Random Perturbation of PDEs and Fluid Dynamic Models: École d'Été de Probabilités de Saint-Flour XL - 2010

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris Subseries: École d’Été de Probabilités de ...

Author: Franco Flandoli

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris Subseries: École d’Été de Probabilités de Saint-Flour

2015

Saint-Flour Probability Summer School

The Saint-Flour volumes are reflections of the courses given at the Saint-Flour Probability Summer School. Founded in 1971, this school is organised every year by the Laboratoire de Mathématiques (CNRS and Université Blaise Pascal, Clermont-Ferrand, France). It is intended for PhD students, teachers and researchers who are interested in probability theory, statistics, and in their applications. The duration of each school is 13 days (it was 17 days up to 2005), and up to 70 participants can attend it. The aim is to provide, in three highlevel courses, a comprehensive study of some fields in probability theory or Statistics. The lecturers are chosen by an international scientific board. The participants themselves also have the opportunity to give short lectures about their research work. Participants are lodged and work in the same building, a former seminary built in the 18th century in the city of Saint-Flour, at an altitude of 900 m. The pleasant surroundings facilitate scientific discussion and exchange. The Saint-Flour Probability Summer School is supported by: – Université Blaise Pascal – Centre National de la Recherche Scientifique (C.N.R.S.) – Ministère délégué à l’Enseignement supérieur et à la Recherche For more information, see back pages of the book and http://math.univ-bpclermont.fr/stflour/ Jean Picard Summer School Chairman Laboratoire de Mathématiques Université Blaise Pascal 63177 Aubière Cedex France

Franco Flandoli

Random Perturbation of PDEs and Fluid Dynamic Models ´ ´ e de Probabilités Ecole d’Et´ de Saint-Flour XL – 2010

123

Franco Flandoli University of Pisa Department of Applied Mathematics Via Buonarroti 1 50127 Pisa Italy [email protected]

ISBN 978-3-642-18230-3 e-ISBN 978-3-642-18231-0 DOI 10.1007/978-3-642-18231-0 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011923078 Mathematics Subject Classification (2011): 60H15, 60H10, 60J65, 35R60, 35Q35, 35B44, 76BO3 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface Regularization by Noise

The most obvious interpretation of the title Random Perturbations of PDEs is Stochastic Partial Differential Equations (SPDEs). This is not wrong but the emphasis is that we start from a PDE and want to investigate the changes produced by random perturbations. Although it would be of great interest to discuss perturbation of several qualitative properties and objects (like asymptotic behavior, soliton and other special solutions, and so on), we will concentrate only on the fundamental issue of well posedness. The “normal” behavior is that the PDE is well posed and nothing changes passing to the SPDE, except maybe the technics of proofs. In principle it may also happen that the PDE is well posed and the SPDE is not, but this is not common. Much more interesting is, in my opinion, the case when the PDE is not well posed but the SPDE is well posed. When this happens, we observe what could be called a regularization by noise. Well posedness is not the rule for PDEs arising in fluid dynamics. There are examples of non well posedness and examples where the question is open. Thus regularization by noise in fluid dynamic models would be a very interesting fact, if true. This is the purpose of the research activity reported here. This activity is just at the beginning, since regularization has been proven only for a few simple fluid dynamic models. As a purpose for a series of Saint Flour lectures, trying to prove that noise restores well posedness of a fluid dynamic equation is certainly a very particular aim. Let me justify this choice by saying that: (a) well posedness of 3-dimensional Navier–Stokes equations is one of the millennium Clay Institute problems (see Fefferman [91]), and: (b) we understand interesting and maybe new features of stochastic analysis and stochastic differential equations (ordinary or partial). The lack of well posedness mentioned here is of two types. The main one we shall deal with is lack of uniqueness. The second one, that we address only very partially, is the emergence of singularities. To some extent, the latter phenomenon is even more interesting and intuitive but we have understood it only very partially, until now. Thus it is better to describe some more clear principles behind “uniqueness by noise” and hope the interest in this subject v

vi

Preface

will drive progresses on the more difficult problem of “interaction between noise and singularities”. Often, useful series of lectures are devoted to the exposition of general techniques that can be applied to a large variety of problems. We completely lack such a purpose: each example we are able to treat requires its own ideas and techniques. But we hope some of them will lead some researcher to try to find new ones, even if this is far from a mechanical application of mathematical methods. I would like to thank all those with whom I shared the research work in recent years and whose contribution was essential for the development of the results described in these notes. Finally, I would like to thank my wife Marta, who has more patience than I and she had thought.

Contents

1

Introduction to Uniqueness and Blow-Up . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Non Uniqueness in Ordinary Differential Equations . . . . . . . . . . . . . 1 1.2 Non uniqueness in Partial Differential Equations .. . . . . . . . . . . . . . . 4 1.2.1 Random Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Definitions of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Deterministic ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Stochastic Differential Equations .. . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Zero-Noise and Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Examples of Blow-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2

Regularization by Additive Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Regularization of Functions by Noise: Occupation Measure . . . . 2.1.1 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Is x → µT,x+W (ω) Continuous in Total Variation? . . . . . . . 2.1.3 An Estimate for H¨ older Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 An Estimate for Lp Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Summary on Occupation Measure . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Regularization of SDE by Additive Noise . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Proof of Theorem 2.7 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stochastic Flow of Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . 2.3 Infinite Dimensional Equations with Additive Noise . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Infinite Dimensional Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Uniqueness in Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Pathwise Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Finite Dimensional Ornstein–Uhlenbeck and Kolmogorov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Proof of Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 20 21 25 29 31 31 34 37 40 40 42 46 53 56 62

vii

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3

Dyadic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction: 3D Euler Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Fourier Formulation of Euler Equations . . . . . . . . . . . . . . . . . . 3.2 The Dyadic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Deterministic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Anomalous Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Numerical Picture of the Anomalous Dissipation . . . . . . . . 3.3.4 Examples of Non-Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Uniqueness of Positive Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Summary and Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Random Perturbation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Preliminary Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Definitions and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Auxiliary Linear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Girsanov Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Proof of Theorem 3.4 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Proof of Theorem 3.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 74 77 77 78 82 83 84 86 87 87 89 92 96 96 99

4

Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Linear Transport: Structural Approach .. . . . . . . . . . . . . . . . . . 4.2 Deterministic Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Lipschitz Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Weakly Differentiable Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stochastic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Renormalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Representation of the Law of u (t, x) . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Analogy with Stabilization by Noise . . . . . . . . . . . . . . . . . . . . . . 4.4 Uniqueness by Stochastic Characteristics.. . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Stochastic Flow of SDE with Non Regular Drift . . . . . . . . . 4.4.2 Proof of Uniqueness Using the Flow: Introduction. . . . . . . 4.4.3 Distributional Commutator Lemma . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Convergence of the Commutator Composed with the Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Final Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 103 104 104 105 113 113 115 117 121 122 123 123 123 125

Other Models: Uniqueness and Singularities . . . . . . . . . . . . . . . . . . . 5.1 Negative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Linear Transport Equation with Random Coefficients. . . 5.1.2 Euler Equation with Too Simple Noise . . . . . . . . . . . . . . . . . . . 5.1.3 Inviscid Burgers Equation: Non Uniqueness . . . . . . . . . . . . . . 5.1.4 Inviscid Burgers Equation: Blow-Up . . . . . . . . . . . . . . . . . . . . . .

133 133 133 134 136 137

5

127 130

Contents

5.2 Positive Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Stochastic Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Point Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Singularities of Stochastic Schr¨ odinger Equation . . . . . . . . . . . . . . . . 5.4 Additive Noise in 3D Navier–Stokes Equations . . . . . . . . . . . . . . . . . . 5.4.1 About Singularities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Markov Selections and Strong Feller Property . . . . . . . . . . . 5.5 Conclusions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 SPDEs: The Effects of Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 What We Have Learned on the Effect of Noise on Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Other Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

139 139 143 149 152 153 154 155 155 156 158

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter 1

Introduction to Uniqueness and Blow-Up

1.1 Non Uniqueness in Ordinary Differential Equations We need to keep in mind a number of examples of non-uniqueness for ODEs, first. Let us start with probably the most famous one. Example 1.1. Consider the function b (x) = 2sign (x) |x| and the 1-dimensional ODE dXt = b (Xt ) , dt

X0 = x0 .

The function b is regular for x = 0 and one can prove that the ODE has a unique global solution for every x0 = 0. But for x0 = 0 there are infinitely many solutions. One of them is Xt = 0. Other two ones, plotted in Fig. 1.1, are (±) Xt = ±t2 . One can show that all other solutions have the following form: they are equal 2 to zero on some interval [0, t0 ] and equal to ± (t − t0 ) for t ≥ t0 . The solu(±) are, in a sense, extremal among all possible solutions. The set tions Xt C (x0 ) of solutions is compact in the locally uniform topology. In general, in one dimension, if b is continuous and autonomous, and b (x0 ) > 0 (the case b (x0 ) < 0 is similar) then, at least locally in time, there is a unique solution with initial condition x0 . Indeed, if Xt is a local solution, (necessarily continuous by definition), we have b (Xt ) = 0 on some interval [0, t0 ), by continuity. Then 1 dXt =1 b (Xt ) dt on [0, t0 ). We integrate this identity on any interval [0, t] ⊂ [0, t0 ), change variable and get

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Lecture Notes in Mathematics 2015, DOI 10.1007/978-3-642-18231-0 1, c Springer-Verlag Berlin Heidelberg 2011

1

2

1 Introduction to Uniqueness and Blow-Up

y 4 2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

x

–2

–4 Fig. 1.1 Extremal solutions to Xt = 2sign (Xt )

Xt

x0

|Xt |, X0 = 0

1 dx = t b (x)

which identifies uniquely Xt since the function Hx0 (r) :=

r x0

1 dx b (x)

is strictly increasing. We have the “explicit” formula Xt = Hx−1 (t). 0 Therefore, for continuous functions b in dimension 1, non-uniqueness may happen only when b vanishes at some point x0 where b is also not continuously differentiable. For “usual” functions b this may happen only at isolated points. But less trivial functions b may have infinitely many such points, with accumulations. Example 1.2. Let (Wt )t∈R be a two-sided Brownian motion on a probability space (Ω, F, P ) (a process such that (Wt )t≥0 and (W−t )t≥0 are two independent Brownian motions). Let α ∈ (0, 1) be given. Given ω ∈ Ω, consider the function x → bω (x) defined as bω (x) = |Wx (ω)|α ,

x ∈ R.

Except for ω in a P -null set N ∈ F , the set Z (ω) of all zero’s of the function bω (·) Z (ω) = {x ∈ R : bω (x) = 0} is a non empty perfect set (closed and without isolated points, see Revuz and Yor [177] and Pratelli [173] for further informations), with zero-Lebesgue (−) measure. If x0 ∈ Z (ω), the constant function Xt = x0 is a solution. But the function Hx0 (r) above, or more precisely

1.1 Non Uniqueness in Ordinary Differential Equations

Hx0 ,ω (r) :=

r

x0

3

1 dx bω (x)

is still well α ∈ (0, 1): for every T > 0, it is elementary to prove defined when T 1 that E −T |Wt |α dt < ∞; we have to exclude maybe more ω’s, from a larger

P -null set N ∈ F . The function Hx0 ,ω is strictly increasing (since Z (ω) has zero-Lebesgue measure) and continuous, hence we may set (+)

Xt

:= Hx−1 (t) 0 ,ω

and verify that this is another solution of the ODE (notice in particular that Hx−1 is continuously differentiable, even if Hx0 ,ω is not: at points where the 0 ,ω derivative of Hx0 ,ω does not exist in the usual sense of being a real number, in fact it is equal to +∞ and the derivative of Hx−1 at the corresponding 0 ,ω point exists and is equal to zero). It is the maximal one, of a large family that one can guess. T Remark 1.1. One has P 0 |W1t |α dt = ∞ = 1 for α ≥ 1, see Engelbert and Schmidt [87], Karatzas and Shreve [129] Lemma 6.26. In a preliminary version of these notes the author made a mistake about this fact, corrected thanks to the careful reading and insight of Francesco Caravenna. If we do not ask continuity, we may construct strange examples more easily. First, we may have a sharper separation of trajectories. Example 1.3. Let b (x) = sign (x), x0 = 0. The extremal solutions are (±) Xt = ±t. Then we can easily produce examples with splitting from any initial condition in arbitrarily short times. Example 1.4. Let b (x) be the Dirichlet function b (x) =

0 for x ∈ Q . 1 for x ∈ RQ (−)

From any initial condition x0 ∈ Q we have the minimal solution Xt = x0 (+) and the maximal one Xt = x0 + t. From any x0 ∈ RQ we have at least (+) the solution Xt = x0 + t. But, from x0 ∈ RQ, we may also consider, for every t0 > 0 such that x0 + t0 ∈ Q, the solution equal to x0 + t on [0, t0 ], and then equal to the constant x0 + t0 on [t0 , ∞). The splitting time t0 can be arbitrarily small. Hence, even from x0 ∈ RQ, we cannot say that locally, namely on some time interval [0, ε), there exists a unique solution.

4


In dimension d > 1 the non-uniqueness may be wilder. Even when b is continuous, we do not know easy criteria like b (x0 ) = 0 which guarantee uniqueness. For instance, by a simple 2D modification of Example 1.1 above and a rotation, one can construct a 2-dimensional continuous example without uniqueness from a point x0 such that b (x0 ) = (1, 1).

1.2 Non uniqueness in Partial Differential Equations Let us start with a straightforward extension of the previous examples. Example 1.5. Consider the parabolic equation 1 ∂u (t, x) = Δu (t, x) + b (u (t, x)) , ∂t 2

t ≥ 0,

x ∈ Rd

with initial condition u|t=0 = u0 , where b : R → R is a given function. Assume we investigate L∞ -solutions (with some regularity, but no decay at infinity in space). Assume b (x0 ) = 0 at some point x0 . If u0 (x) ≡ x0 , then the function u (t, x) ≡ x0 is a solution. When b is like in the examples of the previous section, we easily find other solutions. For instance, if b (x) = 2 |x|, then u (t, x) = t2 is a solution from the initial condition u0 (x) ≡ 0 different from the solution u (t, x) ≡ 0. Examples of this kind are of interest for conceptual reasons, and maybe for some application, but not for fluid dynamics. Most fluid dynamic equations have only a quadratic non-linearity, nothing like a square root. But quadratic nonlinearities in infinite dimensions may cause non-uniqueness, for topological reasons which are more subtle than the non-locally Lipschitz examples above. A fluid dynamic equation of central interest is Euler system: ∂u + (u · ∇) u + ∇p = 0, ∂t

div u = 0

(see Chap. 3 for some detail). There exist very non-trivial examples of nonuniqueness for this equations, which originate from causes of different nature with respect to those of the square root example above. We shall try to explain one of them in a simplified example called dyadic model. It is an infinite system of 1-dimensional differential equations, coupled only through certain nearest neighbor terms: dXn (t) 2 = kn−1 Xn−1 (t) − kn Xn (t) Xn+1 (t) dt with n = 1, 2, ..., X0 (t) = 0, k0 = 0, and kn given for instance by kn = 2n . The multiplication by kn is a sort of differentiation in Fourier components,

1.2 Non uniqueness in Partial Differential Equations

5

but this system is only a model of Euler equation, it is not the Fourier formulation of it or even a simplified form of it. Due to its simplicity we can prove the existence of initial conditions with more than one solution and we can understand the “physical” origin of the lack of uniqueness. Having in mind more difficult nonlinear equations of fluid dynamics (like Euler equations or hyperbolic conservation laws), it is perhaps useful to investigate first the linear, so called transport, equation ∂u + (b · ∇) u = 0, ∂t

u|t=0 = u0

to get some feeling about possible troubles arising from transport terms. The vector field b here is given, it does not depend on the solution. When b is sufficiently regular, it is an easy and well posed equation, whose solution is given by

u (t, x) = u0 ϕ−1 t (x) t where ϕt is the flow associated to the ODE of characteristics, dX dt = b (Xt ). But when b is less regular, exactly as in the case of the ODEs of the previous section, troubles start to appear. Example 1.6. Consider again Example 1.1, b (x) = 2sign (x) |x| in one space dimension. Consider the initial condition u0 = 1x>0 . Any solution u (t, x), at given time t, will be equal to 1 above the upper extremal solution (+) (+) Xt , namely for x > Xt , and will be equal to 0 below the lower extremal (−) (−) (+) , there is a solution Xt . But in between, namely for x ∈ Xt , Xt large degree of indeterminacy. For instance, if we set u (t, x) equal to 1 for

(−)

x ∈ Xt

(+)

, Xt

and all t ≥ 0, this function is a solution; call it u(+) : u(+) (x, t) = 1{x≥t2 } .

