in which the admissible laws of motion are all monotone.

(ii) Applications of the theoretical results to a few topics : (a) turnpike theorems in the literature on descriptive and optimal growth under uncertainty: when each admissible law of motion is monotone increasing, and satisfies the appropriate Inada-type 'end point' condition, Corollary 3.1 can be applied directly : see Sections 3.2.1-3.2.2 . (b) estimation of the invariant distribution: as noted above, an important implication of the splitting condition is an estimate of the speed of convergence. This estimate is used in Section 3.2.3 to prove a result on JTi-consistency of the sample mean as an estimator of the expect ed long run equilibrium value (Le., the value of the state variable with respect to the invariant distribution) . Next , in Section 4 we briefly turn to qualitative properties of random it erates of quadratic maps : a growing literature has focused on this theme , in view of th e discussion in Section 1.2 and of the privileged status of the quadratic family in understanding complex or chaotic behavior of dynamical systems. 1.1. The Solow model: A dynamical system with an increas-

ing law of motion. Here is a discrete time exposition of Solow's model [11] of economic growth with full employment. There is only one producible commodity which can be either consumed or used as an input along with labor to produce more of itself. When consumed, it simply disappears from the scene. Net output at the "end" of period t, denoted by yt(~ 0) is related to the input of the producible good Kt (called "capit al" ) and

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

183

labor L; employed "at the beginning of" period t according to the following technological rule ("production function") :

(1.2) where Kt ~ 0, L, ~ O. The fraction of output saved (at the end of period t) is a const ant s, so that total saving St in period t is given by

s, = syt, 0 < s < 1.

(1.3)

Equilibrium of saving and investment plans requires

(1.4) where It is the net investment in period t. For simplicity, assume that capital stock does not depreciate over time, so that at the beginning of period t + 1, the capital stock K t+ 1 is given by

(1.5) Suppose that the total supply of labor in period t , denoted by mined completely exogeneously, according to a "nat ural" law:

(1.6) Full employment of the labor force, requires that

(1.7) Hence, from (1.2)-(1.7), we have

K t+ 1 = Kt

+ sF (K t , Lt) .

Assume that F is homogeneous of degree one. We then have

Kt+l . Lt+l Lt+1 Lt A

A

__

Kt Lt A

+ SF (Kt A"

1)

Lt

Writing k« == Kt/ i; we get

(1.8) where

f(k) == F(Kj L, 1). From (1.8)

kt+l = [kt/(l

+ 1])] + [sf(k t)j(l + 1])]

i; is deter-

184

MUKUL MAJUMDAR

or (1.9) where

a(k) == [k/(I

(1.10)

+ 1])] + s[J(k)/(I + 1])].

Equation (1.9) is the fundamental dynamic equation describing the intertemporal behavior of kt when both the full employment condition and the condition of short run savings-investment equilibrium [see (1.4) and (1.7)] are satisfied. We shall refer to (1.9) as the law of motion ofthe Solow model in its reduced form. For any k > 0, the trajectory T(k) from k is given by T(k) == (aj(k))~o where aO(k) == k, a 1(k) == a(k), aj(k) == a(a j- 1(k)) for j ~ 2. Assume that f(O) = 0, f'(k) > 0, f"(k) < 0 for k > 0; and lim f'(k) =

k!O

00,

lim f'(k) = O. Then, using (1.10), we see that a(O) = 0;

kToo

a'(k) = (1 + 1])-1[1 + s!,(k)] > 0 at k > 0; (1.11)

a"(k) = (1 +1])-l sf" (k ) < 0 at k > O.

Also, verify the boundary conditions: lima'(k) (1.12)

k!O

= lim[(I + 1])-1 + (1 + 1])-1 s!,(k)] = 00 . k!O

lima'(k) = (1 + 1])-1 < 1.

kToo

The existence, uniqueness and stability of a steady state k" > 0 of the dynamical system (1.9) can be proved . Here is a summary of the results: PROPOSITION 1.1 . There is a unique k* > 0 such that

k* = a(k*) ; equivalently, (1.13)

k* = [k* /(1

+ 1])] + s[f(k*)/I + 1]].

If k < k", the trajectory T(k) from k is increasing and converges to k*. If k > k", the trajectory T(k) from k is decreasing and converges to k*. 1.2. The quadratic family in dynamic optimization problems. We consider a family of economies indexed by a parameter /-L, where /-LEA = [1,4] . Each economy in this family has the same production function , f : lR+ ---+ lR+ and the same discount factor OE(O, 1). The economies in this family differ in the specification of their return functions, W : lR~ x A ---+ lR+ [depending on the parameter value of /-LEA that is picked].

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

The following assumptions of

f

185

are used:

(F.1) f(O) = O. (F.2) f is non-decreasing, continuous and concave on ~+ . (F.3) There is K > 0, such that f(x) < x for all x > K, and f(x) for all 0 < x < K. A program from an initial input x 2: 0 is a sequence (xt} satisfying

Xo = x , 0:::; Xt :::; f(Xt-d

>x

for t 2: 1.

We interpret Xt as the input in period t, and this leads to the output f(xt} in the subsequent period. The consumption sequence (c.), generated by a program (Xt) is given by

Ct = f(xt-d - Xtk 0) for t 2: 1. It is standard to verify that for any program (Xt) from x 2: 0, we have Xt, == max(K,x) for t 2: O. Given any flEA, the following assumptions on w(·, fL) are used : (W.1) W(C,X,fL) is non-decreasing inc given x, and non-decreasing in x, given c. (W.2) W(C,X,fL) is continuous on ~~ . (W.3) W(C,X,fL) is concave on ~~. In defining "optimality" of a program, we note that the notion has to be economy specific. Since we can keep track of the economies by simply noting its fL value, we find it convenient to rerer to the appropriate notion of optimality as fL-optimality. Given any flEA, a program (Xt) from x 2: 0 is fL-optimal if Ct+l :::; K(x)

00

00

t=O

t=O

I>5 tw(Ct+l , Xt , fL ) 2: Lotw(Ct+l ,Xt,fL) for every program (Xt) from x. Define a set Y

c

~~ by

Y = {(C,X)E~~ : c:::; f(x)}. For much of our discussion of fL-optimal programs, what is crucial is the behavior of w(" fL) on Y (rather than on ~~). We now proceed to assume: (W.4) Given any WA,W(C,X,fL) is strictly increasing and strictly concave

in

C

given x, on the set Y.

Standard arguments ensure that given any flEA, there is a fL-optimal program from every x 2: O. Assumptions (F .2), (W.3), and (W.4) ensure that a fL-optimal program is unique. Since there is a unique fL-optimal program from every x 2: 0, one can define an optimal transition function h : ~+ x A ~ ~+ by

186

MUKUL MAJUMDAR

where (Xt) is the Jl-optimal program from x 2: O. It is easily checked that this definition also implies that for all t 2: 0, we have

Xt+! = hp(xd . Consider now the family of economies, where f,8 and ware numerically specified as follows:

f(x) = (1.14)

(16/3)x - 8x 2 + (16/3)x 4 {

for

XE[O, 0.5]

for x 2: 0.5

1

8 = 0.0025.

The function w is specified in a more involved fashion . To ease writing, denote L == 98, a == 425. Also, denote by I the closed interval [0,1]' and define the function B: I x A --t I by

B(x, Jl) and u : [2 x A (1.15)

--t

~

=

Jlx(1 - x)

for

xel, JlEA

by

u(x, z , Jl) = ax - 0.5Lx 2 + zB(x, Jl) - 0.5z 2 - 8[az - 0.5Lz 2 + 0.5B(z, Jl)2].

Define a set D C [2 by

D= and a function w : D x A

(1.16)

{(C,X)E~+ X --t

~+

L : c::; f(x)}

by

w(c, x , Jl) = u(x,J(x) - c, Jl)

for (c, x)ED and JlEA.

We now extend the definition of w(" Jl) to the domain Y . For (c, x )EY with x> 1 [so that f(x) = 1, and c::; 1], define

(1.17)

w(c,X,Jl) = w(c, I,Jl)'

Finally, we exend the definition of w(' , Jl) to the domain ~~ . For (c,x )E~~ with c> f(x) , define (1.18)

w(c,X,Jl)

=

w(f(x), X,Jl).

It can be checked [see Majumdar and Mitra [9]] that for the above specifications, f satisfies (F.l)-(F.3), and given any JlEA , w(' ,Jl) satisfies (W.l)-

(W.4) . We observe that w(c, x , Jl) 2: w(O, 0, Jl) [by (W.l)] = u(O, f(O) -0, Jl) = 0, for all (c,x )E~~ . Thus w(', Jl) maps from ~~ to ~+ . Also, for all (C,X)E~~, w(c,X,Jl) ::;w(c, I,Jl) = w(l , I,Jl)'

One can verify [see Majumdar and Mitra [9]] the following: PROPOSITION 1.2 . The optimal tmnsition functions for the family of

economics (f,w( ',Jl),8) are given by (1.19)

hp(x) = Jlx(1 - x)

for all xel

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

187

2. Random dynamical systems. Let S be a metric space and S be the Borel a-field of S. Endow r with a a-field E such that the map b, x) - t b(x)) on (I' x S, E 18> S) into (S, S) is measurable. Let Q be a probability measure on (r,E). On some probability space (n,F,p) let ((}n)~=l be a sequence of random functions from r with a common distribution Q. For a given random variable X o (with values in S), independent of the sequence ((}n)~=l' define

Xl == (}1(XO) == (}IXO

(2.1)

Xn+l = (}n+l(X n) == (}n+l(}n ... (}IXO.

(2.2)

We write Xn(x) for the case Xo = x; to simplify notation we write X n = (}IXO for the more general (random) X o. Then X n is a Markov process with a stationary transition probability p(x, dy) given as follows: for x E S, CES,

(}n •••

p(x,C) = Q(b Er: ')'(x) E Cl)·

(2.3)

The stationary transition probability p(x, dy) is said to be weakly continuous or to have the Feller property if for any sequence X n converging to x, the sequence of probability measures p(x n, .) converges weakly to p(x , .). One can show that if r consists of a family of continuous maps, p(x, dy) has the Feller property.

3. Evolution. To study the evolution of the process (2.2), it is convenient to define the map T* [on the space M(S) of all finite signed measures on (S, S)] by (3.1)

T* p,(C) =

is

p(x, C)p,(dx) =

l

p,b-1C)Q(d')'),

P, E

M(S) .

Let P(S) be the set of all probability measures on (S, S). An element 'IT' of P(S) is invariant for p(x, dy) (or for the Markov process X n) if it is a fixed point of T*, i.e.,

(3.2)

'IT'

is invariant

iff

T*'IT' =

'IT' .

Now write p(n)(x , dy) for the n-step transition probability with p(1) = p(x, dy). Then p(n)(x, dy) is the distribution of (}n .....(}IX . Define 'I?" as the n-th iterate of T* :

(3.3)

T*np,

= T*(n-I)(T* f..L)(n ~ 2), T*l = T* , T*0 = Identity.

Then for any C E S, (3.4)

188

MUKUL MAJUMDAR

so that T:" J.L is the distribution of X n when X o has distribution J.L. To express 'I?" in terms of the common distribution Q of the LLd. maps, let I'" denote the usual Cartesian product I' x r x ... x r (n terms), and let Qn be the product probability Q x Q x ... x Q on (I'", s@n) where s @n is the product er-field on I'". Thus Qn is the (joint) distribution of a = (aI, a2,..., an). For 'Y = ("{b 'Y2, ...,'Yn) € I'" let ;Y denote the composition ~

(3.5)

'Y := 'Yn'Yn-I ···'YI·

We suppress the dependence of i' on n for notational simplicity. Then, since T*nJ.L is the distribution of X n = an...aIXO, one has (T*nJ.L)(A) = Prob "", - 1

rv

(X o € a A), where a = anan-I ....al . Therefore, by the independence of and X o,

a

(A€S, J.L€P(S)) .

(3.6)

Finally, we come to the definition of stability. A Markov process X n is stable in distribution if there is a unique invariant probability measure 7f such that Xn(x) converges in distribution to 7f irrespective of the initial state x, Le., if p(n) (x, dy) converges weakly to the same probability measure 7f for all x . 3.1. A general theorem under splitting. Recall that S is the Borel er-field of the state space S . Let A c S , define

(3.7)

d(p" v) := sup IJ.L(A) - v(A)1

(J.L, V€P(S)) .

AeA

1) Consider the following hypothesis (HI) :

(3.8)

(P(S) , d) is a complete metric space;

2) there exists a positive integer N such that for all 'Y e rN , one has

(3.9) 3) there exists 0 > 0 such that 'v'A€A, and with N as in (2), one has

(3.10)

P(a-I(A) = S or o.

THEOREM 3.1. Assume the hypothesis (HI) ' Then there exists a unique invariant probability 7f for the Markov process X n := an...aIXO, where Xo is indepen dent of {an := n ~ I}. Also, one has

(3.11)

d(T*np"

7f)

:S (1 - 0)[nlNI

(p,€P(S))

where T*n is the distribution of X n when X o has distribution J.L, and [n/N] is the integer part of nfN;

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

189

We now state two corollaries of Theorem 3.1 applied to LLd. monotone maps. Corollary 3.1 extends a result of Dubins and Freedman, [1966 Thm. (5.10)], to more general state spaces in ~ and relaxes the requirement of continuity of an ' The set of monotone maps may include both nondecreasing and nonincreasing ones. Let S be a closed subset or an interval of~. Denote by dK(J.l, v) the Kolmogorov distance on P(S) . That is, if Fp. , F; denote the distribution functions (d.f) of J.l and i/, then

dK(J.l, v) = sup 1Fp.(x) - Fv(x)1 (3.12)

X €s

== sup 1Fp.(x) - Fv(x)l , (J.l , VEP(S)). X€!R

It should be noted that convergence in the distance d« on P(S) implies weak convergence in P(S). COROLLARY 3.1. Let S be an interval or a closed subset of~. Suppose an(n ~ 1) is a sequence of i.i.d. monotone maps on S satisfying the splitting condition (H) : (H) There exist XOES, a positive integer Nand a constant fJ > 0 such that

Prob(aNaN-I aIX Prob (aNaN-I aIx

~

~

Xo \fxES) Xo \fxES)

~

fJ

~ fJ

(aJ Then the sequence of distributions T*nJ.l of X n := an...aIXO converges to a probability measure 7l' on S exponentially fast in the Kolmogorov distance dk irrespective of X o. Indeed, (3.13)

where [y] denotes the integer part of y. (bJ tt in (aJ is the unique invariant probability of the Markov process X n . Proofs of Theorem 3.1 and Corollary 3.1 are spelled out in Bhattacharya and Majumdar [3, 4]. 3.2. Applications of splitting. 3.2.1. Stochastic turnpike theorems. We now turn to the problem of economic growth under uncertainty. A complete list of references to the literature - influenced by ther works of Brock and Mirman - is in Majumdar, Mitra and Nyarko [8]. I indicate how the principal results of this literature can be obtained by using Corollary 3.1. Instead of a single law of motion (1.9), we allow for a class of admissible laws with properties suggested by the deterministic Solow model in its reduced form [see (1.10) and (1.11)]. Consider the case where S = ~+; and r = {FI , F2 , .. . , Fe, ..., FN } where the distinct laws of motion F; satisfy:

190

MUKUL MAJUMDAR

(F.1) Pi is strictly increasing, continuous, and there is some r, > 0 such that Fi(x) > x on (0, r i) and Fi(x) < x fOT X > ri. Note that Fi(ri) = Ti for all i = 1, ..., N. Next , assume: (F.2) r, =I rj for i =I j. In other words , the unique positive fixed points r i of distinct laws of motion are all distinct. We choose the indices i = 1,2, ..., N so that

Let Prob (an = Fi) = Pi > O(i ::; i ::; N) . Consider the Markov process {Xn(x)} with the state space (0,00). Ify ~ rI , then Fi(y) ~ Fi(rl) > rl for i = 2, ...N , and F1(TI) = rI, so that Xn(x) ~ TI for all n ~ 0 if x ~ TI. Similarly, if y ::; TN, then F;(y) ::; F;(rN) < TN for i = 1, ..., N - 1 and FN(TN) = r», so that Xn(x) ::; rN for all n ~ 0 if x ::; rn . Hence, if the initial state x is in [rl,rN], then the process {Xn(x) : n ~ O} remains in h, TN] forever. We shall presently see that for a long run analysis we can consider h ,r N] as the effective state space. We shall first indicate that on the state space h , r N] the splitting 2)(x) ::; F1(x) etc. The condition (H) is satisfied. If x ~ rI, F1(x) ::; x, Fi limit of this decreasing sequence F1(n) (x) must be a fixed point of F1 , and therefore must be rl. Similarly, if x ::; rN , then FfJ(x) increases to r N. In particular,

Thus, there must be a positive integer no such that

Fino)(rN) < F~nO)(rl)' This means that if Zo E

[pino) (r N), Fino) (rd]' then

Prob(Xno(x) ::; Zo 'v'xEh,rN]) ~ Probfo., = F 1 for 1 ::; n ::; no) =

P~o

>0

Prob(Xno(x) ~ Zo 'v'xEh, Tn]) ~ Probfo., = FN for 1 ::; n ::; no) = p';Y > O. Hence , considering h, r N] as the state space , and using Theorem 3.1, there is a unique invariant probability 7r with the stability property holding for all initial XEh , rN] ' Now, define m(x) = . min Fi(x) , and fix the initial .=I ,...,N

state XE(O, rr). One can verify that (i) m is continuous; (ii) m is strictly increasing; (iii) m(rI) = rI and m(x) > x for XE(O, TI), and m(x) < x for x > rI. Clearly m(n)(x) increases with n, and m(n)(x) ::; rI. The limit of the sequence m(n)(x) must be a fixed point, and is, therefore "i- Since Fi(rI) > rI for

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

191

i = 2, ..., N , there exists some e > 0 such that Fi(y) > rl (2 :S i :S N) for all YE[rl - c,rI] . Clearly there is some ne such that mn«x) ~ rl - c. If 71 = inf{n ~ 1 : Xn(x) > ri} then it follows that for all k ~ 1

Prcbf r,

> ne + k) :S p~ .

pt

Since goes to zero as k ~ 00 , it follows that 71 is finite almost surely. Also, X T 1 (X) :S rN, since for y:S rI, (i) Fi(y) < Fi(rN) for all i and (ii) Fi(rN) < rn for i = 1,2, ...,N -1 and FN(rN) = rN. (In a single period it is not possible to go from a state less than rl to one larger than rN) . By the strong Markov property, and our earlier result , XT+m( x) converges in distribution to IT as m ~ 00 for all u(O ,rI) . Similarly, one can check that as n ~ 00, Xn( x) converges in distribution to IT for all x > rN. The assumption that I' is finite can be dispensed with if one has additional structures in the model. Here is a simple example.

3.2.2. Uncountable I': an example. Let F: R+ ~ R+ satisfy: (F.1) F is strictly increasing and continuous. We shall keep F fixed. Consider e = [B 1 , B2 ], where 0 < B1 < B2 , and assume the following concavity and "end point" conditions: (F.2) F(x)/x is strictly decreasing in x> 0, 02~~:;") < 1 for some x" > 0

01F~x') > 1 for some x' > O. Since BF(x) is also strictly decreasing x x in x , F .1 and F.2 implies that for each BEe, there is a unique Xo

>0

BF(xo) . BF(x) such that = 1, i.e., BF(xo) = x «. Observe that - - > 1 for Xo x BF(x) 1£ N B' B'" u O < x < Xo , - - < or x > Xo. ow, > impues Xo' > xo" ; x ()'F(xo") ()"F(xo") ()' F(xo') W. --'--:"'-;" > = 1= Xo' > XO". nte Xo" Xo" Xo'

= U : f = BF, BEe}, I, = ()I F I'

12 =

and

()2F.

e

Assume that B is chosen LLd. according to a density function g(B) on which is positive and continuous on e . In our notation, h (xo1) = xo1; 12(xoJ = xo2 • If x ~ XO\l f(x) == BF(x) ~ f(xoJ ~ h(xo 1) = xo1. Hence Xn(x) ~ x01 for all n ~ 0 if x ~ xo1. If x :S X0 2

Hence , if x E [xo 1, xo2] then the process X n (x) remains in [xo 1, xo2 ] forn)(xo = x0 and lim f~n)(X01) = xo . There must ever . Now, lim fi 2 2) 1 n ~oo

n -lo OO

be a positive integer no such that fino) (xo2 ) < f~no) (xo1). Choose some

192

MUKUL MAJUMDAR

no) (X0 ) , fino)(xo ) ). There exist intervals [rh, 0 + m], [0 -m', O 2 1 2] l 2 such that for all 0 € [01,01 + m] and BE[02 - m , O2 ] Zo € ui

Then the splitting condition holds. Now fix x such that 0

< x < XO I , then

Let m be any given positive integer. Since (OIF)(n)(x) ---+ XO I as n ---+ 00, 1 there exists n' == n'(x) such that (OIF)(n)(x) > XO I - - for all n > n' . This m = 1 implies that Xn(x) > XO I - - for all n 2 n'. Therefore, lim infXn(x) 2 m XOI ' We now argue that with probability one, lim infXn(x) > XOI ' For n~~

.

this, note that

. If we choose

n-+oo

~-~

0 = --2- and e

> 0 such that XO

I

-

e>0

then min{OF(y) - fhF(y) : 0 2 (h + 0, y 2 XO I - e} 2 of(xol - e) > O. Write 0' == of(xo l - e) > O. Since with probability one, the LLd. sequence {e(n) : n = 1,2, ...} takes values in [e l + 0, e2 ] infinitely often so that 1 lim infXn(x) > xO I - 1.. + 0'. Choose m so that - < 0'. Then with n ~ oo m m probability one the sequence Xn(x) exceeds XOI' Since x02 = fJn)(xo2 ) 2 X n(xo2 ) 2 Xn(x) for all n, it follows that with probability one, Xn(x) reaches the interval [xo I , xo2 ] and remains in it thereafter. Similarly, one can prove that if x > X0 2 then with probability one, the Markov process Xn( x) will reach [xo l l x o2 ] in finite time and stay in the interval thereafter. REMARK 3.1. The proof of this result holds for any bounded nondegenerate distributioin of e (if E is the support of the distribution of e, define el == inf c < e2 == sup £). 3.2.3. An estimation problem. Consider a Markov chain X n with a unique stationary distribution 71". Some of the celebrated results on the ergodicity and the strong law of large numbers hold for rr-almost every initial condition. However, even with [0,1] as the state space the invariant distribution 71" may be hard to compute explicitly when the laws of motion are allowed to be non-linear, and its support may be difficult to determine or may be a set of zero Lebesgue measure [see Bhattacharya and Rao [2]] . Moreover, in many economic models, the initial condition may be historically given, and there may be little justification in assuming that it belongs to the support of 71". Consider then a random dynamical system with state space [e, dJ (without loss of generality for what follows choose e > 0). Assume r consists of a family of monotone maps from S with S, and the splitting condition (H ) holds. The process starts with a given x. There is, by Corollary 3.1, a unique invariant distribution (a stochastic equilibrium) 71" of the random dynamical system, and (3.13) holds. Suppose

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

we want to estimate the equilibrium mean ~

n-1

L: X j .

Is Y7f'(dy) by

193

sample means

We say that the estimator ~ L: X, is vn-consistent if

j=O

j=O n-1

~ L x, =

(3.14)

j=O

J

Y7f'(dy)

+ Op(n- 1/ 2 )

where Op(n- 1/ 2 ) is a random sequence en such that len/n- 1/ 21 is bounded in probability. Thus, if the estimator is vn-consistent, the fluctuations of the empirical (or sample-) mean around the equilibrium mean is Op(n- 1/ 2 ) . We shall outline the main steps in the verification (3.14) in our context. For any bounded (Borel) measurable f on rc, dj, define the transition operator T as:

Tf(x) = hf(y)p(x,dY) By using the estimate (3.13), one can show that (see Bhattacharya and Majumdar [4]' pp. 217-219) if f(z) = z - I Y7f'(dy) then 00

00

sup

L

x

n=m+1

ITnf(x)1 ~ (d - c)

L

(1 - 8)[n/N]

--->

0

as m

---> 00

n=m+1 00

Hence, 9 = -

L: T N f

[where TO is the identity operator I] is well-defined,

n=O 00

and g, and Tg are bounded functions. Also, (T - 1)g

= - L: T" f + n=l

00

L: TN f = f . Hence, n=O n-1

n-1

L f(Xj) = L(T - I)g(Xj) j=O

j=O n-1

= L((Tg)(Xj) - g(Xj)) j=O n

= L[(Tg)(Xj-d - g(Xj)] + g(Xn )

-

g(Xo)

j=l

By the Markov property and the definition of Tg it follows that

where Fr is the a-field generated by {Xj : 0 ~ j ::s; r} . Hence, (Tg)(Xj_I)g(Xj)(j ~ 1) is a martingale difference sequence, and are uncorrelated,

194

MUKUL MAJUMDAR

so that n

k

(3.15)

EL [(Tg(Xj-d - g(X j ))]2 = LE((Tg)(Xj-d - g(X j ))2. j=1

j=1

Given the boundedness of 9 and Tg, the right side is bounded by n .a for some constant a . It follows that for all n where r/ is a constant that does not depend on X o. Thus, n-1

E(~ ~ x, J=O

J

Y1r(dy))2 ::; r/ln

which implies, n-1

~ L x, = j=O

J

Y1r(dy)

+ Op(n- 1/ 2).

For other examples of vn-consistent estimation, see Bhattacharya and Majumdar [4] . 4. Iterates of quadratic maps. On the state space S = (0,1) consider the Markov process defined recursively by

X n +1 = a g n + 1 X n (n = 0, 1,2, ...)

(4.1)

where {en : n ~ 1} is a sequence of LLd. random variables with values in (0,4) and , for each value (}£(0,4), a() is the quadratic function (on S):

(4.2)

aox

== ao(x) = Ox(l

- x)

o<x 0, there exists a compact K g C S such that J.Ln(K g) ~ 1 - e for all n ~ 1. THEOREM 4 .1. Assume that the distribution of e1 has a nonzero absolutely continuous component (w.r .t . Lebesgue measure on (0,4) whose density is bounded away from zero on some nondegenerate interval in (1,4). If, in addition, E~=1P(n)(x ,dy) : N ~ I} is tight on S = (0,1) for some x, then (i) {Xn : n ~ O} is Hams recurrent and has a un ique invariant

{*"

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

195

-k

probability 1T" and (ii) 'L,:=1 p(n)(x, dy) converges to 1T" in total variation distance, for every x, as n ~ 00 . COROLLARY 4.1. If Cl has a nonzero density component which is bounded away from zero on some nondegenerate interval contained in (1,4) and if, in addition,

(4.3)

E log

Cl

>

°

and E Ilog(4 -

cd/ < 00,

then {Xn : n 2: O} has a unique invariant probability 1T" on S = (0,1) and (l/N) 'L,:=1 p(n)(x, dy) ~ 1T" in total variation distance, for every X€(o, 1). REMARK 4.1. Under the hypothesis of Theorem 4 .1, the Markov process is not in general aperiodic. For example, one may take the distribution of En to be concentrated in an interval such that for every () in this interval no has a stable periodic orbit of period m > 1. One may find an interval of this kind so that the process is irreducible and cyclical of period m . If Cn has a density component bounded away from zero on a nondegenerate interval B containing a stable fixed point, i.e., Bn(O, 3) -I cP, then the process is aperiodic and p(n)(x, .) converges in total variation distance to a unique invariant 1T". Assumptions of this kind have been used by Bhattacharya and Rao [2] and Dai [6] .

REFERENCES [1] BHATTACHARYA RN. and E.C . WAYMIRE: Stochastic Processes with Applications, John WHey, New York (1990).

[2] BHATTACHARYA R.N . and B.V. RAO: Random Iterations of Two Quadratic Maps .