If we set it equal to 0 for x ∈

(−) (+) Xt , Xt and all t ≥ 0, it is again a

solution, that we call u(−) : u(−) (x, t) = 1{x>−t2 } . The solutions u(+) and u(−) are not extremal as in the 1-dimensional ODE (−) (+) case: we can also put u equal to +2, for instance, in Xt , Xt , for all t > 0. See Sect. 4.2.2.1 of Chap. 4 for more details. Notice that, being the PDE linear, when we have an initial condition with a multiplicity of solutions, the same happens for all initial conditions. Nonuniqueness hold for all initial conditions. The first nonlinear generalization of linear transport equations are perhaps the hyperbolic conservation laws. Let us describe an example of

6


The solution u(−) (x, t) = 1{x>−t2 }

The solution u(+) (x, t) = 1{x≥t2 }

non-uniqueness for the (inviscid) Burgers equation. Let us advise of a possible misunderstanding: opposite to Examples 1.1–1.6 that will be regularized by suitable noise, the next example is not (we do not know how to regularize it by noise). But we include it because we may perform explicit computations, useful to understand what happens, although having a negative final result. Example 1.7. Consider the equation ∂u ∂u +u = 0, ∂t ∂x

u|t=0 = u0

in one space dimension. Take u0 = 1x>0 . The solution is uniquely identified by the initial condition where characteristics travel in a bijective way, namely for x ≤ 0 and x > t (see Sect. 5.1.4 of Chap. 5 for more details). The solution u (t, x) is equal to zero for x ≤ 0 (the characteristics here are the lines x = const) and, writing any x > t in the form x0 + t with x0 > 0 (these lines are the characteristics in the region x > t), we have u (t, x0 + t) = u0 (x0 ) = 1. But for 0 < x ≤ t there is a large degree of indeterminacy. For instance, one has the two solutions u1 (t, x) = 1x>0 and u2 (t, x) = 1x>t. Let us also remark, however, that uniqueness is restored by a selection principle: the so-called entropy solutions are unique, see for instance Lax [146].

1.2.1 Random Perturbations The question is: are there intuitive reasons to believe that random perturbations could restore uniqueness, for ODEs or PDEs which do not have it? Two research lines on this problem are known. One of them simply considers random perturbations of initial conditions. The initial condition is a random variable, and one looks for a theorem of uniqueness for a.e. value of this random variable. The intuition behind this approach is that

1.2 Non uniqueness in Partial Differential Equations

7

non-uniqueness, in certain examples, is an uncommon phenomenon, which may hold for special initial conditions but not generically. This is the case in Examples 1.1, 1.2 (even if here the set of bad initial conditions is quite rich), 1.3, and likely 1.5. But it is not the case in Examples 1.4 and 1.6. It is presumably true for the dyadic model, but we do not have a proof. What happens for the dyadic model is common to other difficult examples: one has instances of initial conditions with more than one solution, but how large is the set of such initial conditions is unknown. A variant of the idea of random initial condition is the notion of Lagrangian flow, also described below. The second main approach to random perturbations is the one given by stochastic differential equations, SDEs or SPDEs. For instance, a typical question is: if (Wt )t≥0 is a Brownian motion in Rd defined on a filtered probability space (Ω, Ft , P ) (by a filtered probability space (Ω, Ft , P ), on a finite time horizon [0, T ], we mean a probability space (Ω, FT , P ) and a right-continuous filtration (Ft )t∈[0,T ] ) and b : [0, T ] × Rd → Rd is a vector field with some degree of regularity but less than locally Lipschitz continuous, does the SDE dXt = b (t, Xt ) dt + σdWt ,

X0 = x0 ∈ Rd

(1.1)

have a unique solution? Here σ is a positive real number. The intuitive reason is initially the same one as in the case of random initial conditions: there could exist special initial conditions with branching, or special trajectories that branch sometimes, but this is a sort of rare event, these trajectories are somewhat unstable. Even a very small random perturbation move outside them and restores uniqueness. This idea is certainly a good first approximation, but we shall see that: • •

Looking closer to the motion around branching solutions, we realize that it is not so obvious that noise moves away, in the most na¨ıve fashion. There is more, because suitable noise restores uniqueness also in Examples 1.4 and 1.6 above, where branching is everywhere!

About the first remark, think to Example 1.1. Very close to t = 0 any solution fluctuates like Brownian motion (the drift contribution is infinitesimal compared to Brownian motion, see also Fig. 1.2). Then any solution, with initial condition equal to 0, in any short time interval [0, ε] crosses the axes x = 0 infinitely many times. We cannot say that, due to noise, we immediately move apart from the dangerous region. In a sense, we move apart and we come back, infinitely may times. Is it sure that in this up-and-down motion around x = 0 there is no possibility of splitting? A good intuitive idea is that, even if the trajectory crosses x = 0 infinitely many times, the time spent at x = 0 is negligible. The concept of occupation measure is a way to capture this fact. Less easy is the intuition when splitting occurs at every point.

8


Fig. 1.2 Random solution close to the origin

2

W

1

0

–1 0

200

400

600

800

1000

Time

1.3 Definitions of Uniqueness In this section we give the relevant definitions in the finite dimensional case. The analogous definitions for PDEs and SPDEs will be given for each specific model at due time.

1.3.1 Abstract Scheme The following abstract scheme may help to classify different concepts. Let (X, d) be a set with its Borel σ-field, (Ω, F , P ) a probability space, F : X × Ω → R a measurable map. Consider the equation depending on the parameter ω F (x, ω) = 0 where x is the unknown. We have in mind two main examples, discussed below: when ω is the initial condition of a differential equation, and when ω is the random parameter of a stochastic differential equation. For each ω ∈ Ω, denote by S (ω) ⊂ X the set of all solutions. Let us list a few concepts of uniqueness. 1. It may happen that S (ω) has at most one element, for every ω ∈ Ω. 2. It may happen that S (ω) has at most one element, for P -almost every ω ∈ Ω. 3. It may happen that, when x(i) : Ω → X, i = 1, 2, are measurable mappings such that P ω ∈ Ω : F x(i) (ω) , ω = 0 = 1

1.3 Definitions of Uniqueness

9

(we may call them measurable selections) then P ω ∈ Ω : x(1) (ω) = x(2) (ω) = 1. 4. It may happen that, when x(i) : Ω → X, i = 1, 2, are measurable selections, then the image laws of x(1) and x(2) on X coincide. The first two notions, 1 and 2, are ω-wise (we say pathwise in certain applications) notions of uniqueness. The second ones, 3 and 4, involve measurable selections x(i) : Ω → X, i = 1, 2 and do not claim anything about S (ω) for single values of ω ∈ Ω (in fact 3 and 2 are quite close). We could call 3 strong stochastic uniqueness. Notion 4 could be called weak stochastic uniqueness but it is different from uniqueness in law defined below for SDE’s, where the

probability space Ω(i) , F (i) , P (i) may depend on the solution (we cannot give such general concept at the abstract level). Moreover, we advise that there is not a precise correspondence between these general concepts and the definitions of uniqueness below, also because of essential details as the progressive measurability imposed below; so the discussion here is only to point out certain structural differences. Condition 1 implies 2, which implies 3, which implies 4 (the relation 3 ⇒ 4 in the case of uniqueness in law for SDE’s is deeper and given by Yamada–Watanabe theorem). The converse implications do not hold in general. About the converse, under some additional assumptions, condition 3 (or a variant where x(1) (ω) = x(2) (ω) for all ω ∈ Ω) could imply something about 1 and 2, but this kind of argument does not solve the problem mentioned at the end of Sect. 1.3.3 below.

1.3.2 Deterministic ODEs Consider the ODE in Rd dXt = b (t, Xt ) , dt

X0 = x0 ∈ Rd

(1.2)

where b : [0, T ] × Rd → Rd is a measurable vector field. We insist that b is a given function, not an equivalence class, otherwise we should investigate the meaning of the composition b (t, Xt ) (at the end, one should understand if the results depend on modifications of b). In integral form the equation simply is Xt = x0 +

t 0

b (s, Xs ) ds,

t ∈ [0, T ] .

Given x0 , we call solution any continuous function X : [0, T ] → Rd such that

10


T 0

|b (s, Xs )| ds < ∞

and the integral equation is satisfied for all t ∈ [0, T ]. A posteriori, since t → t b (s, Xs ) ds has some regularity (for instance it is absolutely continuous), 0 the same regularity holds true for solutions. The definition of solution on a local interval [0, ε) is the same. Less obvious is a definition of generalized flow solution. Let us give the definition of regular Lagrangian flow (see DiPerna and P.L. Lions [81], Ambrosio [8], Ambrosio and Crippa [9]). It is a measurable function (t, x0 ) → Xtx0 , from [0, T ] × Rd to Rd , such that: (a) For Lebesgue a.e. x0 ∈ Rd , the function t → Xtx0 is a solution in the integral sense above. (b) For a.e. t ∈ [0, T ], the function x0 → Xtx0 maps Lebesgue measure into an image measure μt which is absolutely continuous with respect to Lebesgue measure, with bounded density. This definition was successful to solve uniquely certain classes of equations. Let us give different notions of uniqueness; notice the similarity with 1, 2 and 3 of Sect. 1.3.1. Definition 1.1. (i) We say there is uniqueness for the ODE (1.2) when there is a unique continuous solution, for every given x0 ∈ Rd . (ii) We say there is almost everywhere uniqueness when the same is true for Lebesgue a.e. x0 ∈ Rd . (iii) We say there is uniqueness of regular Lagrangian flows when any two tx0 have the property such flows Xtx0 and X x0 for all t ∈ [0, T ] Xtx0 = X t for Lebesgue a.e. x0 ∈ Rd . Condition (i) implies (ii), which implies (iii), but the converse is not true.

1.3.3 Stochastic Differential Equations Let us consider now the following stochastic differential equation in Rd dXt = b (t, Xt ) dt +

∞

σk (t, Xt ) dWtk ,

X0 = x0

k=1

where (Wtk )t≥0 , k ∈ N, is a sequence of independent real valued Brownian motions defined on a filtered probability space (Ω, Ft , P ), and b, σk : [0, T ] × Rd → Rd , k ∈ N, are measurable vector fields. We understand this equation

1.3 Definitions of Uniqueness

11

in integral form Xt = x0 +

t 0

b (s, Xs ) ds +

∞

k=1

t 0

σk (s, Xs ) dWsk ,

t ∈ [0, T ]

(1.3)

where the last integrals are Itˆ o integrals, when well defined. Definition 1.2. A continuous Ft -adapted process X is called a solution when T ∞ T

|b (s, Xs )| ds < ∞, |σk (s, Xs )|2 ds < ∞ 0

k=1

0

with probability one and the integral equation (1.3) is satisfied for all t ∈ [0, T ], with probability one. If X is adapted to the completion of the natural filtration associated to the family Wtk t≥0 , k ∈ N, then we say that X is a strong solution. About uniqueness, we use the following definitions. Definition 1.3. We say we have pathwise uniqueness for the SDE (1.3) when, given any filtered probability space Ft , P ), any sequence of inde (Ω, pendent real valued Brownian motions Wtk t≥0 , k ∈ N, on (Ω, Ft , P ), given

x0 ∈ Rd and any two solutions X (i) , i = 1, 2 of (1.3) with respect to these (1) (2) data, we have Xt = Xt for all t ∈ [0, T ], with probability one.

Definition 1.4. We say we have weak uniqueness for the SDE (1.3) when, d for every respect to any (Ω, Ft , P )

k x0 ∈ R , any two solutions (defined with

and Wt t≥0 , k ∈ N) have the same law on C [0, T ] ; Rd . Sometimes we shall say strong uniqueness instead of pathwise uniqueness, and uniqueness in law instead of weak uniqueness. Let us also give a less classical definition. Let us restrict the attention to a very simple class, to avoid a discussion of the meaning of stochastic integrals for given single paths. Thus let us concentrate on the simple (1.1) with additive noise. The theory about this notion is so small, that we do not know whether it is better to prescribe x0 ∈ Rd before or after ω, and thus for sake of simplicity we give it a priori. Let x0 ∈ Rd be given. Assume b is continuous and bounded. Given ω ∈ Ω such that the path Wt (ω) is continuous, there exists at least one continuous function y such that yt = x0 +

t 0

b (s, ys ) ds + Wt (ω) .

The proof is the same as the one of classical Peano theorem. Denote by S (x0 , ω) the set of all such solutions y. The set S (x0 , ω) is non empty and compact in the locally uniform topology. The question is whether this set is

12


a singleton or not, for P -a.e. ω ∈ Ω. Let us express the definition even in the case of more general b (namely when a priori we do not know that S (x0 , ω) is non empty). Definition 1.5. We say we have path-by-path uniqueness for the Cauchy problem (1.1) when there exists a full P -measure set Ω0 ⊂ Ω such that for all ω ∈ Ω0 the following statement is true: there exists at most one continuous T function yt , t ∈ [0, T ], which satisfies 0 |b (s, ys )| ds < ∞ and yt = x0 +

0

t

b (s, ys ) ds + Wt (ω)

for all t ∈ [0, T ]. This concept is, a priori, stronger than strong uniqueness. Even if we do not know counterexamples, at least conceptually it is much stronger than strong uniqueness: indeed, the uniqueness of strong solutions means the P -a.s. (i) coincidence between any two families Xt (ω), i = 1, 2, both having special measurability properties. Path-by-path uniqueness requires that a single ω is prescribed a priori (no measurable family with respect to ω), and for such ω there is only one solution y; this property must hold at least for a.e. ω ∈ Ω. Let us make a conceptual comparison. Compare the deterministic theory and the stochastic one by looking at the role of Wt (ω) in the latter similarly to the role of x0 in the former. This apparently strange association (Wt (ω) ↔ x0 ) is motivated by the remarks of the previous section, where we argued about random perturbations of only the initial conditions with respect to noise perturbations of the equation itself. Under this association, we see that the notion of regular Lagrangian flow and its uniqueness is in the direction of the notion of strong solution and pathwise uniqueness. While the notion of path-by-path uniqueness is in the direction of the more classical notion of almost everywhere uniqueness for the ODE (1.2). See Sect. 1.3.1 for an abstract version of these ideas. In this sense, perhaps the notion of path-bypath uniqueness could deserve more investigation. The proof of path-by-path uniqueness results (of course outside locally Lipschitz assumptions) looks at present very difficult; see Davie [68]. Let us state the problem in a more general form, which shows it is in a sense a fundamental problem about ODEs. Consider the equation in Rd yt = x0 +

0

t

b (s, ys ) ds + γt

where γ is a continuous function. If b is Lipschitz in y, uniformly in t, with uniform linear growth, there is existence and uniqueness of solutions; for uniqueness, locally Lipschitz is sufficient. But when b is not locally Lipschitz, could we have uniqueness if γ has special properties? Which paths γ regularize the equation, in the sense of these lectures? Davie result [68] states that an

1.4 Zero-Noise and Selection

13

L∞

field b is regularized (in the sense that uniqueness holds), by all γ ∈ Ω0 ⊂ C [0, T ] ; Rd , where Ω0 is a set of full Wiener measure. Strong uniqueness of SDEs is another kind of result. It would be very interesting to have analogs of Davie result for single deterministic paths γ, having special properties. The intuition is that γ must be strongly spread. In the language of occupation measure introduced in Chap. 2, the occupation measure of γ should be very diffused. Maybe the concept of maximally curved path used by Tao and Wright [192] could be relevant (this potential relation has been pointed out to the author by T. Lyons).

1.4 Zero-Noise and Selection One of the most interesting problems related to the subject of these lectures is the zero-noise limit when the deterministic equation has more solutions and the stochastic equation has only one solution (for the fundamental classical case see Friedlin and Wentzell [110]). The hope is that it produces an interesting selection criterion for the deterministic equation. This program is very difficult and only very few examples are known. The main one is the case of 1-dimensional equations treated by Bafico and Baldi [16, 17], developed further by Gradinaru et al. [120] (large deviations) and extended by Attanasio and Flandoli [14] and Attanasio [13] to linear PDEs of transport type. See also Menoukeu-Pamen et al. [159] and references therein. Zero-noise selection for nonlinear PDEs is of course even more difficult. A remarkable example in the field of geometric motions by mean curvature, is the result by Dirr et al. [82] on the selection with positive probability of different evolutions, in a problem of fattening; and see also the remarkable uniqueness-by-noise result of Souganidis and Yip [189] on this problem; on this subject see also Weber [197]; in another direction, zero-noise and entropy solitions for conservation laws have been treated by Mariani [156]. Let us see the result of [16] on Example 1.1. First, as we shall see later on, the stochastic equation dXt = b (Xt ) dt + εdWt ,

X0 = x0

has a unique solution, both in law and pathwise, for all ε > 0, when b is Hölder continuous with some control at infinity, which includes b (x) = 2sign (x) |x|. In particular the solution is unique for x0 = 0. Call Pε the law on C ([0, T ] ; R) of this solution (x0 = 0). It is not difficult and a general fact for many SDE that the family (Pε )ε>0 is tight, hence by Prohorov theorem it has subsequences which converge weakly to measures P on C ([0, T ] ; R), and such probability measures P are carried by the set of all solutions of the ODE with x0 = 0.

14


The non trivial zero-noise result is that the full family Pε weakly converges, or that there is only one limit measure P . This is proved by Bafico and Baldi [16] for a class of Hölder continuous functions b having an isolated singular point at some x0 (smooth outside). The measure P selects suitable solutions of the ODE, always extremal. Sometimes it is carried by both the extremal solutions, sometimes by only one of them. And when it is carried by both, the weight given to each of them is not trivial and computable. In the simple, symmetric, Example 1.1, P is given by P =

1 1 δ (+) + δX (−) . 2 X 2

Elaborating this fact, for Example 1.6 of linear transport equation, Attanasio and Flandoli [14] prove that in the zero-noise limit the two solutions u− (t, x) and u+ (t, x) are selected, similarly to the one-dimensional case. Following a terminology used in the finite dimensional case, see Ambrosio [8] (see also Flandoli [96]), one may call superposition solutions of the deterministic Cauchy problem (1.2) all probability measures μ on C ([0, T ] ; R) such that μ (C (u0 )) = 1, where C (u0 ) is the set of all solutions of the ODE. One could also call true superposition solution a superposition solution which is not a delta Dirac. The previous result states that the limit in law of solutions to the stochastic equation, in the example above, is a true superposition solution. If we accept the general viewpoint that the ‘physical’ objects (solutions, invariant measures, see Eckmann and Ruelle [85]), in case of non uniqueness, are those obtained in the zero-noise limit, then we see that true superposition solutions are the right objects for certain ODEs.

1.5 Examples of Blow-Up The examples of equations having or suspected to have blow-up are very many. To catch the variety of phenomena, see Eggers and Fontelos [86]. Several examples are related to fluid dynamics, to some extent. What happens to singularities, blow-up, when there is noise in the system? Certain kinds of blow-up are so “monotone”, robust, that noise does not interferes. This happens in 1D inviscid Burgers blow-up of derivatives, when the initial condition has a region of negative slope and the noise is of a class which was regularizing for other equations. Sometimes else, noise may prevent blow-up. This may happen when blowup is a special phenomenon that requires a great degree of organization, geometric perfection, unstable under perturbations. We have detected two examples where this happens (with no general ideas or theory behind them, except the similarity of the noise). Thus let us describe these two deterministic examples, that we shall perturb in Chap. 5. More difficult and intriguing is the picture that emerges for Schrödinger equations, see Sect. 5.3.

1.5 Examples of Blow-Up

15

Example 1.8. The continuity equation is dual to the transport equation and thus it is not strange to observe dual phenomena, under the same choice of drift b. Non-uniqueness is dual to non-existence, and blow-up is a special form of non-existence: regular solutions (starting from regular initial data) exists only locally in time, before they blow-up. We may easily create an example of this kind by taking the continuity equation ∂u + div (bu) = 0, ∂t

u|t=0 = u0

with b (x) = −2sign (x)

|x|

(notice the change of sign in the drift with respect to Example 1.6). Trajectot ries of the ODE dX dt = b (Xt ) coalesce in finite time: they are the time-reversal of the trajectories of Example 1.1 which split. Take, for instance, the initial condition u0 (x) equal to zero for x ∈ [−1, 1], positive otherwise, smooth, with u0 (x) dx = 1, thus a probability density. Until t = 1, only the trajectories Xt with initial condition X0 ∈ [−1, 1] coalesce, the others not. At any time t ∈ [0, 1], is a smooth probability density, equal to the solution u (t, x) 2 2 zero for x ∈ − (1 − t) , (1 − t) , outside being the image density of u0 (x) / [−1, 1]. But after t = 1, under the flow map of the ODE restricted to X0 ∈ a positive initial mass reached x = 0 and stay there, if we consider measurevalued solutions: so the solution is still a probability measure but with an atom at x = 0, of increasing mass; otherwise we could trow away the mass which reaches zero and consider function-valued solutions, but they loose the property u (t, x) dx = 1 and may be discontinuous at x = 0 if u0 (x) is not symmetric. In any case, a form of singularity emerges at time t = 1 or after. Example 1.9. Some details of this example require more definitions, so we address Chap. 5 for them. In space dimension 2 the notion of point vortex of an inviscid fluid looks relevant from the physical viewpoint and also for mathematical approximations. A number of reasons (including theorems) tell us that, if we idealize a very concentrated vorticity field ω (t, x) as the sum of finite number of delta Dirac vortices ω (t, x) =

n

i=1

ωi δxit

at points xit ∈ R2 , i = 1, ..., n, then this structure of the vorticity field persists in time and the following system of ODEs holds

dxit = ωj K(xit − xjt ), dt j=i

i = 1, ..., n

16


P3

P1

P2

Fig. 1.3 Initial configuration which collapses in finite time

where K(x) =

x⊥ |x|

2

with the notation x⊥ = (−x2 , x1 ) if x = (x1 , x2 ). The drift of this system is singular at x = 0: if two (or more) point vortices meet, the equations loose meaning and it is not clear what should happen afterwards. Well, there are examples of initial conditions such that this happens in finite time: there is coalescence of some point vortices. An example is provided by three point vortices initially settled at the points P1 = (−1, 0) ,

P2 = (1, 0) ,

√ P3 = 1, 2

with intensities ω1 = 2,

ω2 = 2,

ω3 = −1.

This triangular configuration (Fig. 1.3) evolves in a self-similar way, rotating and collapsing to one point in finite time (the explicit computation can be found in the book of Marchioro and Pulvirenti [155]). We consider this coalescing phenomenon as an emergence of singularity. In strict mathematical terms, it deals with an ODE, but behind there are Euler equations (however, they have no meaning for point vortices with variable sign, so the PDE relevance of this example is not a priori clear).

Chapter 2

Regularization by Additive Noise

Although the final aim of these lectures is to understand the effect of noise on PDE, it is very important to investigate more and more the finite dimensional case. Even more basically, we have to understand how Brownian motion regularizes functions, when it acts on them. Thus, the first section will be devoted to the action of noise on non-regular functions and the concept of occupation measure which, in my opinion, is the deep reason of the strong uniqueness results described later on. At least, it is an interesting intuitive way to understand “geometrically” the regularization by noise. The second section deals with finite dimensional SDE with additive noise. Finally, in the last two sections, we generalize some of the ideas to a class of SPDE with additive noise. The results of the second section on SDE will also be used in Chap. 4 to solve uniquely an SPDE by a method involving characteristics.

2.1 Regularization of Functions by Noise: Occupation Measure 2.1.1 Examples Example 2.1. Let (Wt )t≥0 be a d-dimensional Brownian motion, defined on a filtered probability space (Ω, Ft , P ). Let ϕ : Rd → R be a measurable bounded function. Then u (t, x) = E [ϕ (x + Wt )] ,

t ≥ 0, x ∈ Rd

is smooth in x, for each t > 0. This well known regularization fact can be easily proved by the identity


17

18

2 Regularization by Additive Noise

|z|2 dz u (t, x) = (2πt) ϕ (x + z) exp − 2t Rd 2 |y − x| −d/2 = (2πt) dy. ϕ (y) exp − 2t Rd −d/2

Remark 2.1. Notice that, opposite to other cases developed later, no regularization occurs without the expected value, namely for the function x → ϕ (x + Wt (ω)) at given (t, ω). This example is based on the smooth x-dependence of the image measure N (x, tI) (the d-dimensional Gaussian distribution with mean x and covariance tI). We deal with the image measure (push forward) under the map: ω → x + Wt (ω) Ω → Rd P → N (x, tI) and its dependence on the parameter x. It is a regularity property of the law of Wt , for given t > 0. The fact that x → u (t, x) is continuous (t > 0) when ϕ is only bounded measurable is called Strong Feller property. Problem. Is there some kind of smooth x-dependence of the following image measure μT,x+W (ω) t → x + Wt (ω) [0, T ] → Rd Leb → μT,x+W (ω) at (almost every) given ω ∈ Ω? If yes, this is a regularity property of single paths. We call μT,x+W (ω) the occupation measure of the path x + W (ω) up to time T . In general, given a measurable path γ : [0, ∞) → Rd , we call occupation measure of the path γ up to time T the measure μT,γ on Borel sets of Rd defined as

T

μT,γ (ϕ) =

ϕ (γt ) dt,

ϕ ∈ Cb R d .

0

Remark. One way to capture a form of regularity of a Borel measure in Rd is to shift it and observe it by a non-smooth function; namely, if the measure

2.1 Regularization of Functions by Noise: Occupation Measure

19

is the image law of a map f : (X, B, λ) → Rd , B Rd , we consider the smoothness of the function x → ϕ (x + f (a)) λ (da) , x ∈ Rd , X

under the observable ϕ : Rd → R . This is what is done in Example 2.1, with (X, B, λ) = (Ω, F, P ), and will be done again in the next examples, with (X, B, λ) = ([0, T ] , B ([0, T ]) , Leb). If, at one extreme, the measure is a delta Dirac, when we shift it, it moves continuously in the weak topology (namely when observed by continuous functions), but not in the topology of total variation (namely when observed by bounded Borel functions): when the measure crosses a discontinuity, we observe a jump. On the opposite side, if we shift a non degenerate Gaussian distribution, its motion is extremely smooth even when observed by an L∞ function. Now, what happens to the occupation measure of a curve? If the curve is smooth, we may expect a poor level of regularity; this happens for sure when the curve has zero derivative at some point: the occupation measure concentrates and such concentration points detect discontinuities of L∞ observables. See also Sect. 2.1.5 for an explanation of this point. When, on the contrary, the curve is like the path of a Brownian motion, it is never at rest, its oscillations are very regular in a sense, the measure spreads to some extent and thus we may hope that, when we shift it, we do not see anymore discontinuities. Unfortunately, this hope cannot be uniform in the L∞ observable, as we shall see. Indeed, given a Brownian trajectory, there are L∞ functions which have discontinuity set which “meet” the support of that trajectory; when we shift the Brownian trajectory and its support overlaps the discontinuity set, we loose regularity. But this is very special. In a sense, the usual behavior is very good. Example 2.2. In dimension one, d = 1, μT,x+W (ω) is absolutely continuous with respect to Lebesgue measure, with density given by (we also write intermediate formal expressions to help the intuition) dμT,x+W dμT,x+W (a) = δa (a ) (a ) da = μT,γ (δa ) da da R T T δa (x + Wt ) dt = |x + WT | − |x| − sign (x + Wt ) dWt . =

LaT,x =

0

0

The random field Lat,x is called the local time of x + W . See Revuz and Yor [177]. Remark. In dimension d ≥ 2, for a.e. ω, the support of the curve x+W (ω) has zero Lebesgue measure, hence μT,x+W (ω) is singular with respect to Lebesgue

20


measure. Indeed,

E Rd

1x+W (y) dy =

Rd

P (y − x ∈ W ) dy = 0

because points are polar. However, the Hausdorff dimension of the support is 2. Thus the measure μT,x+W (ω) has some degree of spreading, which hints at regularity. When we move x by continuity, the measure μT,x+W (ω) cannot change so smoothly as N (x, tI), but maybe it has some degree of smoothness. Let us try to capture it. Example 2.3. A.M. Davie [68] proved the following result. It is a sort of Lipschitz estimate for an L∞ -function ϕ, but composed with Brownian motion and in the average. Theorem 2.1. Assume ϕ ∈ L∞ [0, T ] × Rd . Then, for every p ≥ 2 and T >0

p

T

(ϕ (t, x + Wt ) dt − ϕ (t, y + Wt )) dt ≤ (CT p)p/2 ϕL∞ |x − y|p E

0 for all x, y ∈ Rd .

Corollary 2.1. With the notations above, for ϕ ∈ L∞ Rd , we have p

p

E [|μT,x+W (ϕ) − μT,y+W (ϕ)| ] ≤ (CT p)p/2 ϕL∞ |x − y| . Apparently this is a result in the average, about the law of Wt , but we may apply Kolmogorov regularity criterion to get the following consequence. Corollary 2.2. Given ϕ ∈ L∞ Rd and T > 0, the random field

T

x → μT,x+W (ω) (ϕ) =

ϕ (x + Wt (ω)) dt 0

has an α-H¨ older continuous modification, for all α < 1.

2.1.2 Is x → µT,x+W (ω) Continuous in Total Variation? The result of Corollary 2.2 suggests that the map x → μT,x+W (ω) could be continuous in total variation or at least in a topology similar to total


21

variation where measures are tested on H¨ older continuous functions instead of L∞ -ones. In expressive terms, the question is: for a.e. given ω ∈ Ω, does the

family of measures μT,x+W (ω) ; x ∈ Rd have an analog of the Strong Feller property of Example 2.1? The answer is negative. The provided modification by Corollary 2.2 depends on ϕ. For every ϕ ∈ L∞ Rd , there exists a full measure set Ωϕ ⊂ Ω such that for all ω ∈ Ωϕ the function x → μT,x+W (ω) (ϕ) defined on rational points x ∈ Rd is uniformly continuous (in fact α-Hölder continuous for all α < 1), and thus admits a unique uniformly continuous extension to Rd . But the set Ωϕ depends on ϕ: ∩ϕ∈L∞ (Rd ) Ωϕ is negligible. Proposition 2.1. There is no measurable set A ⊂ Ω with P (A) > 0 such that for all ω ∈ A and all ϕ ∈ L∞ Rd the function x → μT,x+W (ω) (ϕ) defined on rational points x ∈ Rd is uniformly continuous. Proof. Let us prove it by contradiction. Let A be such a set. Being A of positive measure, there exists ω0 ∈ A such that the support of W (ω0 ) has zero Lebesgue measure (we remarked above that it is true for a.e. ω). Take ϕ equal to 1 on the support of W (ω0 ), zero elsewhere. Then μT,x+W (ω0 ) (ϕ) is equal to one for x = 0, but, denoting by μT,x+W (ω0 ) (ϕ) the uniformly continuous extension to the whole Rd , Rd

μT,x+W (ω0 ) (ϕ) dx =

T

0

Rd

T

ϕ (x + Wt ) dxdt =

ϕ (x) dxdt = 0 0

Rd

because the Lebesgue measure of the support of W (ω0 ) is zero. Hence μT,x+W (ω0 ) (ϕ) = 0 for all x ∈ Rd (recall it is uniformly continuous). Thus we must have μT,W (ω0 ) (ϕ) = 0, which contradicts μT,W (ω0 ) (ϕ) = 1 found above. The proof is complete.

2.1.3 An Estimate for H¨ older Functions For α ∈ (0, 1), denote by Cbα Rd the space of all continuous f : Rd → R such that uC α (T ) := sup |u (x)| + sup b

x∈Rd

x=y

|u (x) − u (y)| < ∞. α |x − y|

The proof given by Davie of his theorem is quite tricky. Let us give a rather elementary proof of a simplified statement in the case ϕ ∈ C [0, T ] ; Cbα Rd . The drawback of this elementary proof is that we miss the constant of Davie estimate but still it allows us to prove the analog of Corollary 2.2. The proof is taken from Flandoli [97].