[3] [4] [5J [6] [7]

[8] [9] [10] [11]

In Stochastic Processes (eds. S. Cambanis, J.K Ghosh, RL. Karandikar, and P.K Sen), Springer Verlag, New York (1993), pp. 13-21. BHATTACHARYA R.N. and M. MAJUMDAR: On a Theorem of Dubins and Freedman , Journal of Theoretical Probability, 12 (1999), pp. 1067-1087. BHATTACHARYA RN . and M. MAJUMDAR: On a Class of Stable Random Dynamical Systems: Theory and Applications, Journal of Economic Theory, 96 (2001) , pp. 208-229. BHATTACHARYA R.N. and M. MAJUMDAR: Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes, CAE Working Paper 02-03, Cornell University, (2002) [to appear in Annals of Applied Probability.] DAI J .J .: A Result Regarding Convergence of Random Logistic Maps , Statistics and Probability Letters, 47 (2000), pp. 11-14. DUBINS L.E. and D. FREEDMAN (1966): Invariant Probabilities for Certain Markov Processes, Annals of Mathematical Statistics, 37, pp. 837-858. MAJUMDAR M., MITRA T ., and Y. NYARKo: Dynamic Optimization under Uncertainty; Non-convex Feasible Set . In Joan Robinson and Modern Economic Theory (ed. C.R. Feiwel), MacMillan, London (1989), pp. 545-590. MAJUMDAR M. and T . MITRA: Robust Ergodic Chaos in Discounted Dynamic Optimization Models, Economic Theory, 4 (1994), pp. 677-688. MITRA K : On Capital Accumulation Paths in a Neoclassical Stochastic Growth Model, Economic Theory, 11 (1998), pp. 457-464. SOLOW R.M.: A Contribution of the Theory of Economic Growth, Quarterly Journal of Economics, 70 (1956), pp. 65-94.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION OF THE NAVIER-STOKES EQUATIONS M. ROMITO· Abstract . We explain some ideas contained in some recent pa pe rs, concerni ng the st at ist ical long time behaviour of the spectral approximation of the Navier-Stokes equations, driven by a highly degenerate white noise forcing. The analysis highlights th at the ergodicity of the stochastic system is obtained by a geometric cascad e. Such a cascade can be int erp reted as the mathematical counterpart of th e energy cascade , a well-known phenomenon in turbulence. In th e second pa rt of the paper, we analyse the results of some numerical simulations. Such simulations give a hint on the beh aviour of the system in the case where th e whit e noise forcing fails the assumptions of the main theorem . Key words. Navier-Stokes equations, ergodicity, hypo-ellipticity, cont rollability. AMS(MOS) subject classifications. 76M35, 76F55 .

Primary 76D05;

Secondary 35Q30,

1. Introduction. The article aims to explain a few ideas related to the analysis of the statistical long time behaviour of a model of isotropic homogeneous turbulence, that have been presented in some recent papers. We will refer mainly to Romito [33], of whom we will present the approach, but also to E and Mattingly [11], who originally solved the problem in the 2D setting. Moreover, we will also refer to Agrachev and Sarychev [1], for the section on the irreducibility property (Section 5). The model equations are the 3D Navier-Stokes equations in a cube , with periodic boundary conditions, driven by a white noise random force. Due to the well-known analytical difficulties related to the problem , we will examine the spectral approximation of the equation.' Namely, we will consider the Fourier series of the solution up to a fixed, arbitrary but finite, threshold. The approach has anyway a qualitative interest for the general problem. Indeed , according to Kolmogorov's theory of turbulence (Kolmogorov [21]) , the cascade of energy, responsible of the transport of the energy through the scales, is mainly effective only in the inertial range , and it becomes negligible at smaller scales, where the dissipation ends up to be the only relevant phenomenon. Hence the long-time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field. In some sense, if the ultraviolet cut-off threshold is sufficiently large , in order to capture all the important modes, the corresponding invariant measure, • Dipartimento di Matematica, U niversita di Firenze, Viale Morgagni 67/a, 50134 Fi renze , Italia (romito~ath.unifi.it). 1 In a recent paper by Da Prato and Debussche [6] , a proof of ergodicity is given for the infinite dimensional stochastic PDE. We just remark that, among other assumptions, the noise acts on all Fouri er modes . 197

198

M. ROMITO

E(k)

[ne~ange

~

Dissipation

----, ~_-~~

r

~

injection of energy by large-scale

noise forcing

forcing

•• •• • ~~'lY ca.lde

. . .... .. . ~ Ikl

FIG . 1.

Th e energy spectrum .

which is the mathematical object representing the asymptotic behaviour, should give t he real picture of the dynamics of the fluid (the analysis we have described t urns out to be true in a purely deterministic setting, as it has been proved by Constantin, Foias and Temam [4]). A very rough picture of the Kolmogorov spectru m is given in Figure 1. Energy is injected by a deterministic forcing in th e system at the largest scales. The rough forcing injects energy in th e system at scales much smaller but larger than the dissipation scale. One should imagine th at the deterministic components correspond to forcings like gravity, while th e stochastic forcing corresponds to small scales forcings like mechanical vibrations, or temperature fluctuations , etc. In our model the noise injects energy at the level of the largest lengthscales, since we neglect the deterministic forcing. It means that the noise acts only on a few of the total number of modes taken into account, up to the spectral threshold, for the description of the syst em. In principle , one could expect that not all of the component of the system are dumped by the noise. The main point of the paper is that indeed all components of the system are, either directly or indirectly, dumped by th e noise. We call this phenomenon the geometric cascade, because we will trace the noise dumping, component by component, from the forced Fourier modes to all the ot hers (this is the heurist ic interpretation of Lemma 4.1). E and Mattingly [11] showed, in t he 2D case, that this mechanism proves effective to deduce that the transition probabi lity densities of the Markov process , solut ion to the model equation, are regular. In Romito [33], it is proved (see also Theorem 5.1) that the geometric cascade is effective also in showing that the process is steered by the noise to any part of the state space , with positive probability; such a property is known as irreducibility. A toy model, that is, a very simple SDE, is given in Section 6, to show how the concepts and methods explained in the article apply practically. An open question related to the analysis above concerns the sit uation in the non hypo-elliptic case. Assume that the noise does not excite

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 199

(directly or indirectly) all the modes. Heuristically, the system should experience a decoupling between the forced modes and the modes not excited by the noise. On one hand, the modes forced should behave as in the hypoelliptic situation, hence converging to a statistical steady state, which will be necessarily concentrated on a hyper-plane. On the other hand, the nonforced modes should behave as in the case of absence of noise, converging to zero. In the last part of the paper (see Section 7), we try to obtain a hint on this heuristic picture from the analysis of some numerical simulations. The aim is to verify that some modes, the ones that supposedly are not forced, either directly or indirectly, by the noise, converge to zero. Our analysis aims to give just a qualitative answer to our conjecture, without any pretension of a precise quantitative analysis. 2. The Navier-Stokes equations in the Galerkin approximation. Consider the stochastic Navier-Stokes equations (2.1)

du, = (v.6.u - (u· V')u - V'p) dt + dB t , divu = 0

in the domain x E [0, 27f]d, where d = 2,3, with periodic boundary conditions. Following the classic approach, that dates back to the pioneering works of Leray (see [23]' [24], [25]), we will consider the equations projected onto the space of divergence-free vector fields. Such a projection reads easily in our case, namely of periodic boundary condit ions, since it is diagonal with respect to the Fourier basis (e1k.x)kE71d . Indeed , the Helmholtz-Leray projection is defined as (2.2)

where &k is the projection in IR d onto the orthogonal space to the vector k and it is given by d= 2, d= 3,

where kl. = (-k 2 , kI) is the vector perpendicular to k. If we write the solution u(t,x) in the Fourier basis, namely

u(t,x) =

L

uk(t)e 1k'x ,

kE71d

the deterministic dynamics is given by (2.3) 9"(v.6.u - (u · 'V )u - 'Vp)

=-

L (v 1k I + L 2

kE71 d

i

(k · Uh) 9"k(ul))eik'X,

h+l=k

200

M. ROMITO

the pressure term being disappeared because of the projection onto the divergence free vector space . We will assume also that Uo == 0, that means that we observe the system in a reference frame centred on the centre of mass of the fluid. There is no loss of generality, since the forcing has zero average, hence the centre of mass moves with constant velocity. 2.1. The noisy forcing term. As it has been remarked in the introduction, the forcing term is a white noise, the time derivative of a Brownian motion. It is the simplest, yet quite significant, model for rough and unpredictable forcing. We will assume that the covariance of the noise is diagonal with respect to the Fourier basis , so that

e, = L

(2.4)

qk . j3fei k .x

kEZ d

where (j3fh~o, k E tl d are independent d-dimensional Brownian motions and the qk are d x d matrices such that q~ . k = O. The last condition means that the Brownian motion has trajectories which are divergence free. We will assume that most of the qk are zero, so that we define the set of forced modes as

N

(2.5)

= { k E tl

d

Iqk ~ 0 } .

For the sake of simplicity, we will assume that for each kEN, the matrix qk has rank d - 1. 2.2. The Galerkin approximation. The Navier-Stokes equations, specially in three dimensions, are difficult to be managed. Hence, we will consider a spectral approximation of the equation. Let N E IN be an integer, and consider the truncated representation UN (t, x)

=

L

Uk(t)e

ik x .

Ikloo:$N

where JkJ oo = max(lkd, · . . , Ikdl) . The equations for the truncated system can be derived by projecting the representation for the deterministic dynamics (2.3), so that at the end 7Lk=-[vlkI2+i (2.6)

:L (k 'Uh).9'k(uI)]dt+qk dj3f , h+l=k

Uk '

k

= 0,

with Ikloo ::; N , and the sum is extended to all hand 1 smaller than the threshold N.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 201

3. The ergodic properties of the process. The system of stochastic differential equations (2.6) defines a Markov process on the state space

where D = 2d[(2N + l)d -1] and we understand that Uk = rk generator of the Markov process is given by (3.2)

2" = F

+~

L (Xk:

2

+ iSk. The

+ Xk2 )

kEN

where F = Llkl~N Fk: a~k +Fka~k is the vector field given by the deterministic dynamics and Xk: , Xk are the vector fields given by the noise forcing. Finally, we will denote by (Pd t20 the Markov semigroup associated to the Markov process U = (uk)lkl~N' The main result we want to show says that, under suitable assumptions on the number of forced modes (see Definition 4.1), the Markov process is ergodic: it has a unique invariant measure and the distributions of th e process converge exponentially to the unique invariant measure. The main tool is the following theorem. THEOREM 3.1 (Doob [8]) . Given a Marko v semigroup (Pdt2 0 and an invariant measure f.l for the semigroup, if there is a time to such that the probability measures Pt(u, ·) are mutually equivalent for all t ;::: to and u E U, then f.l is the unique invariant measure, it is equivalent to all Pt(u, ') , for t ;::: to, and it is strongly mixing. The assumptions of Doob 's theorem can be fulfilled under the assumptions of the following theorem, due to Has'minskif. THEOREM 3.2 (Has'minskii [16]). If the Markov semigroup (Pdt2 0 is strongly Feller and irreducible, then all probability m easures Pt (u , .) are mutually equivalent. The previous result is the key point for the proof of the main theorem of the paper. The claim on the support of the invariant measure is a direct consequence of the irreducibility (see Stroock and Varadhan [36]) . THEOREM 3.3. A ssume that the set of indices N corresponding to the forced modes is a determining set of indices (as defined in Definition 4.1) . Then the Markov process solution to the stochastic differential system (2.6) is ergodic. Moreover, the unique invariant measure is fully supported on the state space U . We will see in the next section the precise definition of a determining set of indices. In the two dimensional case, some examples are the set of modes {(I , 0), (1, I)} (see E and Mattingly [11]) or the set of modes

{(M

+ 1,0), (M, 0), (0, K + 1) , (0 , K)} ,

for suitable integers M and K (see Mattingly and Pardoux [27]) . In the three dimensional case, the set of modes

202

M. ROMITO

(1,0,0),(0,1 ,0),(0,0,1) is given in Romito [33] . REMARK 3.1. It can be proved that the convergence stated in the previous theorem is indeed exponentially fast (see Romito [33J) . Such a claim follows by applying Theorem 6.1 of Meyn and Tweedie [31]. The key point is to show that there is a Lyapunov function, that is, a positive function V, defined on the state space, such that

if V

:s; -cV + d

°

for some constants c > and d. A Lyapunov junction for the problem under examination is the kinetic energy 2:k IUkI 2 . 4. Regularity of the transition probabilities. The first main point of the proof of Theorem 3.3 is to prove that the transition semigroup has the strong Feller property. It means that for each bounded measurable function ep defined on the state space U, the function Ptep is bounded continuous. First assume that the noise excites all modes. In our notations, K contains all indices Ikl oo :s; N or, equivalently, the covariance f5j of the noise is a non-singular matrix. The Bismut-Elworthy-Li formula (see Bismut [2] and Elworthy and Li [10]) gives

(D(Pt'P)(UO),V)

=

1 -E['P(u(t,uo» t

it 0

(..P-1Du(s,uo)v,dB s ) ] ,

where u(t, uo) is the process , solution to (2.6), starting at Uo (one can see also Cerrai [3]). It turns out that P; has a regularising effect, in particular it implies the strong Feller property. In the general case, where K contains just a few modes.f one needs to have better information on the non-linear dynamics, which is ultimately responsible of the spreading of the noise forcing to all modes. Here we need the probabilistic version of Hormander celebrated theorem. THEOREM 4.1 (Malliavin [26], Stroock [35]). Consider the Stratonovich SDE in IRffi n

«x, =

F(Xt ) dt +

L Fi(X

t)

0

dW;,

i=l

Xo=x, 2 As it has been proved in E and Mattingly [11], for the spectral approximation of 2D Navier-Stokes equations, two modes, namely (1,0) and (1,1) are enough. In the 3D case, as it can be found in Romito [33], the three modes (1,0,0), (0,1,0), (0,0,1) are sufficient .