22


Theorem 2.2. For every ϕ ∈ C [0, T ] ; Cbα Rd , consider the random field Xϕ (t, x) defined as

t

Xϕ (t, x) :=

ϕ (s, x + Ws ) ds. 0

For every p ≥ 2, α, α ∈ (0, 1) α < α, there is a constant C = Cp,T,α,α , independent of ϕ, such that E E

sup |Xϕ (t, x) − Xϕ (t, y)|

0≤t≤T

p

p

≤ C ϕC α |x − y|

p

b

sup |∇Xϕ (t, x) − ∇Xϕ (t, y)|p ≤ C ϕpC α |x − y|α p b

0≤t≤T

(∇Xϕ (t, x) is the gradient in the space variable, computed at (t, x)). Hence there is a continuous version of the field Xϕ (t, x), such that ∇Xϕ (t, x) is of d α for all α < α. R class C [0, T ] ; C Proof. Consider the backward heat equation ∂u 1 + Δu = −ϕ on [0, T ] , u (T, x) = 0. ∂t 2 It has a solution of class C [0, T ] ; Cb2,α Rd ∩ C 1 [0, T ] ; Cbα Rd and the solution in these topologies is bounded by a constant times ϕC α (see b Theorem 2.3 below). By Itˆ o formula du (t, x + Wt ) = −ϕ (t, x + Wt ) + ∇u (t, x + Wt ) · dWt hence

t

Xϕ (t, x) = u (0, x) − u (t, x + Wt ) +

∇u (s, x + Ws ) · dWs .

(2.1)

0

We get E

p

0≤t≤T

≤

sup |Xϕ (t, x) − Xϕ (t, y)|

Cp ϕpC α b

p

|x − y| + Cp,T E

T

p

∇u (s, x + Ws ) − ∇u (s, y + Ws ) ds

0

p

≤ Cp,T ϕC α |x − y|

p

b

because even the second derivative of u is uniformly bounded by a constant times ϕC α . This proves the first inequality. b


23

Applying classical arguments (see for instance Kunita [143]) we may differentiate (2.1) and get (denote by ∂i the derivative in x, in the direction i)

t

∂i Xϕ (t, x) = ∂i u (0, x) − ∂i u (t, x + Wt ) +

∇∂i u (s, x + Ws ) · dWs

0

which implies, by the uniform boundedness of D2 u and its uniform α-Hölderianity, E

sup |∂i Xϕ (t, x) − ∂i Xϕ (t, y)|p

0≤t≤T

p

p

≤ Cp ϕC α |x − y| + Cp,T E b T p × ∇∂i u (s, x + Ws ) − ∇∂i u (s, y + Ws ) ds 0

p p p αp ≤ Cp ϕC α |x − y| + Cp,T D 2 uC α |x − y| . b

b

The proof of Theorem 2.2 is complete, with the last claim following by Kolmogorov regularity theorem. We have used the following simple and classical result (some of the claims will be used only in the next section). The best classical result (see Krylov [136]) includes uniqueness and maximal regularity: u ∈ C [0, T ] ; Cb2,α Rd ∩ C 1 [0, T ] ; Cbα Rd but we do not need them. let us denote by ϕC α (T ) the norm in C([0, T ]; b Cbα (Rd )). Theorem 2.3. For all ϕ ∈ C [0, T ] ; Cbα Rd there exists at least one solution u to the heat equation 1 ∂u = Δu + ϕ, ∂t 2 of class

u|t=0 = 0

u ∈ C [0, T ] ; Cb2,α Rd ∩ C 1 [0, T ] ; Cbα Rd

for all α ∈ (0, α) with 2 D u α ≤ Cα ϕC α (T ) C (T )

(2.2)

∇uC α (T ) ≤ C (T ) ϕC α (T ) with lim C (T ) = 0.

(2.3)

b

b

and b

b

T →0

24


Proof. Let us give a probabilistic proof of the most difficult estimates. Let (Wt )t≥0 be a d-dimensional Brownian motion, defined on a filtered probability space (Ω, Ft , P ). Consider the function

t

E [ϕ (s, x + Wt−s )] ds,

u (t, x) =

t ≥ 0, x ∈ Rd .

0

We have the identity 2 |z| dz ϕ (s, x + z) exp − E [ϕ (s, x + Wt−s )] = (2π (t − s)) 2 (t − s) Rd |y − x|2 −d/2 = (2π (t − s)) ϕ (s, y) exp − dy 2 (t − s) Rd −d/2

which allows us to differentiate E [ϕ (s, x + Wt−s )] in the x-variable, for all x ∈ Rd and t − s > 0, arbitrarily many times. By easy computations, we get DE [ϕ (s, x + Wt−s )] = − D2 E [ϕ (s, x + Wt−s )] =

1 (t − s)2

1 E [ϕ (s, x + Wt−s ) Wt−s ] t−s

E [ϕ (s, x + Wt−s ) (Wt−s ⊗ Wt−s − (t − s) Id )]

where Id is the identity matrix in Rd . Since E [Wt−s ⊗ Wt−s ] = (t − s) Id we can rewrite D2 E [ϕ (s, x + Wt−s )] 1 = 2 E [(ϕ (s, x + Wt−s ) − ϕ (s, x)) (Wt−s ⊗ Wt−s − (t − s) Id )] . (t − s) Therefore 2 D E [ϕ (s, x + Wt−s )] ≤

1 2

α 2 ϕC α E |Wt−s | |Wt−s | + |t − s|

b |t − s| 1 1 1+α/2 ϕC α + ≤ ϕC α C |t − s| 2 b b |t − s| |t − s| C ϕC α b ≤ 1−α/2 |t − s|


25

(2+α)/4 2+α 4 1+α/2 ≤ E |Wt−s | since E |Wt−s | ≤ C |t − s| . This implies that t 2 D E [ϕ (s, x + Wt−s )] ds ≤ C ϕ α tα/2 . Cb

0

The H¨ older continuity of D2 u can be proved, from the identity for D2 E [ϕ (s, x + Wt−s )], by applying to x → ϕ (s, x) the following inequality (left as a simple exercise): if g is an α-Hölder continuous function, then |g (x + z) − g (x) − (g (y + z) − g (y))| ≤ C |x − y|α−ε |z|ε for all ε ∈ [0, α]. We leave the rest of the proof to the reader; the function u is a solution. Remark 2.2. A number of important constants in the above proof depend on the dimension d, for instance because E |Wt |2 = d · t. Below we shall treat an infinite dimensional generalization, where one of the fundamental properties is the independence of the estimates from the dimension. We need a drift term of the form Ax, Du (t, x) with suitable operator A to reach such a result.

2.1.4 An Estimate for Lp Functions Our main aim is to prove uniqueness and flow properties for the SDE (2.7) with non-regular coefficients. We shall give the details in the case of H¨ older continuous drift, but some results hold true also in the case of Lp-drift. However, instead of reporting all the details about the SDE with Lp -drift (the interest reader may see Fedrizzi [88], Fedrizzi and Flandoli [89, 90]), we just give the proof of part of theorem 2.2 under Lp -regularity, proof which is similar to the proof of uniqueness for the SDE but easier and shorter (for instance we save some detail about Kolmogorov equations with Lp-coefficients) see Flandoli [97]. When ϕ ∈ L∞ [0, T ] × Rd we do not have any more a maximal regularity result for the heat equation and in particular we cannot say that D 2 u is uniformly bounded. We do not have a short proof in the case, essentially different from the one given by Davie as the one above in the H¨ older case. When ϕ ∈ Lq 0, T ; Lp Rd

for some p, q ∈ (1, ∞) with

d 2 + 0 such

that |f |1+δ/2 ∈ Lqp (T ) with new p , q satisfying

inequality of step 1 holds for |f |

+

2 q

< 1. Then the

in place of f . Choose ε > 0 such that

T

sup E x∈Rd

1+δ/2

d p

εf

2+δ

(s, x + Ws ) ds < 1.

0

Then, by Khas’minskii result, T 2+δ sup E e 0 εf (s,x+Ws ) ds < ∞.

x∈Rd

From Young inequality, there exists a constant Cε,δ > 0 such that f 2 ≤ εf 2+δ + Cε,δ . Then T 2 T 2+δ sup E e 0 f (s,x+Ws )ds ≤ sup E e 0 εf (s,x+Ws )ds eCε,δ < ∞.

x∈Rd

The proof is complete.

x∈Rd

2.1 Regularization of Functions by Noise: Occupation Measure Fig. 2.1 Occupation measures of constant and Brownian curves

x

29

x

y

+1

–1

+1

y

–1

2.1.5 Summary on Occupation Measure Let us summarize the previous achievements in terms of the occupation measure. It is only a reformulation, but it has a stronger intuitive impact if we figure out the “shape” of this measure. The analytical forms given above of the statements are more abstract, in a sense. Figure 2.1 illustrates what we also said in Remark 2.1.1. Let us describe it. The picture on the left describes a function ϕ : R2 → R, which is equal to +1 on the left of the vertical line, and −1 on the right (we just draw the vertical line and the two values). Then we have shown two points, x and y, close to each other. Think that we have a delta Dirac unitary mass μx at x, or μy at y. Then μx (ϕ) = 1, μy (ϕ) = −1. Close points x, y give rise to very different values. On the right of the same figure there is the picture of what could be the occupation measure μx+W (ω) at x, and μy+W (ω) at y, same trajectory W (ω) (we omit the time T ). We intuitively see that μx+W (ω) (ϕ)

and

μy+W (ω) (ϕ)

are close to each other, since the amount of +1 and −1 moved from one case to the other is relatively small. After this intuitive explanation, which hopefully motivates the insistence here to reformulate results in terms of occupation measure, let us introduce the augmented occupation measure of a continuous curve γ : [0, ∞) → Rd up to time T : the finite Borel measure μ T,γ on [0, ∞) × Rd defined as μ T,γ (ϕ) = 0

∞

Rd

T

ϕd μT,γ =

ϕ (s, γs ) ds,

∀ϕ ∈ Cb [0, ∞) × Rd .

0

This extended concepts helps to generalize our statements. The relation with occupation measure is simply

30


∀ϕ ∈ Cb Rd

μT,γ (ϕ) = μ T,γ (ϕ) ,

so, when the test function ϕ does not depend on time, it is the same concept. Theorem 2.5. For every r ≥ 2,α ∈ (0, 1) there is a constant Cr,T,α such that for all ϕ ∈ C [0, T ] ; Cbα Rd we have E

μt,x+W (ϕ) − μ t,y+W (ϕ)| sup |

r

r

b

0≤t≤T

E

r

≤ Cr,T,α ϕC α |x − y| r

sup |∂i ( μt,x+W (ϕ) − μ t,y+W (ϕ))|

r

αr

≤ Cr,T,α ϕC α |x − y| b

0≤t≤T

.

In particular, given ϕ ∈ C [0, T ] ; Cbα Rd , there is a continuous version (depending of ϕ) of the random field (t, x) → μ t,x+W (ω) (ϕ) this version is of class C [0, T ] ; C 1+α Rd for all α < α. Theorem 2.6. Assume ϕ ∈ Lq 0, T ; Lp Rd

for some p, q ∈ (1, ∞) with

d 2 + < 1. p q

For every r ≥ 2, p, q ∈ (1, ∞) such that dp + 2q < 1, there is a constant Cr,T,p,q such that for all ϕ ∈ Lq 0, T ; Lp Rd we have E

μt,x+W (ϕ) − μ t,y+W (ϕ) ds| sup |

r

0≤t≤T

r

r

≤ Cr,T,p,q ϕLqp |x − y|

for every x, y ∈ Rd . As a consequence, the random field (t, x) → μ t,x+W (ω) (ϕ) has a continuous version, of class C [0, T ] ; C α Rd for all α < 1. Example 2.4. Let us see another deterministic example, less extreme than the delta Dirac masses μx and μy above. Take any smooth curve γ in the plane with γ (t0 ) = 0 at some value t0 of the parameter. By a rotation, a dilation and a re-parametrization, assume γ (0) = (1, 0). This is a generic condition. Assume it is locally of the form γ (t) ≈ t, at2 with a > 0 (similarly for a < 0; a = 0 is again a generic condition). Take ϕ equal to 1 in the half-plane (x1 , x2 ) with x2 < 0, equal to zero in the half-plane x2 > 0. Shift γ by vectors of the form x = (0, −ε). Then μT,(0,0)+γ (ϕ) = 0,

2.2 Regularization of SDE by Additive Noise

31

μT,(0,−ε)+γ (ϕ) ≈ 2

ε . a

We see that μT,x+γ (ϕ) − μT,y+γ (ϕ) is of the order |x − y| for such points x = (0, −ε), y = (0, 0) and test function ϕ. Thus the degree of smoothness of the map x → μT,x+γ (ϕ) is less than in the Brownian motion case.

2.2 Regularization of SDE by Additive Noise 2.2.1 Main Result We have seen in Chap. 1 (Example 1.1) that ODEs in Rd of the form dXt = b (t, Xt ) , dt

X0 = x ∈ Rd

(2.5)

where b : [0, T ] × Rd → Rd is of class b ∈ C [0, T ] ; Cbα Rd

(2.6)

for some α ∈ (0, 1), may lack uniqueness. And on the contrary, by Girsanov type arguments, the SDE Xt = x +

t

b (s, Xs ) ds + Wt

(2.7)

0

has uniqueness in law (this extremely interesting but classical fact can be found in many books, like [129, 151, 177] and thus it is not discussed here in detail). We want to understand better this regularization phenomenon here, by means of the regularity properties of occupation measure. The result will be also an improvement of the uniqueness in law given by Girsanov: we get strong uniqueness and existence of a stochastic flow. All the results reported in this Chapter are written here for simplicity in the case of additive noise. However, conceptually, the key assumption is the non-degeneracy of the noise (the degenerate case is essentially open). Since our results depend on regularity theory of parabolic equations, the key question is under which assumptions of regularity of the diffusion coefficients, assumed non degenerate, we keep the same results. There are investigations in this direction, but we do not report them here. t The idea is very simple: the difficult term 0 b (s, Xs ) ds is equal to μ t,X (b). If we can prove an inequality similar to those of the previous section for the occupation measure, with x + W and y + W (there) replaced by two solutions X (1) and X (2) (here), we should get a Lipschitz property of the integral

32


t

b (s, Xs ) ds in the argument X, which will lead us to prove uniqueness. Let 0 us state the result on the occupation measure, proved in the next section. Remark 2.3. In the next two theorems we compute moments of solutions to (2.7). If the initial condition X0 is deterministic, or also if it is an r

r

F0 -measurable r.v. with E [|X0 | ] < ∞, r ≥ 2, then E supt∈[0,T ] |Xt | < ∞. This can be proved by a stopping time argument and Doob theorem. Theorem 2.7. For every r ≥ 2, α ∈ (0, 1), given ϕ ∈ C [0, T ] ; Cbα Rd , 0 depending on ϕC α (T ) such that we there are constants Cr,T,α and CT,r,α b have the following properties:

(i) for every pair of solutions X (i) , i = 1, 2,of (2.7) with initial conditions

(i) r (i)

< ∞, solutions defined x which are F0 -measurable r.v. with E x on the same filtered probability space (Ω, Ft , P ), we have E

r t,X (1) (ϕ) − μ sup μ t,X (2) (ϕ)

0≤t≤T

r

r

(1)

(2)

0 ≤ Cr,T,α E x(1) − x(2) + CT,r,α E sup Xt − Xt

0≤t≤T

r T

(1)

+ Cr,T,α E

Xs − Xs(2) ds 0

0 = 0. (ii) limT →0 CT,r,α

The proof will be given in the next subsection. The theorem states that processes close to each other in the usual topology E ·rC 0 (0,T ) have occupation measures which are closer to each other for small T , in a sort of average α d total variation topology (in fact weaker also because of C R instead of b L∞ Rd ). The processes must be solutions of an SDE with additive noise (presumably generalizable to non-degenerate noise). If they are solutions of SDEs with different drifts, there will be an additional term related to the closedness of the drift, which here is absent. Remark 2.4. Property (ii) of the previous theorem is not true for smooth (i) paths, solutions of the same equation. Take two solutions X , i = 1, 2, of (i)

the ODE

dXt dt

(i)

= b Xt

(i)

, X0 = x. It is not true that

μT,X (1) (ϕ) − μT,X (2) (ϕ) ≤ CT ϕ α d sup

X (1) − X (2)

t t Cb (R ) 0≤t≤T

with lim CT = 0. T →0

Indeed, if b (x) = 2sign (x) α sign (x) |x| , and get

(1)

|x| we can take Xt

(2)

= t2 , Xt

= −t2 , ϕ (x) =


33

T

T 2α+1

2α

μT,X (1) (ϕ) − μT,X (2) (ϕ) = 2

t dt =

0

2α + 1

(1) (2)

sup Xt − Xt = T 2 . 0≤t≤T

For instance, in the most relevant case when ϕ = b, we have α = 12 and 1 thus CT = 2α+1 . We do not have limT →0 CT = 0, absolutely essential for the uniqueness. If we could take ϕ (x) = sign (x) (as in the figure above) the effect would be even more clear. The consequence of Theorem 2.7 is uniqueness for the SDE. Corollary 2.3. Strong uniqueness holds for (2.7). Moreover, if X x denotes the solution with initial condition x, then E

μt,X x (ϕ) − μ t,X y (ϕ)|r ≤ Cr |x − y|r sup |

0≤t≤T

E

sup 0≤t≤T

|Xtx

−

r Xty |

r

≤ Cr |x − y| .