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 203

where the vector fields F , F i , . . . , Fn satisfy suitable boundedness conditions, and assume that Hormander's condition holds: the Lie algebra generated by the vector fields F , F i , ... , F n , once evaluat ed at x , spans IR m . Then, for all t > 0, the random variable X, has an absolutely continuous distribution with a sm ooth COO density. Such a result has been established by Malliavin [26], and in its full strength by Stroock [34], [35]. A simplified argument is given in Norris [32]. In order to analyse how the non-linearity spreads the effect of the noise, we notice that the state space U, defined in (3.1), can be written as a direct sum of linear subspaces, namely,

where each point of Uk has all coordinates equal to zero , but the coordinates Uk corresponding to k, and k . Uk = O. In the same way we can define the Lie algebra U of vector fields 9 = L Gk a~k such that k . Gk = O. Moreover, we will consider the subspaces Uk of U of constant vector fields whose coefficients are in Uk . Now the main technical lemma follows. LEMMA 4.1. Let m and n be indices, such that Iml oo , [n] 00 ~ N , and consider two vector fields V E Urn , WE Un' Then (i) if m 1\ n = 0, then [[F,V], W] = 0; (ii) [[F,V] , W] = ~[[F, V + W], V + W] . Moreover, if m 1\ n i= 0, [m], i= Inl2 and [m + nl oo ~ N ,3 then the Li e algebra Urn +n is contained in the Lie algebra generated by the vecto r fields [[F, V], W], with V E Urn and W E Un . The last part of the lemma is the main point. We can interpret it in the following way: if the modes m and n are forced by the noise, the mod es corresponding to m + n are in the Lie algebra generated by the vector fields F and Xk. It means that the (m + n)-component of the system of SDEs is forced indirectly by the noise. Those components, again, can transmit the noise force to other components, and so on. If the set N of directly forced mod es generates the set of all modes, the equ ations behave as if some kind of forcing happens at each component. This is truly a geometric cascade for the Fourier modes of the Navier-Stokes equations. In order to substantiate the idea, we introduce the set A(N) with the following rules:

Ne A(N) , if k E A(N) , -k E A(N) , [> if m , nE A(N), m 1\ n i= 0, Iml2 i= Inl2 and [m + nl oo ~ N , then m+n E A(N). DEFINITION 4.1 (Determining sets of indices) . We say that a set [> [>

le c {k Ilkl oo ~ N} is a determining set of indices for the threshold N if the set A(N) contains all indices Ikloo ~ N . 3Here

Irnl2

is the Euclidean norm of the vector

rn,

namely Irnl ~ =

L rn;.

204

M. ROMITO

Consequently, we have the following result. THEOREM 4.2 . If the set N is a determining set of indices, then the Markov semigroup associated to the SDE (2.6) has the strong Feller property. 5. Irreducibility and the control problem. A Markov process is said to be irreducible if for each time T and each open set A of the state space, the probability that the process is in A at time T is positive. The irreducibility property is implied (see for example Stroock and Varadhan [36]) by a controllability property of the control problem which is obtained from the original system (2.6) by replacing the noise with some controls . In our case, we obtain the system

where Pk is a quadratic polynomial in the Uk and Vk are the controls. We say that the system is controllable if for each time T and each pair of points 1 2 U , u , there exists a control v such that the corresponding solution of the control problem starts at time 0 in u 1 and stops at time T in u 2 • Intuitively, the irreducibility is related to the control problem because, if the system is controllable, then this means that there are realizations of the noise such that the Markov process solution of the SDE can move from u 1 to u 2 . A necessary condition for such a property is indeed the hypo-ellipticity (see Jurdjevic [18] for generalities on the geometric control theory) . Heuristically, if at each point of the state space , our system is allowed to follow any direction in any way (only the first assumption is true in the hypoellipticity proof) , then the system is controllable. Roughly speaking, the Lie brackets that have been evaluated in the previous section represents all the possible directions that the process can follows . Indeed, the drift of the equations would lead the process to follow the field :F of the deterministic dynamics . The kicks of the noise allow the process to follow other directions, which are a combination of the original dynamics :F and the directions Xk of the kicks. If the polynomial P has odd degree, the hypo-ellipticity condition is also sufficient (see Jurdjevic and Kupka [19]), since by changing u ----+ -u , the system can always follows both ways of each direction. The problem is, the Navier-Stokes system gives rise to a polynomial of even degree. Such quadratic terms in some sense give a preferential direction to be followed by the system. In order to give a clearer idea of such phenomena, one can see the following section , where an extremely simple SDE is presented, which gives a hypo-elliptic diffusion, and such that the upwards direction is preferential. The key property that the spectral approximation of the Navier-Stokes equations enjoys is the absence of such dangerous quadratic terms, and it is a consequence of properties (i) and (ii) of Lemma 4.1.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 205

/

" " , , ..,. ""

""

."

... . . ".

tI ; ~""., of If'

,

.,., ,,

of".,"

, , • , ,. ,,, t

..

, ,

..

r ,.

"

..

", , ,. ,,, , ,.

..

" "",,,.,,,,, .,,,,,,, ,,,,. ...... i.," , "".,,, , , , ,, ,, , t, ,,.,. "\. .. ... , "''', ,''' " , f' .. ::~ :::::::;:;: :: ~ ~ :: : ~ ~ ~ ~ ~ ~ : ~ : ...

... ;

...

·"

jj

· ,·"."

I I

11 '

·

'

..

,.

'

,.

\\

,

... ", " " ,,,,,., " " 1"""""'4",,,,, ,·,1" """, •• , ,"' ".. 1"""".,,· ."." ••" .,, ". ' """ . . "'1."" ", ", ,."

..

~ • I I ' 'r ••••• ••• I' ~ . t j ".,. ,' r

........

• • • • 14"

.,."""""""

J "'''''i''"ij .,..,.",,,,,,,, " i I. ' iI j

j

j

j

j j

j

t

.,..,."'''''·,, · ' , ; ,

Jjji~~~~~~~~~~~~~~~~~~~~~~Il~~ FIG. 2. Th e directions of the deterministic dynamics F. THEOREM 5.1. If the set N is a determining set of indices, then the Markov semigroup associated to the SOE (2.6) is irreducible. REMARK 5.1. In [1], Agrachev and Sarychev prove something more for the 2D case. Namely, they prove controllability in observed projection, that is, given a spectral threshold N , an initial point uO E L2([0, 27r]2) , with divergence equal to zero, and a time T > 0, there is a control that steers the solution of the infinite dimensional Navier-Stokes equations from uO to a point with assigned finite dimensional projection . As a consequence, approximate controllability holds for the infinite dimensional stochastic POE.

6. A toy model. As an example of how such things go on, we shall examine the following toy model, where we will see how the ideas of th e previous section can work. Consider th e system of stochastic differential equations dx; = - Xt + y; - XtYt + dB t , { dYt = -Yt + x; - XtYt,

(6.1)

where B, is a one dimensional Brownian motion . We will see that the diffusion given by the SDE is hypo-elliptic (so that strong Feller prop erty holds), but the syst em is not controllable. Hence the controllability results on the Navier-Stokes dynamics depend heavily on the geometrical structure of the non-linearity. We can have a rough idea of the deterministic dynamics from Figure 2. It is easy to see that the system has a global in time solution (just apply + Y;). Define the following vector fields Ito formula to

x;

2

a

a

2

F= (-X+Y - xY)ax +(-y+x -xY)ay

and

x=

a ax '

where F is the vector fields given by the deterministic dynamics (the drift) and X is the one given by the direction of the noise. Since

UT,Xl, x]

a

= 2 ay'

206

M. ROMITO

it follows that that the Lie algebra generated by F and X has full rank, once it is evaluated at each point of the state space IR 2 . In other words, Hormander's condition holds. On the other hand, the system is not globally controllable. Indeed, if one consider the solution at the starting point (xo, Yo) = (0,0), it is easy to solve the equation for Yt (it is a linear equation with random coefficients), thus obtaining

which is almost surely non negative. In view of the control theory issues explained in the previous section, one can explain the above phenomenon in the following way: since the direction Ox is the one forced by the noise, both Ox and -Ox are directions that the system is allowed to follow, and Oy = [[F, X], X] as well. On the contrary, the vector field -Oy (which is in the Lie algebra, thus ensuring the regularity of the transition probability densities) is a forbidden direction for the system. REMARK 6.1. The Markov process solution of the system (6.1) is anyway ergodic. Hence, controllability arguments are just sufficient for the proof of uniqueness of the invariant measure. One can use the same argument that is used in E and Mattingly [l l], namely it is sufficient to show that the neighbours of the origin, which is the stable point aitracior of the dynamics, are recurrent for the process. 7. Some numerical results. In this last section we aim to investigate what happens in the case where the hypo-ellipticity condition does not hold. The Navier-Stokes equations, without any external forcing, converge to 0 as the time goes to infinity. If we have an external forcing, namely a white noise forcing, which excites only a few number of modes, it may happen that the other modes are indirectly forced by means of the nonlinearity. This is what essentially happens under the assumptions of the previous sections. Assume now that the forced modes do not constitute a determining set of indices (as defined in Definition 4.1). Intuitively, we believe that in this case the system experiences a sort of decoupling, where the (directly and indirectly) forced modes behave as in the former case, while the others behave as if there were no forcing, hence collapsing to zero. If this picture turns out to be true, it follows that the invariant measure concentrates on a hyper-plane of the state space . 7.1. The numerical simulation. In order to lower the number of equations of the system to be numerically solved, we have chosen to examine the approximation of the 2D Navier-Stokes equations in the simplified

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 207

vorticity" formulation that has been introduced in E and Mattingly [11] . Namely, we consider the following system of SDEs 2

dCk = - vJk l ck +

h.L · I ""' h.L · I k W(ShSl - ChCi) + L W (ChCl + shsd + O"k d{3~' , h+l=k h-l=k ""'

L

2 ""' h.L · I h.L · I k dsk=-vlkl Sk- L W(rhS1+S hCi)+ ""' L W(ShCi-ChSd+'Ykd.B:' , h+l=k h-cleek

where the vorticity is given by

~(t, x) =

L Ck(t) cos(k· x) + Sk(t) sin(k · x) k

and the sums are extended to all pairs k = (k l , k 2 ) of positive integers, each one being less than the spectral threshold N . In order to solve numerically the system of stochastic equation, we have used the backward Euler method (a standard reference for the approximation of stochastic equations is Kloeden and Platen [20]) . In few words , say that the equation to be approximated is y' = f (y) + a dWt , where y is a vector. Then at each time-step,

where b. Wn are independent centred Gaussian random variables, having variance equal to the size of the time step. The implicit equation in the unknown Yn+! is solved using the Newton's method. The problem is stiff, due mainly to the linear part (there are the large negative coefficients -vlkI 2 ) . In order to obtain relevant results we have taken a small viscosity and a comparable intensity of the noise. Different realizations of the simulation have been run , with different initial conditions. Each initial condition has been chosen with a centred st andard normal distribution. Notice th at, even though we know from the theoretical analysis that the Markov process converges for each init ial condition, the same claim can be incorrect if the size of the initial condition is too large (see for example Higham, Mattingly and Stuart [17]) . All the numerical experiments in the following have been run with the following value of the parameters: t> t> t>

Grashof number'' Gr = 50, spectral threshold N = 8, tim e increment b.t = 0.05.

4The vorticity, in the 2D case , is defined as the scalar field given by € = curl u , wher e u is the velocity field. In the Fourier coord inates, if (Uk)kE71 d are the Fourier coefficients of the velocity field, the coefficients of the vorticity are given by €k = ik X Uk . 5The Grashof number is defined (in the 2D case) as the ratio between the intensity of the force and the square of the viscosity (see for example Foias , Manley, Rosa and Tem am [13]).

208

M. ROMITO

'"

17~

'os

'"

'2 5

~ tl t

t

tP

t 11 11

J I

JlJ5

175

t

10!

FIG. 3. The errors graphic of the (1,1) are forced.

Ck

coefficients. The three modes (1,0) , (0,1) and

The first data obtained have been discarded, they should correspond to the transient regime and they don't give any useful information concerning the invariant measure. The time increment is rather small , because of the stiffness of the problem. REMARK 7.1. For the implementation of the numerical method (especially for the random number generator), we have heavily used some libraries from the GNU Scientific Library [37]. 7.2. Conclusions. According to the results of the numerical simulations, the picture we have figured out , seems to be reasonable. The first case we examine is the one where the three modes (1,0), (0,1) and (1,1) are excited. According to the theoretical result (See Theorem 3.3), the invariant measure is fully supported. The empirical averages of the process hence converge to the expectations with respect to the invariant measure. We see that all the modes are excited (see Figures 3 and 4), even though with different intensity (Notice that the size of the standard deviations has been rescaled by a factor .3 for the sake of readability). In the second case, only the mode (1,0) is excited. Most of the energy then stays in the forced mode and in a few others (see Figures 5 and 6). Notice that in this case the dissipation scale is larger than in the previous case, so that the system dissipates the energy faster than in the previous case. Figure 7 summarises and compare the two cases. It represents the intensity of each J c~ + component in the two cases examined, depending on the position in the Fourier space . The largest values correspond to the modes being excited by the noise.

sa

A GEOMETRIC CASCADE FOR THE SP ECT RAL APPROXIMATIO N... 209

t It. tt 1

F IG . 4. Th e errors gmphic of the Sk coefficients. Th e three modes (1, 0), (0, 1) an d (1, 1) are f orced.

FIG. 5 . Th e error s gm phic of the

Ck

coeffi cien ts. Only the mode (1, 0) is f orced.

M. ROMITO

210

FIG. 6. The errors graphic of the

Sk

coeffi cients. Only th e mode (1,0) is fo rced.