It follows that the two random fields (t, x) → μ t,X x (ϕ) ,

(t, x) → Xtx

have continuous modifications, α-H¨ older for every α < 1. Proof. Take two solutions X (i) , i = 1, 2, of (2.7) with initial conditions x(i) , defined on the same filtered probability space (Ω, Ft , P ). From (2.7) and the theorem we have, for T small enough, E

r

(1) (2)

sup Xt − Xt

0≤t≤T

r

r

≤ C x(1) − x(2) + CE sup μ t,X (1) (b) − μ t,X (2) (b)

0≤t≤T

r 1

r

(1)

(1) (2)

(2)

≤ C x − x + E sup Xt − Xt

2 0≤t≤T

r T

(1)

+ CE

Xs − Xs(2) ds 0

where we have denote generically with C a constant depending on r, T , α, but not on the solutions. This implies E

r

r

(1)

(2)

sup Xt − Xt ≤ C x(1) − x(2)

0≤t≤T

34


if T is small enough. Using this inequality inside the previous theorem, we prove the estimate on the difference of occupation measures. The size of T does not depend on the solution, so the argument can be iterated on successive intervals (using random non-anticipative initial conditions). The proof is complete. This approach to SDEs is taken from Flandoli et al. [100], where it is developed via a transformation, more in the spirit of Zvonkin [201] (here, on the contrary, we insist on the underlying concept of occupation measure). In dimension d = 1, an idea which is somewhat similar was developed by Flandoli and Russo [108]. Let us also mention that in d = 1 the understanding of influence of noise on uniqueness and singularities is very advanced, see Cherny and Engelbert [49].

2.2.2 Proof of Theorem 2.7 Using Theorem 2.3 on the heat equation we may prove a similar result for the Kolmogorov equation. Again we do not give the maximal regularity and the uniqueness, that can be found in the book of Krylov [136]. Theorem 2.8. For all ϕ ∈ C [0, T ] ; Cbα Rd there exists at least one solution u to the backward Kolmogorov equation 1 ∂u + b · ∇u + Δu = −ϕ on [0, T ] , ∂t 2 of class

u (T, x) = 0

(2.8)

u ∈ C [0, T ] ; Cb2,α Rd ∩ C 1 [0, T ] ; Cbα Rd

for all α ∈ (0, α) with 2 D u α ≤ Cα ϕC α (T ) C (T ) b

b

and ∇uC α (T ) ≤ C (T ) ϕC α (T ) with lim C (T ) = 0. b

b

T →0

Proof. (An alternative elegant proof can be done by the method of continuity). Consider a usual Picard type iteration scheme for the Kolmogorov equation, based on the heat equation: u(0) = 0 and, for n ≥ 0, 1 ∂u(n+1) + Δu(n+1) = − (b · ∇) u(n) − ϕ on [0, T ] , ∂t 2

u(n+1) |t=T = 0.


35

By Theorem 2.3, at each iteration step we have a solution u(n+1) ∈ C [0, T ] ; Cb2,α Rd ∩ C 1 [0, T ] ; Cbα Rd (n) (n) for every α < α ; notice that, when u has such regularity, then (b · ∇) u α d is C [0, T ] ; Cb R , so we may continue the iteration. By Theorem 2.3 we precisely have 2 (n+1) ≤ Cα (b · ∇) u(n) + ϕ α α D u

(n+1) ∇u

Cb (T )

Cbα (T )

≤ C (T ) (b · ∇) u(n) + ϕ

Cb (T )

Cbα (T )

with lim C (T ) = 0. T →0

But (n) + ϕ (b · ∇) u

Cbα (T )

≤ ϕC α (T ) + 2 bC α (T ) ∇u(n) b

b

Cbα (T )

≤ ϕC α (T ) + 2 bC α (T ) C (T ) (b · ∇)u(n−1) + ϕCbα (T ) . b

b

Choose T such that 2 bC α (T ) C (T ) ≤ 1/2 and set v (n) := (b · ∇) u(n) + ϕ. b We have 1 (n) ≤ ϕC α (T ) + v (n−1) α v α b 2 Cb (T ) Cb (T ) 1 1 1 ≤ ... ≤ 1 + + ... + n−1 ϕC α (T ) + n v (0) α b 2 2 2 Cb (T ) (n) v

hence

Cbα (T )

This proves (n+1) ∇u

2 (n+1) D u

Cbα (T )

By the equation itself,

≤ 2 ϕC α (T ) . b

Cbα (T )

≤ 2Cα

≤ 2C (T ) ϕC α (T ) with lim C (T ) = 0. T →0

b

(n+1) ∂u ∂t

Cbα (T )

≤ C.

By Ascoli–Arzel` a theorem, one can extract a subsequence which converges uniformly in (t, x) to some u with its first and second space derivatives; and by definition of Cbα spaces, one can check that u has the same regularity and (n) bounds as u(n) , except for the estimate on ∂u∂t . We can pass to the limit in

36


the identity u(n+1) (t, x) = t

T

1 (b · ∇) u(n) + Δu(n+1) + ϕ ds 2

(n+1) and u(n) , which implies and get the same equation with u in place of u ∂u ∈ C [0, T ] ; Cbα Rd and the fact that u is a solution of (2.8). The proof ∂t is complete.

Consider now the vector valued analog of (2.8): 1 ∂UΦ + b · ∇UΦ + ΔUΦ = −Φ on [0, T ] , ∂t 2

UΦ (T, x) = 0

(2.9)

where Φ : [0, T ] × Rd → Rd has components Φk ∈ C [0, T ] ; Cbα Rd . Let us use similar notations for the spaces of vector fields, so we simply write Φ ∈ C [0, T ] ; Cbα Rd , Rd . The solution UΦ : [0, T ] × Rd → Rd of this decoupled system has the properties described in the last theorem. By Itô formula, as described above, we get an interesting occupation measure identity. Corollary 2.4. Let X be a solution of (2.7). Then, for every Φ ∈ C [0, T ] ; Cbα Rd , Rd , we have μ t,X (Φ) = UΦ (0, x) − UΦ (t, Xt ) +

t

∇UΦ (s, Xs ) · dWs

(2.10)

0

From identity (2.10) we see that Φ-observation of the (augmented) occupation measure μ t,X involves terms which are more regular than the initial one, because UΦ and ∇UΦ are more regular than Φ. We can now prove the theorem. From (2.10) we have (by Young inequality)

r

E sup μ t,X (2) (Φ) t,X (1) (Φ) − μ 0≤t≤T

r

≤ Cr E sup UΦ 0, x(1) − UΦ 0, x(2)

0≤t≤T

+ Cr E

r

(1) (2)

− UΦ t, Xt

sup UΦ t, Xt

0≤t≤T

t

r t

(1) (2) + Cr E sup

∇UΦ s, Xs ∇UΦ s, Xs · dWs − · dWs

. 0≤t≤T 0

0

r The first term is bounded by Cr C (T ) E x(1) − x(2) , by the gradient estimate of Theorem 2.8, where limT →0 C (T ) = 0 (but we do not use this fact here). Similarly, the second term is bounded by


Cr C (T ) E

37

r

(1) (2)

sup Xt − Xt

0≤t≤T

(here the property limT →0 C (T ) = 0 is very important). Again similarly, by Burkholder–Davis–Gundy inequality, the last term is bounded by ⎡

⎤

T 2 r/2

Cr E ⎣

∇UΦ s, Xs(1) − ∇UΦ s, Xs(2) ds ⎦

0

r T

(1)

≤ Cr E

Xs − Xs(2) ds . 0

Summarizing, we have proved E

r

sup μ t,X (2) (Φ)

t,X (1) (Φ) − μ

0≤t≤T

r

r

(1)

(2)

≤ Cr E x(1) − x(2) + Cr C (T ) E sup Xt − Xt

0≤t≤T

T

(1)

r + Cr E

Xs − Xs(2) ds 0

with limT →0 C (T ) = 0. The proof is complete.

2.2.3 Stochastic Flow of Diffeomorphisms Corollary 2.3 gives us uniqueness for the SDE but also the existence of a map ϕt (x) = ϕt (x, ω), measurable in all arguments, continuous in (t, x) for a.e. ω, even α-Hölder continuous in x for every α ∈ (0, 1), such that ϕt (x) = Xtx a.s., for every given (t, x). Working on the generic time interval [s, T ] instead of [0, T ], one can define a similar map ϕs,t (x). Using pathwise uniqueness, it is easy to check that ϕr,t (ϕs,r (x, ω) , ω) = ϕs,t (x, ω) a.s., for every given x and s ≤ r ≤ t; moreover, ϕs,s (x) = x, a.s., hence ϕs,t (x) is a sort of stochastic semi-flow. Under the same assumptions of the Corollary, one can prove more, namely that ϕt (x) is a stochastic flow of diffeomorphisms, namely ϕt is a diffeomorphism of Rd for every t ∈ [0, T ]. All the details can be found in Flandoli et al. [100] and Fedrizzi [88]. Let us explain just a few elements. A simple idea behind the homeomorphism property is that one can introduce the backward equation associated to the forward one, which is very

38


similar (additive noise and drift −b), prove strong well posedness, the existence of an Hölder modification in the initial (or better final) conditions, and prove that the forward and backward maps, composed, are equal to the identity. Unfortunately, the rigorous realization of this natural idea contains a major technical difficulty, the fact that the inverse flow is measurable with respect to the future. Following Kunita [143], there is a rigorous but very long way to circumvent such difficulty. It is too long for these lectures, so we ask the reader to accept the result, thanks to the intuition given by this simple idea. A full proof following a slightly different route can be found in Flandoli et al. [100]. Let us give some details about the differentiability of the flow. Proposition 2.2. Under assumption (2.6), P -a.s. the map x → ϕt (x) is differentiable for every t ∈ [0, T ], and (t, x) → Dϕt (x) is continuous (x → Dϕt (x) is also α -H¨ older continuous, every α < α). Proof. Let Ub be the solution of (2.9) with Φ = b. We have (see the beginning of Sect. 2.7) Xtx

= x + Ub (0, x) −

Ub (t, Xtx )

t

+ 0

[I + ∇Ub (s, Xsx)] · dWs

where Ub and ∇Ub are differentiable. Consider the linear equation in ξtx ξtx

= I + ∇Ub (0, x) −

∇Ub (t, Xtx )

·

ξtx

t

+ 0

D2 Ub (s, Xsx ) ξsx · dWs

(it will be the variational equation of the previous one). It has a unique solution (we omit the details) and for every q ≥ 2 E

sup r∈[0,t]

|ξrx |q

≤ Cq +

Cq ∇Ub q∞

E

sup r∈[0,t]

|ξrx |q

t

+ Cq 0

D 2 Ub q E [|ξsx |q ] ds ∞ q

which implies, for T small enough (Cq ∇Ub ∞ can be made arbitrarily small, and then apply Gronwall lemma), E

sup r∈[0,T ]

q |ξrx |

≤ Cq∗

(uniformly in x). We have been slightly informal, since in order to apply Gronwall lemma one needs to know in advance that the q-moment is finite. This can be proved by stopping times and Doob inequality.


39

Moreover, from the equation for ξtx , we have E

sup r∈[0,t]

|ξrx

p ξry |

−

p

≤ Cp |x − y| +

p Cp ∇Ub ∞

p + Cp D2 Ub ∞ E

E

sup r∈[0,t]

sup r∈[0,t]

|ξry |p

|Xrx

|ξrx

−

−

p ξry |

Xry |p

D 2 Ub p E [|ξsx − ξsy |p ] ds ∞ 0 t 2 p p pα D Ub α E |ξsy | |Xsx − Xsy | + Cp,α ds. C (T )

+ Cp

t

b

0

p

older We can make Cp ∇Ub ∞ ≤ 1/2 by proper choice of T . Moreover, by H¨ inequality, E

sup r∈[0,t]

p |ξry |

|Xrx −

p Xry |

≤E

1/2 sup

r∈[0,t]

2p |ξry |

1/2

E

sup |Xrx −

r∈[0,t]

2p Xry |

∗ 1/2 1/2 p ≤ C2p (C2p ) |x − y|

by the previous estimate and the bound of Corollary 2.3. Similarly p pα 1/2 pα ∗ 1/2 ≤ C2p E |ξsy | |Xsx − Xsy | (C2pα ) |x − y| . Summarizing, by Gronwall lemma,

E

sup r∈[0,t]

|ξrx

−

ξry |p

pα . ≤ Cp |x − y|p + Cp,α |x − y|

By the arbitrarity of p ≥ 2 and α ∈ (0, α) and Kolmogorov regularity theorem, we deduce that there exists a modification of ξtx which is continuous in (t, x), α -Hölder continuous in x for every α ∈ (0, α). By means of classical but arguments one can prove that the map lengthy x → Xtx , from Rd to L2 Ω; Rd is differentiable and the (space) derivative Dx Xt satisfies the linear equation above. The proof of the proposition is then easily completed. A detailed proof by a slightly different approach is given by Fedrizzi [88].