FIG. 7. The values of the means for Jc~ + s~. On th e left , only the mode (1, 0 ) is forced. On the right, the three mod es (1,0), (0,1) and (1, 1) are forced.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION... 211

7.2. It has to be remarked that the numerical simulations we have run , are of little interest from the point of view of turbulence , and their only aim is to give a support to our conjecture. Indeed, the spectral threshold is too small and the main parameter, the Grashof number, corresponds to a not so turbulent regime. REMARK

Acknowledgements. The author wishes to thank D. Blomker, P. Constantin, F . Flandoli, M. Hairer, J. Mattingly, A. Sarychev for the helpful conversations on the subject of this paper . The author wishes also to thank the Institute for Matbematics and its Applications, at the University of Minnesota, for its warm hospitality during the Summer Program on Probability and Partial Differential Equations in Modern Applied Matbematics. REFERENCES [1] A.A. AGRACHEV AND A.V . SARYCHEV, Navier-Stokes equation controlled by degenerate forcing: controllability in finite-dimensional projections, to appear on J . Math. Fluid Mech. [2] J.M . BISMUT, Martingales , the Malliavin calculus and hypo-ellipticity under general Hormander 's conditions, Z. Wahrsch. Verw. Gebiete 56 (1981) , 469-505 [3] S . CERRAI, Second Order PDE's in Finite and Infinite Dimension, Lecture Notes in Math. 1762, Springer Verlag, 2001. [4] P . CONSTANTIN , C . FOlAS, AND R . TEMAM , On the large time Galerkin approximation of the Navier-Stoke s equations, SIAM J. Numer. Anal. 21 (1984) , no. 4, 615--634. [5J P . CONSTANTlN, Geometric statistics in turbulence, SIAM Re view 36(1) (1994) , 73-98. [6] G . DA PRATO AND A. DEBUSSCHE, Ergodicity for the 3D stochastic Navier-Stokes equations, preprint (2003) . [7] G . DA PRATO AND J . ZABCZYK, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Not e Series , 229. Cambridge Unive rsity Press, Cambridge, 1996. [8] J .L. DOOB, Asymptotic properties of Markov transition probabilities, Trans. Amer. Math. Soc. 63 (1948) , 393-421. [9] J .P . ECKMANN AND M. HAIRER, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Comm. Math. Phys. 219 (2001) , no . 3, 523-565. [10] D.K. ELWORTHY AND X.M . LI, Formulae for the derivatives of heat semigroups, J . Func. Anal. 125 (1994) , 252-286. [11] W . E AND J .C . MATTlNGLY, Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation , Comm. Pure Appl. Math. 54 (2001), no . 11, 1386-1402. [12] F . FLANDOLI, Irreducibility of the 3-D stochastic Navier-Stokes equation, J. Funct. Anal. 149 (1997), no. 1, 160-177. [13] C . FOlAS, O . MANLEY, R . Rosx , AND R. TEMAM, Navier-Stokes equation s and turbulence, Cambridge University Press, Cambridge, 2001. [14] G. GALLAVOTTI, Foundations of fluid dynamics, translated from the Italian. Texts and Monographs in Physics. Springer Verlag, Berlin, 2002. [15] M . HAIRER, Exponential mixing for a stochastic partial differential equation driven by degenerate noise, Nonlinearity 15 (2002), no. 2, 271-279. [161 R .Z. HAS'MINSKif, Ergodic properties of recurrent diffusion processes and stabilization of the solutions of the Cauchy problem for parabolic equations, Theory Probab. Appl. 5 (1960) ,179-196.

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[17] D.J . HIGHAM, J .C. MATTINGLY, AND A. M. STUART, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185-232. [18] V. JURDJEVIC, Geometric control theory, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, Cambridge, 1997. [19] V. JURDJEVIC AND I. KUPKA, Polynomial control systems Math. Ann . 272 (1985), no. 3, 361-368. [20] P .E. KLOEDEN AND E . PLATEN, Numerical solution of stochastic differential equations , Applications of Mathematics 23, Springer Verlag, Berlin, 1992. [21] A.N . KOLMOGOROV, Local structure of turbulence in an incompressible fluid at a very high Reynolds number, Dokl. Akad. Nauk SSSR 30 (1941), 299-302 ; English transls., C.R. (Dokl.) Acad . ScL URSS 30 (1941), 301-305, and Proc. Ray. Soc. London Ser . A 434 (1991), 9-13 . [22] L.D . LANDAU AND E .M . LIFSHITZ, Fluid mechanics, Course of Theoretical Physics, VoI. 6 . Pergamon Press, London , Paris, Frankfurt, 1959. [23] J. LERAY, Etude de diverses equations integrales non lineaires et de quelques probiemes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82 . [24] J . LERAY, Essei sur les mouvements plans d'un liquide visqueux que limitent des parois, J . Math. Pures Appl. 13 (1934), 331-418. [25] J . LERAY, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248 . [26] P . MALLlAVIN Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res . Inst . Math. ScL, Kyoto Univ., Kyoto (1976), 195-263 , WHey, New York Chichester Brisbane, 1978. [27J J .C. MATTINGLY, E. PARDOUX, Malliavin calculus and the randomly forced NavierStokes equat ions, preprint (2003). [28] J .C . MATTINGLY, On recent progress for the stochastic Navier-Stokes equations, to appear on Journees Equations aux derivees partielles. [29] S.P . MEYN AND R .L. TWEEDlE, Markov chains and stoch astic stability, Communications and Control Engineering Series. Springer-Verlag London, Ltd ., London, 1993. [30] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. Il . Continuoustime processes and sampled chains, Adv . in AppI. Probab. 25 (1993), no. 3, 487-517. [31] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. IlI. FosterLyapunov criteria for continuous-time processes, Adv . in AppI. Probab. 25 (1993), no. 3, 518-548. [32J J . NORRlS, Simplified Malliavin calculus, Seminaire de Probabilites, XX, 1984/85, 101-130, Lecture Notes in Math., 1204, Springer, Berlin, 1986. [33] M. ROMITO, Ergodicity of the finite dimensional approximation of the 3D NavierStokes equations forced by a degenerate noise, J . Stat. Phys. 114 (2004), Nos. 1/2,155-177. [34] D.W . STROOCK, The Malliavin calculus, a function al analytic approach, J. Funct. Anal. 44 (1981), no. 2, 212-257 . [35] D.W. STROOCK, Some applications of stochastic calculus to partial differential equations, Eleventh Saint Flour probability summer school-1981 (Saint Flour, 1981), 267-382, Lecture Notes in Math. 976, Springer, Berlin, 1983. [36] D.W . STROOCK AND S.R.S . VARADHAN, On the support of diffusion processes with applications to the strong maximum principle , Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ . California, Berkeley, Calif., 1970/1971), Vol. Ill: Probability theory, pp. 333-359. Univ. California Press, Berkeley, Calif. , 1972. [37] VV . AA ., The GNU Scientific Library, http;//VWIo7.gnu.org/software/gsl/ .

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS BJORN SCHMALFUSS· Abstract . The intention of the article is t o show the existence of inert ial manifolds for random dyn amical systems generated by infinite dimensional random evolution equations. To find these manifolds we formulate a random graph transform . This transform allows us to introduce a random dynamical system on graphs. A random fixed point of this system defines the graph of the inertial manifold. In contrast to other publications dealing with these objects we also suppose th at th e linear part of such an evolut ion equation contains random operators. To deal with these objects we apply th e multiplicative ergodic theorem. The key assumption for the existence of an inertial manifold is an w-wise gap condition. Key words. Inertial manifolds, random dyn amical systems, stochastic partial de 's, mult iplicative ergodic theorem. AMS(MOS) subject classifications.

Primary: 37HlO ; secondary: 37H15 ,

60H15, 35B42 .

1. Introduction. Inertial manifolds are objects that allow to interpret the dimension of the long-time dynamics of ordinary or partial differential equations. In the case of a partial differential equation this dyn amics will be finite dimensional. Inertial manifolds are Lipschitz continuous manifolds in the phas e space. On the graph of these manifolds the dynamics is given by an ordinary differential equation of smaller dimension than th e dimension of the original equation. States outside these manifolds will be attracted by th e inertial manifolds exponentially fast . A st andard method to show th e existence of th ese manifolds is the Lyapunov-Perron method. Using this technique the graph of the inertial manifold is given as a fixed point of an operator equation related to our original differential equation, see Chow and Lu [7] , Chueshov [8], Constantin et al.[13], Foias et al. [17] and Temam [23] . Similar methods has been used to find inertial manifolds for sto chasti c (partial) differenti al equations, see Bensoussan and Flandoli [3]' Da Prato and Debussche [14]' Chueshov and Girya [10]' [11]' Duan et al. [16] and for delay equations see Chueshov and Scheutzow [12] and references therein. Another technique for non-autonomous dyn amical syste ms has been introduced by Koksch and Siegmund [18] . In contrast to these applications we will use another method. This method is called graph transform method. This transform allows us to introduce a random dyn amical syst em defined on graphs. A random fixed point for this system defines t he random graph of the random inertial manifold. • Mathematical Institute, University of Paderborn, Warburger StraBe 100, 33098 Pad erborn, Germany (schmalfuss0upb . de) . 213

214

BJORN SCHMALFUSS

This method has been introduced by SchmalfuB [211 and by Duan et al. [15] but only for invariant manifolds. In contrast to all these applications we consider more general random evolution equations. Especially, we are able to treat equations with a random linear part and with non-linearities having a random Lipschitz constant. The curial assumption to find invariant manifolds is the gap condition. We will formulate an w-wisegap condition containing the random Lipschitz constant and random coefficients of the exponential dichotomy condition of the linear part. As a main tool to achieve our results we have to apply the multiplicative ergodic theorem. Since this article is base on techniques from the theory or random dynamical systems we introduce in the next section basic terms from this theory. In Section 3 we introduce the random evolution equation for which we will study random inertial manifolds. Section 4 contains the definition of the random dynamical system on graphs . The fixed point argument giving us the random inertial manifold is described in Section 5. The last section contains examples. 2. Random dynamical systems. In the following we are going to describe the dynamics of systems under the influence of random perturbations. Such a perturbation is given by a noise. The mathematical model for a noise is a metric dynamical system. DEFINITION 2.1. Let (0, .1', JP» be a probability space. Suppose that the measurable mapping B : (IR x 0, B(IR) 0 F)

-+

(0, F)

forms a flow :

Bo = ido , for t, r E IR. The measure JP> is supposed to be invariant (ergodic) with respect to (J. Then the quadro-tuple (0, .1', JP>, (J) is called a metric dynamical system. Metric dynamical systems are generated by the Brownian motion (see Arnold [2] Appendix A) or by the fractional Brownian motion (see Maslowski et al. [19]) or by random impulses (see Schmalfuf [22]). For instance we can choose for the set Co(lR, U) of continuous functions on IR which are zero at zero with values in a separable Hilbert space U. For .1' we choose the associated Borel-a-algebra with respect to the compact open topology, for JP> we choose the Wiener measure Le. the distribution of a Wiener process for some covariance Q and B is given by the Wiener shift :

°

Btw( ·) = w(· + t) - w(t)

for w E 0.

Then we obtain the metric dynamical system (0, .1', JP>, B) which is called the Brownian motion. Indeed JP> is ergodic with respect to B.

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 215

In the following we will always suppose that we have a metric dynamical system such that JPl is ergodic with respect to B. Let H be some topological space. A random dynamical system with phase space H and with respect to a metric dynamical system is a measurable mapping

cjJ : (]R+ x D x H,B(]R+) ®F ®B(H)) ---. (H,B(H)) satisfying the cocycle property

cjJ(t + T, W, x) = cjJ(t, (}rw, cjJ( T, W, x)),

cjJ(O ,W, x) = x.

The eocycle property is a generalization of the semi-group property. It reflexes the fact that the deterministic dynamics is perturbed by a noise. For our applications we will always suppose that H is a separable Hilbert space with norm 11 . 11 . Later we will allow a slight modification of the measurability of such a system. Generators for random dynamical systems are solution operators for random or stochastic differential equations. For instance , consider the random differential equation with phase space: H = ]Rd

dcjJ dt = f((}tw, cjJ), Suppose that this equation has a globally unique solution for any noise path wand for any initial condition x E ]Rd. This solution at time t is denoted by cjJ(t, w, x). These operators form a cocyclejrandom dynamical system . The measurability of this mapping then follows by some regularity assumptions about the right hand side f . Another tool we need is the multiplicative ergodic theorem. We consider the linear differential equation (2.1) where B is a random matrix. Then this equation generates a linear random dynamical system with time set ]R, see Arnold [2] 3.4.15. We can describe the dynamics of this random dynamical system by the following theorem which is the multiplicative ergodic theorem: THEOREM 2.2. Let B E ]Rd ®]Rd be a random matrix contained in L 1 (D, F, JPl, (}) where (} is supposed to be ergodic. Then there exists a {(}t hEIR -invariant set of full measure, non random numbers pEN, P ::; d, -00 ::; Ap < Ap-l < ... < Al and d 1 , ' " , dp E N and random linear spaces E 1(w),·· · ,Ep(w) of deterministic dimension d1 , ' " ,dp such that

n

(2.2)

]Rd =

E 1(w) EB,, · EB Ep(w)

for wEn. The spaces E, are invariant with respect to the random dynamical system 'IjJ generated by (2.1): 'IjJ(t,w,Ei(w)) = Ei((}tW)

fori = 1,'" ,p,

wEn,

t E]R

216

BJORN SCHMALFUSS

and

)\{} 11'm logll7/J(t,w)xll __ /I,.. if an d only if x E E( i W 0

t_±oo

t

for wEn . The spaces E; depend mea surably on w. In particular, there exist measurable projections on these spaces.

The numbers Ap , ' " ,AI are called Lyapunov exponents to (2.1). The spaces E, are the Oseledets spaces to (2.1). The assertions of the multiplicative ergodic theorem are only true with respect to a {B t hEIR-invariant set of full measure. We can restrict our original metric dynamical system to this set such that B is B(IR) ® F n,F n measurable where the F n trace (J algebra with respect to We will denote this new metric dynamical system by the old symbols (0, F , IP', B). If we can modify our original metric dynamical system in this sense such that some property holds for the new metric dynamical system we say that a property holds B-almost surely, B-a.s. The intention of this article is to find sufficient conditions for the existence of random inertial manifolds for random dynamical systems . We start with basic notations for this purpose . We call a family of random parametrized sets M {M(W)}wEn, M(w) cH a random set if the mapping

n

n.

0:3 w -+ dist(y, M(w)):=

inf

xEM(w)

Ilx -

yll

is a random variable for any y EH. Such a set is called a positively invariant random set with respect to a random dynamical system cP if

cP(t,w ,M(w)) c M(Btw) for t

~

O,W E O.

Let HI, H 2 be a splitting of the Banach space H :

In addition we suppose that there exist continuous projections HI, H 2 . Let

1fl , 1f2

onto

such that w -+ ')'(w ,x) is measurable for fixed x E HI. For fixed w the mapping x -+ ')'( w, x) is Lipschitz continuous . Then we call the set

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS

217

a random Lipschitz manifold. Indeed M is a random set what follows from Castaing and Valadier [6] Lemma 111.14. In addition, we call a random positively invariant set M exponentially attracting with respect to the random dynamical system 4> if lim

inf

t-oo zEM(8 t w)

114>(t,w,x) - z] = 0

with exponential speed for any x E H. DEFINITION 2.3. An exponentially attracting random Lipschitz manifold which is positively invariant for a random dynamical system 4> is called a random inertial manifold for 4>. The dimension of this manifold is defined by the dimension of HI . Let X be a mapping on 0 with values in jR+. Such a mapping is called tempered if the mapping t --t X (Btw) is subexponentially growing for t --t ±oo B-a.s.: lim log+ X(Btw) = O. t

t-±oo

Note that if X is a random variable on the ergodic metric dynamical system (0, F , P, B) then there exists only one alternative for the above property. Suppose that the random variable X is not tempered. Then we have

(2.3)

. log+ X(Btw) 1im sup =

B- a.s.

00

t

t-±oo

The following lemma is not hard to prove . LEMMA 2.1. (i) Sufficient for temperedness of the random variable X is that lE sup log" X(Btw)
if 4>(t,w,X(w)) for t 2: 0 B-a.s.

= X(Btw)

218

BJORN SCHMALFUSS

3. random evolution equations. The objective of our interest will be the dynamics of the following random evolution equation du dt + A(Btw)u = F(lhw, u).

(3.1)

Our task is to formulate conditions which ensure the existence of an inertial manifold . We start with the random linear differential equation

du dt

(3.2)

+ A(Btw)u =

O.

To treat the non-linear evolution equation by the variation of constant formula we have to assume that t ---? -A(Btw) generates a fundamental solution. For the definition see Amann [11 Chapter 11.2. More precisely, we assume that A generates a random dynamical system of linear continuous mappings on H:

U(t + r,w) = U(t,Brw)U(r,w). Details about the existence of such a random dynamical system can be found in Caraballo et al. [41 . For the following we need the exponential dichotomy condition: There exist continuous projections 7rl, 7r2 related to the splitting of the phase space H = HI Ef7 H 2 commuting with U:

We suppose that U (t, W)7rl defined on the finite dimensional space HI is invertible where these inverse mappings are denoted by a negative first argument:

In addition, we assume that there exist random variables aI, a2 E i, (n, F , JID) such that (3.3)

IIU(t,w)7rl ll

::; eJ~ al(O.w)ds

for t ::; 0

IIU(t, W) 7r2 11

::;

eJ~a2(O.w)ds

for t ~ O.

(It follows from assumptions given below that al(w) > a2(w).) We now consider the complete non-linear equation (3.1). Let us assume that W ---?

and that

F(w, x)

is measurable for any x

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 219

is Lipschitz continuous

(3.4)

117r IF (w, x) - 1r IF(w , y)11

:::; L(w)llx - YII,

111r2F(W, x) - 1r2F(W, y)11

:::; L(w) llx - yll

for any wEn where the Lipschitz constant L(w) depends on w. In particular we assume that this Lipschitz constant is locally integrable:

l

b

L(Bsw)ds < 00 for -

< a < b < 00 .

00

The Lipschitz continuity and the existence of a fundamental solution of the linear problem ensure that the equation

<jJ(t) = U(t,w)x +

I

t

U(t - s, Bsw)F(Bsw , <jJ)ds,

<jJ(O)

=

x EH

has a unique solution for every initial condition <jJ(O) = x . This solution is the mild solution for (3.1). THEOREM 3.1. Suppose that -A is the generator of the random fun-

damental solution U. In addition, we suppose that F satisfies the above measurability and continuity assumption. Then the solution of (3.1) generates a random dynamical system <jJ with phase space H. For our consideration we also need that

(3.5)

sup 111r I F (w, x )1I :::; h(w) , xEH

where

i- , h

sup

111r2F(W,x) 11:::; h(w)

xEH

are tempered random variables.

4. The random graph transform and inertial manifolds. We now introduce a mapping which transforms Lipschitz continuous mappings from HI to H2 into Lipschitz continuous mappings. That transform is defined under the action of the random dynamical system generated by (3.1). Using this transform we will introduce the graph transform. We are going to construct the random inertial manifold as a random fixed point of this mapping. For the following let "Y be a global Lipschitz continuous and bounded mapping from HI into H 2: "Y E C~,I(HI' H2) where

II"Ylle =

sup lI"Y(x+)1I
. 0 We can conclude directly from the proof of the last lemma: COROLLARY 4.3 . Suppose that the assumptions of Lemma 4.2 are satisfied. Then I1>(Tl + T2,w,l') is generated by the solution of (4.1) on [0, t, +T2] which exists. In addition 7rl4J(Tl +T2, w, +1'(w, .)) has an inverse function on HI which will be denoted by 2(Tl + T 2, BTt +T2W, l'(w))( ·). We now consider the two dimensional random linear differential equation

d

(4.7)

dt 'Ij;(t)

d(W(t)) (W(t)) V(t) = B(Btw) V(t)

= dt

where

B(w) =

al (W) - L(w) ( L(w)

-L(W)) , a2(w) + L(w)

The dynamics of this equation can be described by the multiplicative ergodic Theorem 2.2. LEMMA 4.3. Suppose that BELl (0, F, JP» . In addition we suppose that (4.8)

and that the gap condition (4.9)

holds B-a.s. Then the dynamical system 'Ij; generated by (4.7) has two different Oseledets spaces B-a.s. The Oseledets space El (related to the biggest Lyapunov exponent) has an angle between zero and 7r /4. The second Lyapunov exponent >'2 is negative. Proof We consider the dynamics of (4.7) projected onto the unit sphere . The angle a with respect to the axis (1,0) satisfies the following differential equation a~

(4.10)

= (B(Btw)(cos at, sin atf,(-sin at, cos atf) = ~ (a2(Btw) -

al (Btw)

+ 2L(Btw)) sin(2at} + L( Btw)

see Arnold [2] Page 278. For a = 0 we have a~ > 0 and for a = i we have by the gap condition (4.10) a~ < O. Therefore [0, 7r/ 4] is forward

224

BJORN SCHMALFUSS

invariant for (4.10). Hence we are able to prove that the system (4.7) has a random attractor A(w) C [0,71"/4]. (Conditions for the existence of a random attractor can be found in ([20]).) Since our state space is one dimensional the attractor is an interval [6:(w) , £1(w)] where 6: = & is possible. Especially 6:, & are random fixed points . Since (4.7) defines a linear random dynamical system 6:, & represent linear invariant spaces for (4.7). If we reverse the time in (4.10) we find another random attractor given by the random interval [6:-(w),£1-(w)] C [71"/4,71"/2] where the boundary points 6:-, £1- are random fixed points representing invariant spaces for (4.7). By the multiplicative ergodic theorem we have not more than two one dimensionale invariant spaces. Hence one of these spaces is represented by 6:(w) = £1(w) E [0,71"/4] and the other in 6:-(w) = £1-(w) E [71"/4,71"/2] . (By the gap condition a == 71"/4 does not represent an invariant space.) According to Birkhoff's and Liouville's theorem we have that ),1+),2= lim t-+oo

~logldet"p(t,w)l= t

lim t-+oo

~t lattrB(Orw)dr=lE(a1+a2) < 0

such that ),2 is negative . It follows that at tends to £1- (Otw) for t ---. -00 and hence at tends to £1(Otw) for t ---. 00 . 0 By the conclusions of the last lemma we can represent the Oseledets spaces E 1 (w), E2 (w) by unit vectors

with positive elements because the angles for these spaces are contained in (0,71"/2) O-a.s. In particular we can suppose that eu(w) E [1/.J2,1] 0 a.s. On El, E 2 we have linear one dimensional random dynamical systems U1, U2 such that

(4.11)

for i = 1, 2.

(We can suppose that Ui(t,W) > O. Ui(t) never crosses zero.) The system (4.7) can be interpreted as a version of system (3.1) with phase space H = JR 2 = JR x JR if we set

- L (w)W - L(W)V) F(w, W, V) = ( L(w)W + L(w)V . The operator F(w, .) is Lipschitz continuous. The Lipschitz const ants are the same as in (3.4). We now formulate the graph transform for the problem (4.7). Since this is a linear problem we transform instead of Lipschitz continuous graphs the graphs of linear mappings from JR into lR characterized by elements r from JR. We consider the problem (4.7) with boundary conditions

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 225

W(T)

(4.12)

= Y E ]R,

V(O)

= fW(O) + C.

We obtain the graph transform generated by (4.7) setting C = 0 denoted by W. LEMMA 4.4. (i) The system (4.7), (4.12) has a unique solution on [0, T(w, r)] where T(w , f) is defined by (4.4). (ii) Suppose that 0 ~ f l f 2 and W, V are solutions of (4.7), (4.12) for f = f l , f2 on [O,T(w,r)]. Then

s

W(t,w ,fd

~

W(t ,w ,f2),

V(t ,w ,fd

~

V(t,w,f2)'

(iii) Suppose that C = 0 in (4.12). Then we have //(T,w,,)/ILip ~ W(T,w, Ii'YIILip)(l) for 0 ~ T ~ T(w, Ii'YIILip). Proof (i) To see the existence of a solution we can apply exactly the same technique we used to find a fixed point for the operator defined in (4 .2). We obtain the same contraction constant k(T,w,f) as in the proof of Lemma 4.4 (see (4.3)) if we replace Ii'YIILip by f. The constant C has no influence on this contraction constant. Hence we can define T(w , .) by (4.5) .

(ii) can be seen by direct calculation. (iii) Set I' = Ii'YI ILip, C = O. We construct the solution of (4 .1) and (4.7) , (4.12) by successive iteration on [0, T(w, Ii'Y IILip)] starting with

WO(t)O == 1, VO(t) ==

Ii'YIILip,

wO(t) == v", vO(t) == ,

and so on W i , V i and wi ,Vi . To get (wi ,Vi) we have to iterate the operator Ty,T,w,y+ defined in the proof of Lemma 4.1 and similarly for (W i, Vi). We have for i = 0 i + i + W

1r

Ilv (T,W")(~ll-

" )(Y2 )11

v

YI - Y2

~ V i(T,w , Ii'YIILip)(l).

The structure of the matrix B ensures that this inequality remains true for every i E N. Hence we obtain (iii) for i -; 00 . 0 Our knowledge of the Oseledets spaces for (4.7) allows us to determine a random invariant linear manifold for (4.7) without the random graph transform. LEMMA 4.5.

Let /'i;(w) := eI2(w)jell(W). Then the graph of ]R 3

Y -; /'i;(w)Y E ]R

defines a random invariant manifold for (4.7) in the phase space ]R2. Proof Note that ell (w) > 0 such that /'i;( w) is well defined. According to the invariance of the space El we have that

( 1)

'I/J(T,w) /'i;(w)

=

UI

(T, w)

(1)

ell(w) ell(thw) /'i;(thw) .

226

BJORN SCHMALFUSS

We multiply the last equation by

Ul

(T

eu(w) ) (0 )Y=W(O,w,~(w))Y=W(O). ,w en TW

o Now we are in a position to show the global eocycle property for the gr aph transform

(n + 1, {}-n-1W, "'/({}-n-lw))lle (5.2)

:::; 11iI>(n, (}-nw,"'/({}-nw)) - iI>(n, {}-nw, iI>(1 , {}-n-1W , "'/({}-n-lw))lle

:::; K(n ,(}-nw)II"Y({}-nw) - iI>(1 , {}-n-1W, "'/ ({}- n- l w))lle -. O. We have used the temperedness of W -. 11iI>(1, (}-IW,"'/({}-IW)) lie, see Lemma 5.2. The limit of this sequence is denoted by "'/* such that ",/*(w) E G(w). We note that by the properties of F we have

232

BJORN SCHMALFUSS

but the right hand side defines a tempered random variable, see Lemma 5.3. The convergence in (5.2) also holds for T E lR:

IlrP(T, O_TW, ')'(O_TW)) - rP([T], O_[TjW, ')'(O_[TJW)) lie

:::; K([T], O_[TjW) IiI'(O-[TjW) - (T - [T], O-T+[TjW, ')'(O_T+[TjW))

lie

where [.] is the integer part of a real number . To see that ')'* is a random fixed point we can apply the eocycle property and the continuity of ')' ---+

(S,w, ')') E C(H l , H2): (S,w,')'*(w)) = (S,w, lim (n,O_nw,')'(O_nw))) n-oo

=

lim (S + n, O-S-nOSw, ,),(O-n-sOSw)) =

n-oo

')'* (Osw)

for S 2: 0 where we have to use the convergence in (5.2). As a limit in Cb(Hl, H2 ) we have that ,),*(w) E Cb(Hl , H2 ) . In addition for any x+ E H, the mapping W ---+ ')'*(w,x+) is the pointwise limit of random variables, hence measurable. We obtain directly from Lemma 5.1:

11(T,w,')'(w)) -')'*(OTW)I/e = 11(T,w,')'(w)) - (T,w,')'*(w)))lle :::; K(T,w)lII'(w) -')'*(w)I/e

o Applying Lemma 5.2 we have the following main result: 5 .2. Suppose that the inequalities (4.8), (4.9) are satisfied. Then the random dynamical system generated by (3.1) has a random THEOREM

inertial manifold. Proof. We just know that the random fixed point for the graph transform defines a graph of an invariant manifold for the random dynamical system rP. We have to show that this manifold is exponentially attracting. Let w ---+ X(w) be a (tempered) random variable in H . We introduce the graph ')'x(w,x+) == 7T2X(W) which is contained in g. We set for every

x+ E H, y+=y+(T,w, x+) =7TlrP(t , w, x++7T2X(W)),

Y(T,w)=y+(T,w, 7Tl X (W))

such that

7T2rP(T,W, 7Tl X (W) + 7T2X(W)) = (T, w, ')'X )(Y(T, w)). Notice that by Corollary 4.3 the mapping and the other hand we have

:=: exist for any')' E g. On

I/7T2rP(T, W, 7Tl X (W) + 7T2X(W)) -')'*(OTW ,Y(T, w))1/ :::; 11(T,w,')'x) -')'*(OTw)lle where the right hand side tends to zero exponentially fast by Lemma 5.3. 0 This formula is also true for X (w) == x EH.

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 233

6. Examples. In this section we will study two examples . For the first example a similar equation is just formulated in Chueshov and Girya [9] (see also Chueshov and Scheutzow [12] for delay equations and Duan et al. [16]). These considerations are based on the Lyapunov-Perron method. However, our first example demonstrates how the graph method works for those equations. Our second example is completely different to these equations because the linear part of the evolution equation is given by a stationary process. We consider the sto chasti c reaction diffusion equation (6.1)

au at

= t::.u + g(Btw, 1/, u) + ~(t, 1/),

u(O ,1/)

= x(1/) ,

1/ E (0, n) .

with homogeneous Dirichlet boundary conditions on the interval [0, n]. To formulate this equation as an evolution equation we introduce the phase space H = L 2 (0, n) . It is well known that t::. can be interpreted as the generator -A of a strongly cont inuous semigroup U(·) . The eigenvalues, eigenvectors of A are (i 2 ,sin(i1/)) for i E N. Suppose that ~ is a noise term given as the generalized temporal derivative of a two-sided Wiener process W with values in H defined on a filtered probability space (0, F , {Ft}tEIR, JPl) . The covariance of this process is supposed to be of trace class. For the exact definition of such an associated filtered metric dynamical system we refer to Arnold [2] Appendix A. For g we choose a mapping such

(0 x JR, Fo 0 B(JR)) 3 (w, x)

-t

g(w,1/, u)

is measurable for every u E JR, W

sup

-t

Ig(w ,1/, u)1 < 00

1]E[O,7rJ, uEIR

is tempered. We have the Lipschitz condition Ig(w, 1/, u) - g(w,1/, v)1:::; Llu -

vi

for u, v E JR uniformly for w, 1/. Then

H

3

v(·) - t g(w, ., v(·)) E H

defines a Lipschitz continuous mapping on H denoted by G(w, .). In oder to apply our general results we formulate (6.1) as a stochastic evolution equation on H:

(6.2)

du + Audt = G(Btw)dt + dW(t ,w).