40


2.3 Infinite Dimensional Equations with Additive Noise 2.3.1 Introduction The aim of this section is to prove uniqueness for (loosely speaking) “reactiondiffusion” parabolic equations of the form described in Example 1.5 of Chap. 1. We present two results. The first one is a classical result of uniqueness in law. It is based on the existence of a solution to backward Kolmogorov equation, with a suitable gradient estimate. The drift is H¨ older continuous. The exposition is inspired by Gatarek and Goldys [115] and Zambotti [198]; the general technique follows for instance the ideas of the book of Stroock and Varadhan [191]. The second one is pathwise uniqueness, under similar assumptions on the drift. It is also based on backward Kolmogorov equation (non homogeneous), but a much stronger estimate on second order derivatives is needed. The strategy of proof is the same used in the previous section: a reformulation of the SDE where the drift part is regularized by an Itˆ o–Tanaka approach. We do not make explicit use of the concept of occupation measure, but the proof is essentially the same as above. In finite dimensions, additive noise, pathwise unique with non degenerate ness holds under Lq 0, T ; Lp Rd conditions on b, with dp + 2q < 1 and in ockner [138], Veretennikov [193], related to Zvonkin L∞ , see Krylov, M. R¨ approach [201]; see also Zhang [199, 200]; uniqueness in law holds even for certain distributional drifts, see Bass and Chen [30] and references therein. In infinite dimensions, at our present level of understanding, the difference between weak and strong uniqueness is made more by the assumptions on the noise than on the drift, precisely by the assumptions on the pair (A, Q) (see below). Pathwise uniqueness, opposite to uniqueness in law, requires a cylindrical (space-time) noise, or very close to it. The consequence on examples is very strong: for second order parabolic equations similar to Example 1.5 of Chap. 1 (square root nonlinearities), we have uniqueness in law up to space dimension d = 3, but pathwise uniqueness only for d = 1. Pathwise uniqueness for one-dimensional second order parabolic equations with space-time noise can be proved also by other less abstract methods, see Gyöngy and Pardoux [124], Bally et al. [18], Gy¨ ongy [121], Gy¨ ongy and Nualart [123], Alabert and Gy¨ ongy [6], Gy¨ ongy and Mart´ınez [122]. The operator B in these works is mostly of the form B (t, Xt ) (ξ) = b (t, Xt (ξ)) namely pointwise functions of the solution. They may include derivatives in ξ of X but only in a locally Lipschitz way. For this kind of operators, the results of these papers are extremely general, much more than H¨ older continuous as

2.3 Infinite Dimensional Equations with Additive Noise

41

in this section. The techniques are completely different, based on various tools depending on the paper, like Malliavin calculus, comparison principle and also occupation measure, used for different purposes with respect to what is done here. Uniqueness in law by means of a analytic approaches to Kolmogorov and Fokker–Planck equations are perhaps the most promising direction to cover more and more examples. The two usual strategies are to prove uniqueness for the Fokker–Planck equation or existence of sufficiently regular solutions to the backward Kolmogorov equation. A part from the simple result reported here, a considerable amount of work has been done recently on Kolmogorov and Fokker–Planck equations and their applications to uniqueness; let us quote three books and a few papers among many others: Cerrai [46], Da Prato and Zabczyk [67], Da Prato [59], Cerrai [45], Flandoli and Gozzi [99], Priola and Zambotti [175], Barbu et al. [25,26], R¨ ockner and Sobol [178,179], Da Prato and Debussche [62], Stannat [190], Barbu al. [27], Barbu et al. [28], Ambrosio et al. [10], Manca [154], Bogachev et al. [36–38]. Kolmogorov equation applies also to other problems, like control theory and averaging; a full list is not appropriate here, let us mention only Gozzi et al. [118, 119], Fuhrman and Tessitore [112, 113], Cerrai and Freidlin [47]. We hope there will be progresses in this direction. As we shall mention again in Sect. 5.4, Kolmogorov equation has been solved even for 3D stochastic Navier–Stokes equations in the outstanding work of Da Prato and Debussche [61]. However, the regularity of solutions does not allow to apply arguments similar to those written below, and weak uniqueness of the SDE is still open. A last remark concerns Girsanov approach, the most straightforward way to prove uniqueness in law in finite dimensions. In Hilbert spaces the hope of this approach is to relate the nonlinear equation dXt = AXt dt + B (t, Xt ) dt +

QdWt ,

X0 = x

to the linear one dZt = AZt dt +

!t , QdW

Z0 = x

by a change of measure. If W is a cylindrical Wiener process on a filtered probability space (Ω, Ft , P ), Q is injective and

T

P

2

−1/2

B (s, Xs ) ds < ∞ = 1

Q

0

H

(2.11)

consider the local martingale t " ρt := exp 0

2 # 1 t

−1/2

Q−1/2 B (s, Xs ) , dWs − B (s, Xs ) ds .

Q 2 0 H

42


If ρt is a martingale, then !t = Wt − W

t

Q−1/2 B (s, Xs ) ds

0

is a Ω, Ft , P -cylindrical Wiener process, where, on each space (Ω, Ft ), P is

P defined as ddP

= ρt . With this strategy and some further arguments (see Ft

Sect. 3.4.5 for an example), one can transfer the problem of uniqueness in law for the nonlinear equation to the same problem for the linear equation, where it is essentially obvious. This approach has been developed very well by a number of authors, see in particular Kozlov [135], Goldys and Maslowski [116], Ferrario [92]. A Novikov condition is usually required to prove that ρt is a martingale, but following ideas from Liptser and Shiryaev [151], in some cases one can avoid them and ask only condition (2.11) for any solution of the nonlinear equation, see Allouba [7], Ferrario [93]. However, the assumptions needed to apply Girsanov strategy in infinite dimensions are quite demanding. Unless one imposes artificial regularity assumptions on the range of B, it is necessary to √ assume Q invertible, see condition (2.11). But then in applications, being QdWt cylindrical, if A is the Laplacian, only space-dimension 1 is allowed, see Example 2.6 below. In this sense, Girsanov approach is not essentially more general than the one used below to prove pathwise uniqueness (in terms of noise and space-dimension), and gives us less. On the other hand, it allows us to treat L∞ operators B without pain, a fact which is much more difficult with Kolmogorov equation (strong uniqueness for B of class L∞ has been proved by Veretennikov [193] in the finite dimensional case, and by Gy¨ ongy [121] and related works, under various sets of conditions, for 1D parabolic SPDEs with space-time white noise). In principle it is possible to relax invertibility of Q in Girsanov approach: even if Q is not invertible and the range of B is not regular, the function Q−1/2 B (s, Xs ) could be well defined because of additional regularity of Xs and some “transfer of regularity” property of B. But we need Xs more regular in space, and this requires Q more regular; and more regular Q makes more difficult to check that Q−1/2 B (s, Xs ) is well defined. At the end of the story, some improvement on the assumption that Q is invertible may be possible, but it looks small.

2.3.2 Infinite Dimensional Set-Up We denote by H a separable Hilbert space with inner product ., .H and norm $∞ |.|H . Let {en }n∈N be a complete orthonormal system in H and write x = n=1 xn en , xn = x, en H . Let {λn }n∈N be a (weakly) increasing sequence of


43

strictly positive real numbers diverging to infinity and let A be the negative self-adjoint operator with compact resolvent A : D (A) : H → H %

defined as

x∈H :

D (A) =

∞ &

' λ2n x2n

0 (also negative, with suitable extensions of H). We set % α

x∈H :

D ((−A) ) =

∞ &

' 2 λ2α n xn < ∞

n=1

(−A)α x =

∞ &

λα n xn en ,

x ∈ D (A) .

n=1

The operator A is the infinitesimal generator of the analytic semigroup tA

e x :=

∞ &

e−tλn xn en ,

x ∈ H.

n=1

We shall use several properties which can be easily checked, like that α α α (−A) and etA commute (on D ((−A) )), etA maps H into D ((−A) ) and α tA (−A) e is a bounded operator in H and, easy but less trivial, that for every α > 0 one has

(−A)α etA x

where Cα2 :=

H

≤

Cα |x|H , tα

x ∈ H, t > 0

−2tλn sup t2α λ2α = sup s2α e−2s < ∞. n e

t>0,n∈N

s>0

Let W be a cylindrical Wiener process in H, defined on a filtered probability space (Ω, Ft , P ). To be as simple as possible, think that W is the formal expression

44


Wt = where

( ) (n) Wt

n∈N

∞ &

(n)

Wt

en

n=1

is a sequence of independent Brownian motions on

(Ω, Ft , P ) (in a sense, W is such a sequence). The formal series converges in mean square (and more) in a larger space than H, but we do not use this fact. We shall use only certain expressions derived from Wt which are meaningful, as we shall explain. Let {σn }n∈N be a (weakly) decreasing sequence of non-negative real numbers and let Q be the non-negative selfadjoint bounded operator defined as ∞ & σn2 xn en , x ∈ H. Qx = n=1 α

α

The operators (−A) and Q commute (on D ((−A) )). We could develop most of the following theory without such commutativity condition, but we prefer to simplify the exposition. As above, the fractional powers Qα are well defined bounded operators in H. √ When we write QWt we mean the formal expression QWt =

∞ &

(n)

σn Wt

en

n=1

rigorously defined $∞ in a space larger than H, if necessary. When Q is trace class, namely n=1 σn2 < ∞, then this series converges in mean square in H. But we do not assume Q trace class (this would be too restrictive for the uniqueness results, where Q must be the identity or a similar operator). Proposition 2.3. Assume ∞ & σn2 0 (even something less), then assumption (2.12) is fulfilled also in dimension 2.

46


2.3.3 Uniqueness in Law Consider the SDE in the Hilbert space H dXt = AXt dt + B (t, Xt ) dt +

QdWt ,

X0 = x

(2.14)

where A, Wt , Q are defined above, B : [0, T ] × H → H is continuous, H¨ older continuous and bounded in x, uniformly in t. We may interpret this equation in different equivalent ways. One of them is the so called mild form Xt = etA x +

t

t

e(t−s)A B (s, Xs ) ds +

0

e(t−s)A

QdWs

0

which can be found formally by the variation of constant method. We say that X is a strong solution if it is a continuous adapted process (notice that (Ω, Ft , P, Wt ) is given a priori) which satisfies this equation for every t ∈ [0, T ], with probability one. We assume (2.12), so the stochastic integral in the mild formulation is well defined. Let us prove uniqueness of the 1-dimensional marginals of solutions, under rather general conditions. We address to Stroock and Varadhan [191], Theorem 6.2.3 for a general measure theoretic argument to deduce full weak uniqueness: it holds true when uniqueness of the 1-dimensional marginals is proved for all initial conditions and all initial times (the initial time below is always t = 0, conventionally, but it can be any s ∈ (0, T ) because B is arbitrary in the class defined by the assumptions). Theorem 2.9. Let

B ∈ C ([0, T ] ; Cbα (H, H))

for some α ∈ (0, 1). Assume that the operators A, Q introduced in the previous section satisfy assumption (2.12), that Q is injective (σk2 > 0 for every k) and etA (H) ⊂ Qt

1/2

(H) for all t > 0

(2.15) −1/2 tA

and finally that the well defined bounded operator Λt = Qt

T

Λt dt < ∞.

e

satisfies (2.16)

0

Then, if X (i) , i = 1, 2, are two mild solutions, for every t ∈ [0, T ] the laws of (i) Xt are equal. Proof. Let Hn be the span of e1$ , ..., en and πn be the finite dimensional n projection on Hn given by πn x = k=1 xk ek . Let (i,n)

Xt

(i)

:= πn Xt ,

(n)

Wt

:= πn Wt ,

B (n) (t, x) := πn B (t, x) .

2.3 Infinite Dimensional Equations with Additive Noise (i,n)

47

(n)

The processes Xt and Wt live in Hn . The mapping B (n) (t, ·) operates from H or from Hn to Hn . The linear operators A and Q, restricted to Hn , are linear bounded operators in Hn ; we do not change notations for such restrictions. From the mild equation satisfied by X (i) we deduce, by projection, (i,n)

Xt

= etA πn x +

t

0

t e(t−s)A B (n) s, Xs(i) ds + e(t−s)A

QdWs(n) .

0

(i,n)

This is not a closed equation for the projection Xt . Since we are infinite (i,n) dimensions, we can easily check that Xt verifies the identity (in the usual integral sense) (i,n)

dXt

(i,n)

= AXt

(i) dt + B (n) t, Xt dt +

(n)

QdWt

(i,n)

,

X0

= πn x.

Hence (i,n)

dXt where

(i,n)

= AXt

(i,n)

Rt

(i,n) dt + B (n) t, Xt dt +

(n)

QdWt

(i,n)

+ Rt

dt

(i) (i,n) − B t, Xt . = πn B t, Xt

Let τ ∈ (0, T ] be given and u(n) : [0, τ ] × Hn → R be the solution to equation ∂u(n) 1 + Ax + B (n) (t, x) · ∇u(n) + T r QD2 u(n) = 0, ∂t 2 u(n) (τ, x) = ϕ (x)

t ∈ [0, τ ] , x ∈ Hn (2.17) x ∈ Hn

with ϕ ∈Cb1 (H), see Lemma 2.2 below. We have (we drop the argument (i,n) somewhere for shortness) t, Xt ∂u(n) 1 (i,n) (i,n) = dt + ∇u(n) · dXt du(n) t, Xt + T r QD2 u(n) dt ∂t 2 (i,n) (n) = ∇u(n) · Rt dt + ∇u(n) · QdWt hence τ ϕ Xτ(i,n) = u(n) (0, πn x) + ∇u(n) s, Xs(i,n) · Rs(i,n) ds 0 τ ∇u(n) s, Xs(i,n) · QdWs(n) . + 0

48


We get E ϕ Xτ(i,n) = u(n) (0, πn x) +

τ 0

E ∇u(n) s, Xs(i,n) · Rs(i,n) ds.

Hence, from the estimate of Lemma 2.2 below 2 τ

&

E ∇u(n) s, Xs(i,n)

Rs(i,n) ds

E ϕ Xτ(1,n) − E ϕ Xτ(2,n) ≤ i=1

0

≤ C (T ) ∇ϕ∞

2 & i=1

(i,n)

0

τ

E Rs(i,n) ds.

(i)

converges (in n) to Xt by definition, in H, Since B is continuous, and Xt (i,n) uniformly in t, P -a.s., we see that Rt goes to zero uniformly in t, P -a.s.; and it is bounded, hence

τ

lim

n→∞

0

E Rs(i,n) ds = 0.

Similarly

lim E ϕ Xτ(1,n) − E ϕ Xτ(2,n) = E ϕ Xτ(1) − E ϕ Xτ(2) . n→∞

We deduce, in the limit

E ϕ Xτ(1) − E ϕ Xτ(2) ≤ 0. (1)

(2)

This implies that Xτ and Xτ have the same law, since ϕ ∈ Cb1 (H) is arbitrary. The proof is complete. The main technical issue that we have used is the gradient estimate of the following lemma. For every n ∈ N, consider the backward homogeneous Kolmogorov equation (2.17) with ϕ ∈ Cb1 (Hn ). One can prove it has a unique solution u(n) ∈ C [0, τ ] ; Cb2,α (Hn ) ∩ C 1 ([0, τ ] ; Cbα (Hn )) . Even if we do not prove this full claim (that we do not need), we shall recall the basic estimates on second order derivatives in Corollary 2.5 below. What we need here is the following bound, uniform in n: Lemma 2.2. Under the assumptions of Theorem 2.9, there exits C (T ) > 0 (independent of n) such that for all n ∈ N and ϕ ∈ Cb1 (Hn )


(n) ∇u

∞

49

≤ C (T ) ∇ϕ∞ .

Proof. Step 1. A similar preliminary result is true for the non-homogeneous Ornstein–Uhlenbeck equation (B (n) = 0) with ϕ ∈ Cb1 (Hn ), ψ ∈ C ([0, T ] ; Cbα (Hn )): 1 ∂uOU (n) (n) + Ax · ∇uOU + T r QD2 uOU = ψ (t, x) , ∂t 2 (n)

(n)

uOU (τ, x) = ϕ (x) .

The estimate is

(n) ∇uOU (t, ·)

∞

≤ ∇ϕ∞ +

T 0

Λs ds ψ∞ .

Indeed, one can check that (n) uOU

(t, x) = E

where

[ϕ (Ztx )]

+ 0

t

* + x E ψ s, Zt−s ds

Ztx = etA x + πn WQ (t) ,

x ∈ Hn .

For the integral term (the most difficult one) we use the following facts. For each φ ∈ C ([0, T ] ; Cbα (Hn )) and r ∈ [0, T ], E [φ (r, Ztx )] = with

H

φ r, etA x + y pt (y) dy =

H

φ (r, z) pt z − etA x dz

−1/2 1

−1/2

2 − Q y

dim H det Qt e 2 t . pt (y) = (2π)

Since ∂

−1/2

2 1 ∂ pt z − etA x = − pt z − etA x z − etA x

Qt ∂xi 2 ∂x , −1 i tA = pt z − e x Qt z − etA x , etA ei we have ∂ E [φ (r, Ztx )] = ∂xi

H

= H

Ξi t, z − etA x φ (r, z) pt z − etA x dz Ξi (t, y) φ r, etA x + y pt (y) dy.

50


# - " tA∗ −1 , tA Ξi (t, y) = Q−1 Qt y, ei . t y, e ei = e

where

∗

Hence, with the notation Ξ (t, y) = etA Q−1 t y, ∗ ∇E [φ (r, Ztx )] = E φ r, etA x + WQ (t) etA Q−1 W (t) . Q t Notice that a centered Gaussian vector Z in H with covariance Q (diagonal has the property that for with respect to the basis (ek ), without , restriction) - $ every v ∈ H the Gaussian variable Q−1/2 Z, v = σk−1 Zk vk has variance V ar

" " # # 2 &

σk−2 V ar [Zk ] vk2 = |v|2 . Q−1/2 Z, v = E Q−1/2 Z, v = (2.18)

Therefore " #

−1/2

∇E [φ (r, Ztx )] , v ≤ φ∞ E Qt WQ (t) , Λt v

" # 2 1/2

−1/2 WQ (t) , Λt v

≤ φ∞ E Qt = φ∞ |Λt v| namely

∇E [φ (·, Zt· )]∞ ≤ Λt φ∞ .

For the term E [ϕ (Ztx )] we simply use the following facts: * + ∇E [ϕ (Ztx )] = E etA (∇ϕ) (Ztx ) ∇E [ϕ (Z·· )]∞ ≤ ∇ϕ∞ . (n)

Collecting these estimates in the formula above for uOU (t, x) we get (n) ∇u (t, ·)

∞

≤ ∇ϕ∞ + ≤ ∇ϕ∞ +

t

0

Λt−s ψ∞ ds T

0

Λs ds ψ∞ .