We also note that the evolution equation dz + Az dt = dW(t)

234

BJORN SCHMALFUSS

has the unique stationary solution

Z(w) =

[~ U(-T)dW(T,w)

B- a.s.

The random variable Z is tempered. In order to show the existence of an inertial manifold we study the random evolution equation

dv (6.3)

dt

+ Av =

F(Btw ,v)

F(w,v):= G(Btw , v + Z(Btw)). Note that F has the same Lipschitz constant as G above. This equation is defined B-a.s. THEOREM 6.1. Choose an n such that

(6.4)

(n

+ 1)2 -

n2

= 2n + 1> 4L .

Then the random dynamical system

0 such that c2(x) :::: E, where C;(lR) is the set of functions on lR having bounded, continuous derivatives up to order k inclusive. (2) h(x) E Cr'+l(lR)nlHlm+l(lR). (3) O"(x) E Cr'+l(lR) and there exist two positive numbers 0 < o; < O"b such that 0"a :S 0"( x) :S O"b holds for all x E lR. For a given initial function 0, which is independent oft,x,w" and such that

e,

tI

(2.3)

j 2.t aij(t,x,w)eie tail(t,x,w)ef t,J=1 1=1 t=1 d

~oL:leiI2,

VeEjRd,t~O,

i=1 holds (above (2.3) is called superparabolic condition) and for a positive integer m, the following conditions are satisfied:

SOLUTIONS OF A CLASS OF SEMI-LINEAR SPDES

241

(a) The functions ai j , bi , c, (Jil ,h l (i,j = 1,2, .. · , d, l = 1,2,···,dd are B([O, T] x JRd) 18) F measurable, bounded, predictable (for each x E JRd), real functions, and 4> is an Fr-measurable function taking value in L 2(JRd). (b) The functions a i j , v, c, (Jil,h l (i,j = 1,2, ·· ·,d,l = 1,2, .. · , dd are differentiable in x up to order m for all t 2': and w. They, together with

°

their derivatives, are uniformly bounded with respect to t, x, and w by a constant K (m).

(c) cP

E

ILdn,lHIm ) , f

E

L 2([0,T] x n,lHIm -

1 ) , gl E

L 2([0,T] x n,lHIm ) ,

where l = 1,2" . " d 1 . Then, there exists a unique generalized solution u of (2.1), which belongs to the class L 2 ([r, T]; P; IHIm +1 ) n C([r, T]; P; IHIm ) and satisfies equality (2.2) for all t E [r, T] and almost surely with respect to probability JIb. There exists an N > 0 depending only on K(m), d, d 1 , m, r , and T such that

Proof. See the proof of (Rozovskii [5], pp133, Theorem 2).

o

For the classical solution of the Cauchy problem (2.1), first let us give a precise definition. DEFINITION 2.2 . A function v r,.(·,·) mapping from (t ,x,w) E [r,T] x JRd x n to Vr,t(x, w) E JR, which belongs to CO ,2([r, T] x JRd), (JP> - a.s .), is predictable stochastic process for each x E JR d , and satisfies Equation (2.1), is called a classical solution of problem (2.1). THEOREM 2.2. If the conditions of Theorem 2.1 are fulfilled for any mEN, then the classical solution of problem (2.1) is a classical smooth solution or, in other words, it is infinitely differentiable in x (JP> - a.s .) . Proof. See Chapter 4 of Rozovskii [5]. 0

3. Semi-linear SPDE. Based on previous section's results and notations, now we prove that the problem (1.1) has a unique, smooth classical solution. According to Theorem 1.1, for any given r 2': 0 and cP E {Cb(JR)+ n 1HI 1 (JRn, problem (1.1) has a unique Cb(JR) nIHIl (JR)-valued, non-negative, strong solution {'ljJr,t : t 2': r 2': O} if the basic condition holds . Furthermore, for any cP E {Cb(JR)+ W (JRn, II'ljJr,tlla :S IlcPlla holds JIb-a.s. for all t 2': r 2': 0, where 114>lla is the supremum of cP. Now we want to generalized this result such that if we assume that the coefficients and the initial function of (1.1) have better regularity, then the solution of (1.1) also has better regularity. After that, the nonnegativity of the solution with a better regularity can also be derived. First, let us give the 'IjJ-semigroup property of the solution of (1.1). Since the solution of (1.1) depends on the initial function cPU, we can rewrite

n

242

HAOWANG

the solution of (1.1) as 1/Jr,t(X) = 1/Jr,t(x, rjJ) . Based on this new notation, we say that 1/Jr,t(x, rjJ) , the solution of (1.1), defines a 1/J-semigroup if there exists a set N c n such that IP( N) = 0 and for any rjJ E Cb (R) + lHl 1 (R) and 0 :::; r :::; s :::; t,

n

(3.1) holds for all w tJ. N. Remark. (3.1) defines a forward 1/J-semigroup. This corresponds that (1.1) is a forward SPDE. Based on this definition, we have following theorem. THEOREM 3.1. Suppose that the basic condition holds for m ~ 1. Then, for any rjJ E {Cr (R) n lHl m (lR)}, Equation (1.1) with er == 0 has a unique {Cr(lR) lHlm(lR)}-valued, strong solution {1/Jr,t : t ~ r ~ O}. Moreover, the solution defines a 1/J-semigroup. Proof. First, with er == 0, SPDE (1.1) becomes the following linear SPDE:

n

+ jt [~a(X)O;xTr,s(X)] ds

Tr,t(x) = rjJ(x)

(3.2)

+ jt

1

h(y - X)OxTr,s(x)W(ds,dy) ,

t

~

r.

In order to use Theorem 2.1 , here we decompose the Brownian sheet into a sequence of one-dimensional Brownian motions first introduced in [4]. Let {hj : j = 1,2 , ·· ·} be a complete orthonormal system of L2(1R) . Then, for any t ~ 0 and j ~ 1,

defines a sequence of independent standard Brownian motions {Wj 1,2, · ··}. For E > 0 let

:

j =

[l/fJ

Wf(dt, dx) =

L hj(x)Wj(dt)dx,

t ~ O,X E lR,

j=l

where [liE] denotes the maximum integer less than Y]« . By assumption, we have that C E C:+1(lR), c2(x) > C:+ 1 (lR) n lHlm +1 (lR). Now we consider the equation

T:,t(x) = rjJ(x)

+ it

(3.3)

+ it

L

E

> 0, h

[~a(X)O;T:,s(X)]ds

h(y - x )oxT:,s(x )W f(ds,dy),

t

~ r ~

O.

E

243

SOLUTIONS OF A CLASS OF SEMI-LINEAR SPDES

Now we check whether Equation (3.3) satisfies the superparabolic condition (2.3). For Equation (3.3), the left hand side of (2.3) becomes

where ~ E lR and d1 = [l/f] . Then, by Parseval equality and the basic condition, we have

n

Therefore, for any l/J E {Cb'(lR) lHIm(lR)} , by Theorem 2.1, Equation (3.3) has a unique solution T: t(x) E lL 2([r, T], P, lHIm+1 ) CUr, T], P , lHIm) and the following inequality ,

JE

(3.5)

sup

sE[r,Tj

n

IIT:,sll;" s KJEIIl/JII;"

holds. By a limit argument similar to the proof of Rozovskii ([5] p l l l , Theorem 2), we can get that Tr,s(x) , the solution of (3.2), satisfies

JE

(3.6)

sup

sE[r,T]

IITr,sll;" ~ KJEIIl/JII;",

which implies the uniqueness of the solution of (3.2). Now we only need to prove the semigroup property. Let Tr,t be the unique strong solution of (3.2). Then, for any s 2:: rand t 2:: 0, we have

Tr,s+t(x) = l/J(x)

+

t: [1

s+t

(3.7)

+

]

2a(X)o;xTr,u(x) du

1l

r

h(y - X)OxTr ,u(x)W(du, dy),

s 2:: r,

t 2::

o.

We subtract each side of (3.2) from the corresponding side of (3.7), respectively. Then, we get

244

HAOWANG

If we look Tr,s(x) as the initial data in (3.8), since a(x) and h(x) are time homogeneous and the lower limits of the integrals on the right hand side are s, we can reform (3.8) to get

(3.9)

s2:r, t2:0. By the uniqueness of the strong solution of (3.2), for any fixed s 2: r, Tr,s+t(x, '->00

where the limit, for some choice of scaling constants a, b, and c, is understood in the sense of the weak convergence of finite dimensional probability distributions of random fields on the left-hand side to those of the random field on the right-hand side (the convergence of infinite dimiensional distrbutions has not been studied but is a desirable goal). The above result would be useful only if the limiting random field U (t, x) were a sort-of a standard object (like the Gaussian law in the central limit theorem) which had some kind of usable description either through finite-dimensional distributions, characteristics functions, n-point correlations, stochastic integrals, or other similar convenient descriptors. However, before we produce results of this kind let us begin with a statement of a th eorem due to Zuazua [11] which shows, in the case of nonrandom initial data, how a scaling limit of an arbitrary solution of the Burgers equation can be seen as such a standard selfsimilar object. Recall that the Burgers equation (see, e.g., [10]) corresponds to the choice of the Laplacian as a diffusive operator, E = £ 2 = b., and a simple quadr atic nonlinearity Nu = u 2 and is thus , in one dimension, of the form

(5) (6)

u (O, x )

= uo(x ),

In what follows , unless explicitely stated otherwise , we will take IJ = 1. THEOREM 1.1. Let p 2: 1 and let u (t , x ) be a solution of the initial value problem (5-6) with an integrable initial condition with

J

uo(x ) dx = M .

250

WOJBOR A. WOYCZYNSKI

Then lim t!(l-i)lIu(t, .) - U(t , ·)lIp = 0

(7)

t-+oo

where U(t , x) is the unique selfsimilar solution

U(t, x) = t-! e-:: (K(M) -

(8)

['-'I' e- 4 dZ) ,

such that JU(t,x)dx = M and U(t,.) ~ M80 as t It can be immediately verified that

(9)

~ O.

U(t , x ) = C! U(1,t-!x) .

In other words, if the similarity transformation is defined by the formula

(10) then the solution of the Burgers equation described in Theorem 1.1 is selfsimilar, that is,

(11)

U>.=U,

and the theorem implies that, for each t > 0,

J lu>.(t , x) - U(t, x )IP dx

= x-lJ l>.u(>.2 t, >.x) - >.U(>.2t, >.x)IP d(>.x) (12) =

>.p-l J lu(>.2 t, >.x) - U(>.2t , >.x)IP d(>.x)

= C!(P-l)(>.2 t)!(p-l) J lu(>.2 t, >.x) - U(>.2 t, >.x)IP d(>.x)

~0

as >. ~ 00. Thus, if solutions depend on random initial condit ion one gets, under obvious boundedness conditions, that for the random fields u(t, x ) and U(t , x) the following scaling limit result holds true: PROPOSITION 1.2. Let p 21 , and u(t,x) be a solution of the Burgers equation corresponding to integrable random initial data uo. Then, for the rescaled solution random field u>. ,

(13)

lim u>. = U,

>'-+00

where the limit is understood in the following sense: for each t > 0, the expected value (14)

as >.

E

Jlu>.(t,x)-U(t,X)IPdX~O,

~ 00 .

This is just an heuristic start. More subtle scaling limit results will be described in the next section.

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

251

2. Parabolic scaling limits for Burgers turbulence. The Burgers turbulence problem, see, e.g., [10], corresponds to the situation where the initial data Uo in the Burgers equation are gradients uo(x) = \7~(x) , where ~ is a stationary (homogeneous) potential field on JRd which , as a rule , is not integrable on the whole space. Thus a different approach from the one described in Section 1 is necessary. The methods and results described below have been developed in a series of papers with D. Surgailis [9] and with N. Leonenko [4], see also [10]. The fundamental observation is that if the initial potential field satisfies the limit condition (15)

V/3(Y) := B(,6)(exp[~(,6y)jv]- A(,6)) => V(y),

,6

---+ 00,

in the sense of weak convergence (=» of finite-dimensional distributions of the random fields (possibly generalized), then

(16)

,6d+1B(,6) u(,62t ,,6x) => const- (V( .), \7g(t,x - .)),

,6

---+ 00 ,

where (.,.) stands for the usual Hilbertian inner product and g(t,x) (27rt)-1/2 exp[-x 2 j2t] is the standard Gaussian kernel. This result is a direct consequence of the Hopf-Cole formula for the Burgers equation. It follows from the classical Dobrushin's theory that if potential field ~ (x) is strictly stationary and ergodic then, necessarily,

(17)

B(,6) = ,6" L(,6) ,

for some real exponent K, and slowly varying function L. If, additionally, A(,6) == A is independent of ,6 then the limit random field V in (2.15) is selfsimilar and, for each ,6 > 0, and all "smooth" test functions

-(j.

(W( .),\7g(t,x - .)),

where W is the standard white noise.

as

,6 ---+

00,

252

WOJBOR A. WOYCZYNSKI

(ii) Let the initial potential random field ~(x) in the Burgers turbulence problem in JRd be a shot-no ise field of the form

~(x) =

L 1]ih(x - ( i), i

with 1Ji being i.i.d., ((i) being a Poisson point process, and hE £l n Loo. Then the limit behavior (19) also obtains with

and

B(x) :=

J

E(e1) l h (u ) - l)(e1) l h (u + x )

-

1) duo

For initial data with long-range dependence, that is data where the covariance function is not integrable and the spectrum is singular, the situation is more complicated and not completely understood. If the only singularity is at the origin then the following result holds true: THEOREM 2.2. Let the initial potential field ~(x) on md be of zeromean and variance one, and possess a singular spectral density of the form (20)

0
1. This evolution equation has been extensively studied in a joint work with Biler and Karch, see [2], and the main theorem of this section is taken from that paper. Recall that solutions of such equation have to be understood as mild or weak solutions . It is clear that if the similarity transformation is defined now by the formula

a-I "( = --1' r-

oX

> 0,

then, if u is a solution of conservation law (23), then u>. is also a solution of the same conservation law (23). The question of existence of selfsimilar solutions for such conservation laws, that is solutions such that u == u>. , is answered in the following theorem. Observe that the inital data for such selfsimilar solutions have to be homogeneous functions of degree -"(. THEOREM 4 .1. Ifm+"«d,rrJ.N,m~ lrJ. and Uo E

{v E Cm(JRd) : IDfjv(x)1 ~ C!xl--y-Ifjl, LBI ~ m}

for sufficiently small C, then there exists a junction

such that

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

255

is a solution of the fractional conservation law (23). However, the asymptotic large-tim e behavior of solutions of such fractional conservation laws is seldom dictated by th e asymptotics of special selfsimilar soluti ons as was the case, in view of Zuazua's result, for th e Burgers equation. It turns out th at if th e nonlinearity exponent r exceeds the critical value

0: - 1 r e := 1 + -dthen th e asymptotics of solutions is essentially that of th e linearized equation Ut = £Ot u . Indeed, we have THEOREM 4.2. If r > r«, and u(t, x) is a solution of the fractional conservation law (23), then

where ex p[- t £ Ot ] is the exponential semigroup generated by the infinitesimal generator £ Ot . The fundament al solution POt (t , x) of th e linear equation Ut = £ Ot u , th at is th e marginal density of the o:-stable Levy process, satisfies th e selfsimilarity condition

that is, it is invariant under the similarity transformation (24) so that th e solution of the linearized equation U(t, x ) := ex p[- t £Ot l uo(x) =

J

POt(t , x - y )uo(u ) dy

enjoys th e same selfsimilarity prop erty as long as th e initi al condit ion is a homogeneous function of degree - d, that is UO(A X) = A-duo(X). In this case th e assertion of Th eorem 4.2 suggests the following scaling limit result for initi al random dat a: PROPOSITIO N 4 .3 . Let r > r e and u(t,x ) be a solution of the fractional conservation law (23) corresponding to the suitably integrable random initial data Uo with sample paths being homogeneous functions of degree -d. Then, for the rescaled solution random field u>. given by (4.24), (25)

lirn u>. = U, >'-+00

256

WOJBOR A. WOYCZYNSKI

where the limit is understood in the following sense: for each t > 0, the expected value (26)

E

J

lu,\(t,x) - U(t,x)1 2dx ----; 0,

as..\ ----; 00 . Of course, not many interesting initial random fields satisfy stringent conditions of Proposition 4.3 so it is an open question how Theorem 4.2 can be further exploited to get, in case r > re, more subtle scaling limit results for random inital data and , eventually, statistical estimation procedures based on them. In the case when the nonlinearity exponent is equal to the critical value, r = re, the situation becomes totally analogous to that of the Burgers equation. For that reason it is only in this case when we can truly talk about the "fractional Burgers turbulence". The basic result, also found in a joint work with Biler and Karch [2], is as follows: THEOREM 4.4 . Let r = re = 1 + (Q - l)/d and let u(t , x) be a solution of the fractional conservation law (23) with initial data uo(x) such that J Uo (x) dx = M < 00. Then, there exists a unique selfsimilar source solution

U(O, .) = M80 ,

(27)

such that t d/ 2ollu(t, .) - U(t, .)112 ----; 0,

(28)

as

t ----;

00 .

Thus solutions of the critical fractional conservation laws display a true nonlinear asymptotic behavior. In this case Proposition 4.3 can be replaced by a more useful statement since the demand that inital random data are homogeneous of degree -d can be removed. Indeed, condition (4.27) implies that, for any ..\ > 0, under the similarity transformation (4.24) ,

(29) Hence, in view of (4.28), for each t

J

> 0,

lu,\(t, x) - U(t, xW dx

J J

= ,\-d (30) = ,\d =

I..\du('\°t ,'\x) - ,\dU(..\°t, '\x)1 2 d('\x)

lu('\°t, '\x) - U('x°t, 'xxW d('xx)

t-~('x°t)~

J

lu('x°t,'xx) - U('x°t ,'xxWd('xx) ----;

°

257

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

as

>. - t

This calculation suggests the following PROPOSITION 4.5 . Let r = r c and u(t,x) be a solution of the fractional conservation law (23) corresponding to the suitably integrable random initial data Uo. Then, for the rescaled solution random field u,\ given by 00.

(4·24), (31)

lim

u,\

= U,

'\-->00

where the limit is understood in the following sense : for each t expected value

(32) as

E

J

lu,\(t , x) - U(t, x)12 dx

-t

> 0,

the

0,

>. - t 00 .

Of course, the result suggested by Proposition 4.5 does not fully address the issue of finding scaling limit results that would provide convergence in the finite-dimensional distributions, along the lines of Section 2. But it is a step in the right direction. However, the problem of finding statistical estimation procedures analogous to those explained in Section 3, remains a challenge.

REFERENCES [IJ P . BILER AND W .A. WOYCZYNSKI. Global and exploding solutions for non local quadratic evolution problems. SIAM J. Appl. Math., 59 (1999), 845-869.

[2] P. BILER, G. KARCH , AND W .A . WOYCZYNSKI. Critical non linearity exponent and [3]

[4] [5]

[6] [7]

[8]

[9]

self-similar asymptotics for Levy conservation laws. Annales d'Institute H. Poincare-Analyse Nonlineaire (Paris), 18 (2001), 613-637. P . BILER, T . FUNAKI, AND W .A. WOYCZYNSKI. Fractal Burgers equations. Journal of Differential Equations, 148 (1998), 9-46. N. LEONENKO AND W .A. WOYCZYNSKI. Parameter identification for stochastic Burgers' flows via parabolic rescaling. Probability and Mathematical Statistics, 21(1) (2001) , 1-55 . J .A. MANN AND W .A. WOYCZYNSKI. Growing fractal interfacces in the presence of self-similar hopping surface diffusion. Physica A : Statistical Mechanics, 291 (2001) , 159-183. R. METZLER AND J . KLAFTER. The random walk 's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339 (2000), 1-77. S.A. MOLCHANOV, D . SURGAILIS , AND W .A. WOYCZYNSKI. The large-scale structure of the Universe and quasi-Voronoi tessellation of shock fronts in forced inviscid Burgers' turbulence in Rd , Annals of Applied Probability, 7 (1997), 200-228. A. PIRYATINSKA , A.I. SAICHEV, AND W.A . WOYCZYNSKI. Models of anomalous diffusion: the subdiffusive case, CWRU Statistics Department Preprint (2003) , pp. 58. D. SURGAILIS AND W .A . WOYCZYNSKI. Limit theorems for the Burgers equation initialized by data with long-range dependence, in Theory and Applications of Long-Range Dependence, P. Doukhan, G. Oppenheim, and M. Taqqu, Eds., Birkhauser-Boston 2003, pp. 507-524.

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[1OJ W.A . WOYCZYNSKI. Burgers-KPZ Turbulence. Lecture Notes in Mathematics 1700, Springer-Verlag 1998. [11] E . ZUAZUA. Weakly nonlinear large time behavior in scalar convection-diffusion equations. Differential and Integml Equations 6 (1993), 1481-1491.

LIST OF WORKSHOP PARTICIPANTS

• Hassan Allouba, Department of Mathematical Sciences, Kent State University • Anna Amirdjanova, Department of Statistics, University of Michigan • Douglas N. Arnold , IMA, University of Minnesota • Kr ishna Athreya, Operations Research and Industrial Engineering, Cornell University • Siva Athreya, Statistical Mathematical Unit, Indian Statistical Institute • Paul Atzberger, Courant Institute of Mathematical Sciences, New York University • Gerard Awanou , Department of Mathematics, University of Georgia • Michele Baldini, Department of Physics, New York University • Rabi Bhattacharya, Department of Mathematics, University of Arizona • Dirk Blomker, Mathematics Research Center, University of Warwick • Maury Bramson, School of Mathematics, University of Minnesota • Susanne C. Brenner, Department of Mathematics, University of South Carolina • Maria-Carme T . Calderer, School of Mathematics, University of Minnesota • Marco Cannone, Laboratoire d' Analys e et de Mathematiques Applique , Universite de Marne-la-Vallee • Rene Carmona, Operations Research & Financial Engineering, Princeton University • Fernando Carreon, Department of Mathematics, University of Texas - Austin • Panagiotis Chatzipantelidis, Department of Mathematics, Texas A&M University • M. Aslam Chaudhry, Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals • Larry Chen , Department of Mathematics, Oregon State University • Long Chen , Department of Mathematics, Pennsylvania State University • Zhenqing Chen , Department of Mathematics, University of Washington • Lan Cheng , Department of Mathematics, University of Pittsburgh

259

260

LIST OF WORKSHOP PARTICIPANTS

• Erhan Cinlar, Department of Operations Research & Financial Engineering, Princeton University • Michael Cranston, Department of Mathematics, University of Rochester • Ian M. Davies, Department of Mathematics, University of Wales Swansea • Hongjie Dong, School of Mathematics, University of Minnesota • Jinqiao Duan, Department of Applied Mathematics, Illinois Institute of Technology • Valdo Durrleman, Bendheim Center for Finance • Maria Emelianenko , Department of Mathematics, Pennsylvania State University • William Faris , Department of Mathematics, University of Arizona • Mark Freidlin, Department of Mathematics, University of Maryland • Peter K. Friz, Courant Institute of Mathematical Sciences, New York University • Victor Goodman, Department of Mathematics, Indiana University • Priscilla E. Greenwood, Department of Mathematics, Arizona State University • Martin Greiner , Information & Communications, Siemens AG • Ernesto Gutierrez-Miravete, Department of Engineering and Science, Rensselaer Polytechnic Institute • Naresh Jain, School of Mathematics, University of Minnesota • Siwei Jia, Department of Statistics, Oregon State University • Yu-Juan Jien, Department of Mathematics, Purdue University • Yoon Mo Jung, School of Mathematics, University of Minnesota • G. Kallianpur, Department of Statistics, University of North Carolina • Rolf Moritz Kassmann, Department of Mathematics, University of Connecticut • Markus Keel, School of Mathematics, University of Minnesota • Djivede Kelome, Department of Mathematics and Statistics, University of Massachusetts • Eun Heui Kim, Department of Mathematics, California State University, Long Beach • Kyounghee Kim, Department of Mathematics, Indiana University • Panki Kim, Department of Mathematics, University of Washington • Vassili N. Kolokoltsov, School of Computing and Technology, Nottingham Trent University • Yuriy Kolomiyets, Department of Mathematical Sciences, Kent State University • Robert Krasny, Department of Mathematics, University of Michigan • Yves LeJan, Departement de Mathematiques, University Paris Sud

LIST OF WORKSHOP PARTICIPANTS

261

• Seung Lee, Department of Mathematics, Ohio State University • Guang-Tsai Lei, Physiology and Bio-Physics, Mayo Clinic • Runchang Lin, Department of Mathematics, Wayne State University • Yuping Liu, Department of Mathematics, Purdue University • Kening Lu, Department of Mathematics, Michigan State University • Mukul Majumdar, Department of Economics, Cornell University • Rogemar Mamon, Department of Statistics and Actuarial Science, University of Waterloo • Sylvie Meleard, UFR Segmi, Universite Paris X • Oana Mocioalca, Department of Mathematics, Purdue University • Salah Mohammed, Department of Mathematics, Southern Illinois University • Charles M. Newman, Department of Mathematics, New York University • Mahdi Nezafat, Department of Electrical and Computer Engineering, University of Minnesota • Keith Nordstrom, C4-CIRES, University of Colorado , Boulder • Chris Orum, Department of Mathematics, Oregon State University • Mina Ossiander, Department of Mathematics, Oregon State University • Chetan Pahlajani, University of Illinois Urbana-Champaign • Veena Paliwal, Department of Mathematics, Southern Illinois University • Jun Hyun Park, Talbot Laboratory, University of Illinois - UrbanaChampaign • Cecile Penland, NOAA-CIRES, University of Colorado • Lea Popovic, Department of Statistics, University of California Berkeley • Jorge M. Ramirez, Department of Mathematics, Oregon State University • Vivek Ranjan, Department of Mathematics, Indiana University • Marco Romito, Dipartimento di Matematica, Universita di Firenze • Boris Rozovskii, Department of Mathematics, University of California - Los Angeles • David Saunders, Department of Mathematics, University of Pittsburgh • Michael Scheutzow, Fakultiit Il, Institut fur Mathematik Technische, Universitiit Berlin • Bjorn Schmalfuss, Mathematical Institute, University of Paderborn • Rongfeng Sun, Courant Institute of Mathematical Sciences, New York University

262

LIST OF WORKSHOP PARTICIPANTS

• Li-Yeng Sung, Department of Mathematics, University of South Carolina • Michael Tehranchi, Department of Mathematics, University of Texas, Austin • Enrique Thomann, Department of Mathematics, Oregon State University • Ilya Timofeyev, Department of Mathematics, University of Houston • Daniell Toth, Department of Mathematics, J uniata College • Hao Wang, Department of Mathematics, University of Oregon • Jing Wang, The Spectacle Lens Group of Johnson and Johnson • Li Wang, Department of Probability and Statistics, Michigan State University • Lixin Wang, Operations Research and Financial Engineering, Princeton University • Edward C. Waymire, Department of Mathematics, Oregon State University • Hans Weinberger, School of Mathematics, University of Minnesota • Andrew Westmeyer, Department of Mathematics, University of Wyoming • Wojbor A. Woyczynski, Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University • Jian Yang, Department of Mathematics, University of Illinois Urbana-Champaign • Zhihui Yang, Department of Mathematics , University of Maryland • Aaron Nung Kwan Yip, Department of Mathematics, Purdue University • Toshio Yoshikawa, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong • Jianfeng Zhang, School of Mathematics, University of Minnesota • Tao Zhang, Department of Mathematics, Purdue University • Yongcheng Zhou, Department of Mathematics, Michigan State University