Step 2. In the general case (B (n) = 0) the solution u(n) is the uniform limit, with its first derivative, of the iteration scheme in k ∈ N (n is given), k ≥ 1,


51

(n) ∂uk+1 1 (n) (n) + Ax · ∇uk+1 + T r QD 2 uk+1 ∂t 2 (n) (n) (n) = −B (t, x) · ∇uk , uk+1 (τ, x) = ϕ (x)

with

(n)

u1

=0

(see also Corollary 2.5 below). From the estimate of the previous step, (n) ∇uk+1 (t, ·)

∞

≤ ∇ϕ∞ + ≤ ∇ϕ∞ +

T 0 T

(n) Λs ds B (n) · ∇uk

∞

Λs ds B (n)

0

∞

(n) ∇uk

∞

.

T T Choose T so small that ( 0 Λs ds)B∞ ≤ 12 ; hence ( 0 Λsds)B (n) ∞ ≤ 1 2 for all n ∈ N. Then (n) ∇uk+1 (t, ·)

∞

1 (n) ∇uk 2 ∞ 1 1 (n) ≤ ∇ϕ∞ + ∇ϕ∞ + ∇uk−1 2 2 ∞ ≤ ∇ϕ∞ +

≤ 2 ∇ϕ∞ In the limit as k → ∞ we get the result, for small T . By iteration, we get the claim of the lemma for a general T > 0. To prepare examples, let us state the following simple result. Lemma 2.3. Consider the case Q = (−A)−2γ for some γ ≥ 0, namely σn = λ−γ n ,

γ ≥ 0.

Then condition (2.15) is always true. About condition (2.12), it becomes ∞ &

1

1+2γ n=1 λn

d−2 . 4

52


Finally, for a suitable constant Cγ > 0, we have Λt ≤

Cγ 1

t 2 +γ

and thus assumption (2.16) is fulfilled for γ < 12 . Proof. We have

t

Qt en =

e

−2(t−s)λn

0

σn2 ds

en =

1 − e−2tλn en . 2λ1+2γ n

on the range of Qt and have We may define Q−1 t Q−1 t en =

2λ1+2γ n en . 1 − e−2tλn

Define, a priori formally, the operator Λt as Λt en :=

−1/2 tA Qt e en

√ 1/2+γ −tλ 2λn e n = √ en . −2tλ n 1−e √

We see it is a bounded linear operator for every t > 0, since

1/2+γ −tλn 2λn e

√

1−e−2tλn

is

bounded in n; hence assumption (2.15) is fulfilled. Moreover 2

|Λt x| =

∞ ∞ & Cγ2 1 & 2 (tλn )1+2γ e−2tλn 2 2λ1+2γ e−2tλn 2 2 n x = x ≤ |x| n n −2tλn 1+2γ −2tλn 1+2γ 1 − e t 1 − e t n=1 n=1

where Cγ2 := sup s>0

2s1+2γ e−2s 0) Λt ≤

Cγ 1

t 2 +γ

and assumption (2.16) is fulfilled for γ < ples 2.5 and 2.6, we meet the condition & k∈Zd {0}

1 |k|

2+4γ

which is true for 2 + 4γ > d, namely γ >

1 . 2

Finally, in the case of Exam-

0. Thus, by H¨ older inequality,

−1/2

−1/2

≤ Cα Qt−s e(t−s)A v Qt−s e(t−s)A h T race (Qt−s )α for some constant Cα > 0. This is integrable by assumption and thus allow us to transfer the estimate to ∇uϕ The proof is complete. Similarly to what we have done in the previous section on SDEs, we deduce from the theorem on the Ornstein–Uhlenbeck equation a result on the corresponding backward Kolmogorov system of equations ∂U 1 + Ax, ∇U + B, ∇U + T r QD2 U = −Φ, ∂t 2

U (T, x) = 0.

The proof is the same of Theorem 2.8, by the iteration scheme # 1 ∂U (n+1) " + Ax, ∇U (n+1) + T r QD2 U (n+1) ∂t 2 # " (n) , U (n+1) (T, x) = 0. = −Φ − B, ∇U with minor modifications. Corollary 2.5. Under the assumptions of Theorem 2.11, there exists a solution U (t, x) of the backward Kolmogorov system such that each Uk is of class Uk ∈ C [0, T ] ; Cb2 (H, H) ∩ C 1 [0, T ] ; Cb0 (H, H) and for sufficiently small T and all h, v ∈ H we have

&

sup

hk (v · ∇) Uk (t, x) ≤ 2C1 (T ) |v| |h| ΦC α (T ) b

t,x k

2

2

|∇Uk (t, x) − ∇Uk (t, y)| ≤ 8C2 (T ) Φk C α (T ) |x − y|

2

b

where C1 (T ) and C2 (T ) are given by (2.23) and are independent of dim H. Proof. By Theorem 2.11, at each iteration step we have a solution U (n+1) ∈ C [0, T ] ; Cb2 (H, H) ∩ C 1 [0, T ] ; Cb0 (H, H) , Notice that, when U (n) has such regularity, then B, ∇U (n) is C ([0, T ] ; Cbα (H, H)), so we may continue the iteration. By Theorem 2.11 we precisely have ∇U (n+1) ≤ C1 (T ) V (n) ∞

∞

2.4 Pathwise Uniqueness

61

# " V (n) := Φ + B, ∇U (n) .

where But (n) V

Cbα (T )

≤ ΦC α (T ) + 2 BC α (T ) ∇U (n) α b b C (T ) b ≤ ΦC α (T ) + 2 BC α (T ) C1 (T ) V (n−1) b b

Cbα (T )

.

Choose T such that 2 BC α (T ) C1 (T ) ≤ 1/2. We obtain b

(n) V

Cbα (T )

∇U (n+1)

and thus

∞

≤ 2 ΦC α (T ) b

≤ 2C1 (T ) ΦC α (T ) . b

Moreover, by Theorem 2.11 we have

2

(n) 2 (n+1) (n+1) (t, x) − ∇Uk (t, y) ≤ 2C2 (T ) Vk α

∇Uk

Cb (T )

(n)

where Vk

|x − y|

2

" # (n) = Φk + B, ∇Uk . Thus

(n) Vk

Cbα (T )

(n) ≤ Φk C α (T ) + 2 BC α (T ) ∇Uk b

Cbα (T )

b

.

From inequality (2.24) we have (it is sufficient to take all hj equal to zero except for j = k) (n) ∇Uk

Cbα (T )

(n−1) ≤ C1 (T ) Vk

Cbα (T )

hence (n) Vk

Cbα (T )

By iteration

(n−1) ≤ Φk C α (T ) + 2C1 (T ) BC α (T ) Vk b

b

(n) Vk

Cbα (T )

Cbα (T )

.

≤ 2 Φk C α (T ) b

which implies

2

(n+1) (n+1) 2 2 (t, x) − ∇Uk (t, y) ≤ 8C2 (T ) Φk C α (T ) |x − y| .

∇Uk b

62


By Ascoli–Arzel` a theorem, we conclude as in the proof of Theorem 2.8. The proof is complete.

2.4.2 Proof of Theorem 2.10 Let X (i) , i = 1, 2, be two mild solutions. Now H is the infinite dimensional space introduced at the beginning of the section. Step 1 (projection to finite dimensions). Let Hn and πn as in the proof of Theorem 2.9, and let (i,n)

Xt

(i)

:= πn Xt , (i,n)

The processes Xt (i,n)

dXt

(i,n)

= AXt

where

(n)

Wt

(n)

and Wt

:= πn Wt ,

live in Hn and we have

(i,n) dt+ dt+b(n) t, Xt (i,n)

Rt

B (n) (t, x) := πn B (t, x) .

(n)

QdWt

(i,n)

+Rt

dt

(i,n)

X0

= πn x.

(i) (i,n) − b t, Xt . = πn b t, Xt (i,n)

(i)

Since B is continuous, and Xt converges (in n) to Xt by definition, in (i,n) several topologies, it is easy to prove that Rt goes to zero in H uniformly in t, a.s. in ω; and it is bounded, so it converges to zero in several topologies. Step 2 (reformulation). For every positive integer n, consider the backward Kolmogorov equation, in the unknown U (n) (t, x), function from [0, T ]× Hn in Hn : 1 ∂U (n) + Ax + B (n) (t, x) · ∇U (n) + T r QD2 U (n) = −B (n) , ∂t 2 (n) U (T, x) = 0. (n)

(n)

Denote by Uk its components. The system is decoupled: each Uk (n) separate equation, with right-hand-side −Bk . From Corollary 2.5 we deduce the following result.

solves a

Lemma 2.4. Under the assumptions of Theorem 2.11, there exists a solution U (n) (t, x) such that (n)

Uk

∈ C [0, T ] ; Cb2 (Hn , Hn ) ∩ C 1 [0, T ] ; Cb0 (Hn , Hn ) ,

and it satisfies


63

&

(n) sup

hk (v · ∇) Uk (t, x) ≤ 2C1 (T ) |v| |h| B (n) α

Cb (T ) t,x

k

for all h, v ∈ Hn and

2

(n) 2

(n) (n) 2

∇Uk (t, x) − ∇Uk (t, y) ≤ 8C2 (T ) Bk α |x − y| Cb

where C1 (T ) and C2 (T ) are given by (2.23) and are independent of dim H. (i,n) from some of the expressions) We have (we drop the argument t, Xt ∂U (n) 1 (i,n) (n) (i,n) (n) k t, Xt = dt dt + ∇Uk · dXt + T r QD2 Uk ∂t 2 (n) (i,n) (n) (i,n) (n) (n) = −Bk t, Xt dt + ∇Uk · Rt dt + ∇Uk · QdWt

(n)

dUk

hence (i,n)

dXt

(i,n)

= AXt

(n)

(i,n)

dt + QdWt + Rt dt (i,n) (i,n) + ∇U (n) · Rt dt + ∇U (n) · − dU (n) t, Xt (i,n)

(n)

(n)

QdWt

(i,n)

where ∇U (n) · Rt is a vector of components ∇Uk · Rt and ∇U (n) · √ (n) (n) √ (n) QdWt is a vector of components ∇Uk · QdWt . Thus (i,n) (i,n) d Xt + U (n) t, Xt (i,n) (i,n) (i,n) = A Xt + U (n) t, Xt dt − AU (n) t, Xt + (i,n)

+ ∇U (n) · Rt

dt + ∇U (n) ·

(n)

QdWt

(n)

QdWt

.

Therefore (i,n) = etA πn x + U (n) (0, πn x) + U (n) t, Xt t e(t−s)A U (n) s, Xs(i,n) ds −A 0 t t e(t−s)A Rs(i,n) ds + e(t−s)A ∇U (n) s, Xs(i,n) · Rs(i,n) ds + 0 0 t + e(t−s)A ∇U (n) s, Xs(i,n) + In · QdWs(n) .

(i,n)

Xt

0

(i,n)

+ Rt

dt

64


Step 3 (book-keeping of terms to be estimated). For the difference between the solutions we have (1,n)

Xt

(2,n)

− Xt

= I1 + I2 + I3,1 − I3,2 + I4,1 − I4,2 + I5

(2,n) (1,n) I1 = U (n) t, Xt − U (n) t, Xt

where I2 = A

0

t

e(t−s)A U (n) s, Xs(2,n) − U (n) s, Xs(1,n) ds

t

I3,i = I4,i = I5 = 0

0

t

0

t

e(t−s)A Rs(i,n) ds,

i = 1, 2

e(t−s)A ∇U (n) s, Xs(i,n) · Rs(i,n) ds

e(t−s)A ∇U (n) s, Xs(1,n) − ∇U (n) s, Xs(2,n) ·

QdWs(n) .

Let us treat in detail the terms I1 , I2 and I5 . The other are left to the reader, with the remark that we do not need to take advantage of the differences to treat I3,1 − I3,2 + I4,1 − I4,2 , but simply each one of such four terms will converge to zero. Step 4 (estimate for I1 ). Given a smooth g : Rd → Rd we have

&

|g (x) − g (y)| ≤ sup

hk ((x − y) · ∇) gk (z) .

z,h k

where the supremum in h is made over all h ∈ H such that |h| = 1. Indeed,

&

hk (gk (x) − gk (y))

|g (x) − g (y)| ≤ sup

h k

1

&

≤ sup

hk ((x − y) · ∇) gk (αx + (1 − α) y) dα

h 0 k

&

hk ((x − y) · ∇) gk (z) . ≤ sup

z,h

k

Hence


65

2

(2,n) (1,n)

2 − U (n) t, Xt |I1 | = U (n) t, Xt

&

(1,n) (2,n) (n) ≤ sup

hk Xt − Xt · ∇ U (t, z) .

z,h k

By the lemma,

(1,n) (2,n) (n) ≤ 2C1 (T ) Xt − Xt

B

Cbα (T )

.

We have limT →0 C1 (T ) = 0. Then we may choose T such that

(1,n) (2,n)

|I1 | ≤ ε Xt − Xt

with ε > 0 chosen below. Step 5 (estimate for I2 ). The estimate of I2 presents a problem. We have to estimate an expression of the form

t

A

e(t−s)A g (s) ds

0

but we know only that (t−s)A Ae

L(H,H)

≤

C t−s

which is a non-integrable convolution kernel. When the integrand is H¨ older continuous in time, there is a trick:

t

A

e 0

(t−s)A

t

g (s) ds = A

e(t−s)A (g (s) − g (t)) ds + I − etA g (t) .

0

This method requires Hölder estimates in time for U (n) and for the solutions (i,n) Xt . We prefer to use a different idea. We use a “maximal regularity” result for semigroup convolutions. Consider the map t

g → A

e(t−s)A g (s) ds

0

well defined for g ∈ L2 (0, T ; Hn ). As far as we work in a finite dimensional space Hn this integral is well defined. Lemma 2.5. There exists a constant C > 0 independent of the dimension of H, such that

66


T

0

t

2

−(t−s)A

A

e g (s) ds dt ≤ C

0

T

|g (t)|2 dt

0

for all g ∈ L2 (0, T ; Hn). Proof. Introduce the extension g equal to g on [0, T ], zero outside. Let us interpret t λj e−(t−s)λj gj (s) ds 0

as a convolution over the full real line ∞ hj (t − s) gj (s) ds (hj ∗ gj ) (t) = −∞

where hj (t) is λj e−tλj for t ≥ 0, zero otherwise. We have

t

λj

e−(t−s)λj gj (s) ds = (hj ∗ gj ) (t) for all t ∈ [0, T ] .

0

Both functions gj and hj are square integrable, hence we may use the properties of Fourier transform (the identities are correct up to a constant, depending on the precise definition of Fourier transform):

∞

−∞

2

2

hj ∗ gj (ξ) dξ =

2

.

hj (ξ) g.j (ξ) dξ −∞ −∞ ∞

∞

2

. 2 2 = |g.j (ξ)| dξ

hj (ξ) |g.j (ξ)| dξ ≤ C −∞ −∞ ∞ 2 =C |gj (t)| dt

|(hj ∗ gj ) (t)| dt =

∞

∞

−∞

where C > 0 is a constant independent of j, because h.j (ξ) = and thus

∞ −∞

eiξt hj (t) dt =

∞

eiξt λj e−tλj dt =

0

λj iξ − λj

2

.

hj (ξ) ≤ C

independently of j and ξ. We have proved 0

T

2

t

−(t−s)λj

λj e gj (s) ds dt ≤

0

∞

−∞ ∞

≤C

2

|(hj ∗ gj ) (t)| dt

−∞

T

2

|gj (t)| dt = C 0

2

|gj (t)| dt.


67

Therefore

T

0

t

2

−(t−s)A

A

dt = e g (s) ds

0

02 &/ t e−(t−s)A g (s) ds, ej dt A

T

0

j

0

j

0

2 & t

−(t−s)λj

λj

dt e g (s) ds j

T

= 0

T

≤C 0

2

|gj (t)| dt

j T

=C

&

2

|g (t)| dt.

0

The proof is complete. From the lemma we deduce

2

t

−(t−s)A

A e g (s) ds dt ≤ C

T 0

0

T

2

|g (t)| dt.

0

Therefore

T

2

|I2 (t)| dt ≤ C

0

0

T

2

(n) s, Xs(2,n) − U (n) s, Xs(1,n) ds.

U

Since we know that

(n)

U (s, x) − U (n) (s, y) ≤ ε |x − y| uniformly in x, y, s, n, for T small enough, we get

T

2

|I2 (t)| dt ≤ ε

T

2

0

0

2

(2,n)

− Xs(1,n) ds.

Xs

Step 6 (estimate for I5 ). Recall that ∇U (n) · (n) √ (n) of components ∇Uk · QdWt : ∇U (n) ·

(n)

QdWt

=

n & k=1

(n)

∇Uk

·

√

(n)

QdWt

(n)

QdWt

denotes a vector

ek .

Thus the meaning of I5 is I5 =

n & k=1

t

ek 0

(n) (n) e−(t−s)λk ∇Uk s, Xs(1,n) − ∇Uk s, Xs(2,n) · QdWs(n) .

68


Therefore |I5 |2 =

n t &

e−(t−s)λk ∇U (n) s, Xs(1,n) − ∇U (n) s, Xs(2,n) ·

k k 0

k=1

2

(n)

QdWs

which implies E |I5 |2 =

n & k=1

=

E

n &

t

e

t

e

−(t−s)λk

0

(n) ∇Uk

(1,n) s, Xs

(n) − ∇Uk

(2,n) s, Xs

2

·

(n)

QdWs

⎤ n

2 &

(n) (1,n) (n) (2,n) 2 E⎣ s, Xs − ∂j Uk s, Xs

∂j Uk

σj ⎦ ds. ⎡

−2(t−s)λk

k=1 0

j=1

We have proved that n

2 &

(n) 2

(n) (n) 2

∂j Uk (t, x) − ∂j Uk (t, y) ≤ 8C2 (T ) Bk α |x − y| . Cb

j=1

Therefore 2 E |I5 | ≤ C

t

0

n & k=1

(n) 2 e−2(t−s)λk Bk α Cb

2

(1,n) (2,n)

− Xs E Xs

ds.

Step 7 (conclusion). Due to the weak estimate of step 5 we use the integral norm. We have

2

& T

(1,n) (2,n)

2 E Xt − Xt E |Ii,j (t)| dt

dt = 0 0 T

T

2

2

2

(1,n) (2,n)

2 E Xt − Xt E Rs(1,n) + Rs(2,n) ds ≤ 18ε

dt + C∇U (n) 0 0 T t & n 2

2

(n)

e−2(t−s)λk Bk α E Xs(1,n) − Xs(2,n) dsdt +C

T

0

0

k=1

Cb

where C∇U (n) includes a uniform bound on ∇U (n) , which is uniform in n. The last term can be rewritten as T T & n 2

2

(1,n) −2(t−s)λk (n) (2,n)

e − Xs Bk α dt E Xs

ds 0

≤ε 0

s

T

k=1

Cb

2

(1,n) (2,n)

E Xs − Xs

ds


69

for sufficiently small T , since 0

T

n & k=1

(n) 2 e−2tλk Bk α dt < ∞. Cb

We choose now ε and then T , to get

T 0

2

(1,n) (2,n)

E Xt − Xt dt ≤ C

0

T

2

2

E Rs(1,n) + Rs(2,n) ds (1)

(2)

from the main estimate just proved. This readily implies Xt − Xt , by taking the limit as n → ∞. As in previous proofs, to be completely rigorous we have to show first that the second moment of mild solutions used here is finite. But this follows easily from the mild formulation. The proof is complete.

Chapter 3

Dyadic Models

3.1 Introduction: 3D Euler Equations The classical 3-dimensional Euler equations are the system of PDEs ∂u + (u · ∇) u + ∇p = 0 ∂t div u = 0 where u : [0, T ]×D → R3 is the velocity field of the fluid and p : [0, T ]×D → R is the pressure field. This system describes a fluid which occupies the region D ⊂ R3 , is non viscous, non forced, incompressible. Fluid particles (an abstraction of a very small fluid element, not a molecule or microscopic object) move following classical second Newton law: if Xta denotes the position at time t of a fluid particle which started at time zero from position a, d then the velocity dt Xta at time t must be u (t, Xta ), if u denotes fluid velocity as said above. But then the acceleration is d d2 a X = u (t, Xta ) = u (t, Xta ) · ∇u (t, Xta ) . dt2 t dt This expression is the so-called total (or Lagrangian) derivative of u. Newton second law states that this acceleration must be equal to the force acting on the fluid particle (we set mass equal to one). The only force, being the fluid inviscid and not forced, is the pressure force, given by −∇p. This is a short (incomplete) explanation of the equation. In spite of its apparent simplicity, the mathematical analysis of this equation is still a tremendous task. It is not even know a satisfactory theorem of global-in-time existence of solutions! Not to mention uniqueness. At least, for the corresponding viscous equations, the Navier–Stokes equations,


71

72

3 Dyadic Models

∂u + (u · ∇) u + ∇p = νΔu ∂t div u = 0 existence of weak solutions is known. For the Euler (and Navier–Stokes) equations it is known that if the initial condition is sufficiently regular, there is local-in-time existence and uniqueness of a regular solution; but it is open whether this regular solution is global in time, or singularities may appear. A summary of the results on Euler equations is out of the scope of this lectures. Let us only mention that Scheffer [184] and Schnirelman [187] found examples of non-uniqueness. Schnirelman [188], and recently De Lellis and Székelyhidi [78, 79] found solutions which dissipate energy. About the latter issue, let us notice that if u is a sufficiently regular solution, and the boundary conditions allow to avoid boundary terms (consider for instance the case D = R3 ), then d dt

∂u |u (t, x)| dx = 2 u· dx = −2 ∂t 3 3 R R 2

R3

u · (u · ∇) udx − 2

R3

u · ∇pdx

and

R3

2u · (u · ∇) udx =

R3

2

d dt

R3

R3

2

|u| div udx = 0

We get

(u · ∇) |u| dx = −

R3

u · ∇pdx = −

p div udx = 0. R3

2

|u (t, x)| dx = 0, namely R3

2

|u (t, x)| dx =

2

R3

|u (0, x)| dx.

The kinetic energy is constant in time for regular solutions but, as we said, regular solutions are not known to exist globally in time. For weak solutions the previous computations cannot be performed and the authors mentioned above found examples which dissipate energy. Maybe it is interesting to notice that weak solutions require a lot of care, if we want to perform computations. Naively one could think that it is only a matter of regularizing, doing the computations and taking the limit at the end, a strategy which usually works. But a commutator arises, between the regularization and the transport (or inertial) operator (u · ∇) u. Precisely, again the full space, let u be a weak solution (we omit the details of the definition) and let us denote by uε the function uε (t, x) = (θε ∗ u) (t, x) =

R3

θε (x − y) u (t, y) dy

3.1 Introduction: 3D Euler Equations

73

where θε are classical smooth mollifiers, and define pε the same way. One has ∂uε + (u · ∇) uε + ∇pε = Rε ∂t div uε = 0 where Rε = (u · ∇) uε − (θε ∗ ((u · ∇) u)) . As above we have d dt

2

R3

|uε (t, x)| dx = ... = 2

R3

uε · Rε dx.

Which kind of regularity of u is necessary to prove that lim uε · Rε dx = 0? ε→0

R3

This issue received a lot of attention and we cannot summarize the results, but it must be clear that some regularity is needed. For instance, the regularity of the so called weak solutions of the viscous case has not been proved to be sufficient; and thus a fortiori the one of the weak solutions of Euler, which is even weaker. An indeed, energy dissipation for the latter has been proved, showing that R3 uε ·Rε dx cannot go to zero for them. The problem of convergence of the commutator for weak solutions is a central one in Chap. 4. To summarize: 3D Euler equations have problems of existence, uniqueness, anomalous energy dissipation. Does a noise, added to the equations, change this picture? We do not know, and the question looks too difficult at present. But we understand something of a very simplified model of Euler equations, a nonlinear infinite dimensional example which has something in common with Euler system, the so called dyadic model. This is the aim of this chapter.

3.1.1 Fourier Formulation of Euler Equations To compare with the dyadic model, let us see the Euler equations on a torus 3 [0, 2π] formulated in Fourier terms. Write u (t, x) =

k∈Z3

uk (t) eik·x ,

p (t, x) =

pk (t) eik·x

k∈Z3

where, notice, uk (t) takes values in C3 . The condition div u = 0 means 3 j ik·x = 0 which is equivalent to k∈Z3 j=1 uk (t) ikj e

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3 Dyadic Models

k · uk (t) = 0. Substituting the Fourier developments into the Euler equations we get duk (t) eik·x + i ei(k +k )·x (uk (t) · k ) uk (t) dt k∈Z3 k ,k ∈Z3 + pk (t) keik·x = 0. k∈Z3

This is equivalent to duk (t) +i dt

(uk (t) · k ) uk (t) + pk (t) k = 0.

k +k =k

Let us eliminate the pressure by projecting on the plane orthogonal to k. (v,k) k⊗k 3 Given a vector v ∈ R , such a projection is πk v = v − (k,k) k = I − |k|2 v. Hence we get (recall k · uk (t) = 0) duk (t) +i dt

(uk (t) · k ) πk uk (t) = 0.

k +k =k

Each Fourier component uk (t) solves a differential equation which involves all other components uk (t). We remark this fact in view of the simplicity of the model introduced below. Another difference is that here uk (t) ∈ C3 . A third one is that here we do not have “squares” in the quadratic term: if k = k , then uk (t) · k = 0.

3.2 The Dyadic Model Since Euler equation is still too far away from our understanding, we try to see the interaction between noise and unusual phenomena like non-uniqueness and anomalous energy dissipation on a simple nonlinear model, called dyadic model of turbulence. It has a number of features similar to those of Euler equations of fluid dynamics. However, it is much simpler, since, in the language of particle systems, it is a nearest neighbor problem, opposite to the true Euler system where Fourier components interact at every distance in wave space. The dyadic model is the following infinite system of differential equations:

3.2 The Dyadic Model

75

dXn (t) 2 = kn−1 Xn−1 (t)−kn Xn (t) Xn+1 (t) , dt

t ≥ 0,

Xn (0) = xn ,

n≥1

(3.1) with X0 ≡ 0 and positive coefficients kn . We shall add noise at due time. The intuitive idea compared to classical fluid dynamics is that Xn (t) corresponds to some kind of aggregation of Fourier coefficients relative to wave numbers k in some shell, like 2n ≤ |k| < 2n+1 , but this correspondence is not strict. Each level n interacts only with nearest neighbor levels n − 1 and n, in a sort of cascade picture, which will be more clear below. The coefficients kn look like derivations. Very crudely, the dyadic system is like the Fourier system above when we throw away most of the interactions; however, as we have already remarked in the previous section, it is not even so, because 2 (t) appears here, opposite to Euler Xn (t) is real valued and the square Xn−1 equations in Fourier form; and both these facts are essential for the analysis below. So, what do the dyadic model and Euler equation have in common? A few structural facts: • • • • •

The nonlinearity is quadratic They are infinite dimensional system The energy is constant, for sufficiently regular solutions No other positive definite invariant quantity is known A number of a posteriori features look similar

The third fact, formal energy conservation, will be explained a little below. Let us add that Cheskidov et al. [52] provide a phenomenological derivation of the dyadic model from fluid dynamics concepts. Among the relevant literature on this model, see also Friedlander and Pavlovic [109], Katz and Pavlovic [130], Kiselev and Zlatoˇs [132], Waleffe [195], Cheskidov [51], Cheskidov et al. [53], Barbato et al. [24]. Let us also remark that there are other derivations which relate the dyadic model to the space discretization of non-viscous Burgers equation. To fix the ideas, and to reflect the shell structure outlined above, take kn = λn ,

λ>1

but some of the results are more general. 2 The special coupling kn−1 Xn−1 − kn Xn Xn+1 satisfies the following formal property: ∞ d E (t) = 0, E (t) := Xn2 (t) dt n=1 (we call E (t) the energy of the system). Indeed, N d 2 2 X = −2kN XN XN +1 dt n=1 n

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3 Dyadic Models

N N d 2 2 2 (telescoping because dt n=1 Xn = 2 n=1 kn−1 Xn−1 Xn − kn Xn Xn+1 series). Hence, if XN (t) → 0 as N → ∞, sufficiently fast, we have energy conservation. This is true for sufficiently regular solutions (here regularity means fast decay of Xn2 to zero). Initially regular solutions (of class H 1 , for N instance, where H 1 is defined by the condition n=1 kn2 Xn2 ) persists regular only for a local time: all of them loose regularity in finite time. This remarkable fact, open for 3DE, has been proved by Kiselev and Zlatoˇs [132]. Thus, non-regular solutions must be investigated, if we want to see the problem globally in time. For non-regular solutions, it happens that most of them have energy dissipation. Below we limit ourselves to prove energy dissipation for solutions with positive components. Since the system is reversible, there is also a class with increasing energy, but the initial conditions which allow this form a small set, in proper sense. There are also selfsimilar solutions, which blow-up or go to zero, depending on the time-direction we observe them. They provide the existence of solutions with sharper behavior, and perhaps they capture the generic decaying behavior, although this has not been proved. We shall not treat this delicate subject. In the class of non-regular solutions, opposite to Euler equations, there is a class with uniqueness. At present however the set of initial conditions corresponding to that class is still small, we do not know whether for technical or substantial reasons. We give below a particular case of this uniqueness result. For a small set of initial conditions we also know non-uniqueness: there exist both solutions with increasing and with decreasing energy. This is a very important non-uniqueness example (for this reason it is developed below in detail).

The non-linearity is completely different from non-regular functions like |x|, it is just a second order polynomial. Hence the mechanism for non-uniqueness has a different origin. The mechanism here is that energy enters from infinity, there are solutions which pump energy from infinity (in Fourier space). To a certain degree of approximation, it is the same mechanism discovered by Shnirelman [188] for the 3D Euler equations (much more sophisticated in that case). In the next section we present some of these deterministic results. Then in the next section we consider a random perturbation and prove uniqueness for all initial conditions. This is remarkable example of regularization by noise which gives some hope for future research on the 3D Euler equations. The peculiar configurations that may pump energy from infinity are not visible (not stable) under random perturbations.

3.3 Deterministic Results

77

3.3 Deterministic Results 3.3.1 Preliminaries We denote by l2 the Hilbert space of square-summable sequences of real numbers. Definition 3.1. We say that X = (Xn ) is a solution of equation (3.1) if N X ∈ L∞ 0, T ; l 2 ∩ C ([0, T ] ; R) and each equation in (3.1) is satisfied (in integral form, then a posteriori in differential form). We say it is a Laray solution if ∞

Xn2 (t) ≤

n=1

∞

x2n ,

∀t ∈ [0, T ] .

n=1

We say it is a positive solution if Xn (t) ≥ 0 for all n ∈ N and t ∈ [0, T ]. Proposition 3.1. For every x ∈ l2 there exists at least one Leray solution with initial condition x. If x is positive (xn ≥ 0 for all n ∈ N) then all solutions with initial condition x are positive. Proof. We do not give all the details of existence of a solution, purely deterministic (and easy). The idea is simply to consider the finite dimensional system

2 (t) − kn Xn (t)Xn+1 (t) X˙n (t) = kn−1 Xn−1 XN +1 (t) = 0

∀t ≥ 0, ∀t ≥ 0

∀n ∈ {1, 2, . . . , N }

which has a unique solution XnN n∈{1,2,...,N } with constant energy: N N N 2 2 Xn (t) = x2n ≤ x l2

n=1

n=1

and a theorem to each sequence N use this bound to apply Ascoli–Arzel` Xn N >n , for each given n ≥ 1; by a diagonal-in-n argument one can extract a subsequence which converges componentwise to some X = (Xn )n∈N . Since each equation of the system depends only on a finite number of components, this allows one to pass to the limit and show that X is a solution. From this argument we also prove m n=1

Xn2 (t) ≤ x 2l2

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3 Dyadic Models

for every m > 0 and t ∈ [0, T ] and thus, by monotone convergence in m, that X is a Leray solution. Finally, if X is a solution with positive initial condition, then Xn (t) = e−kn

t 0

Xn+1 (r)dr

xn +

t

e−kn

t s

Xn+1 (r)dr

0

2 kn−1 Xn−1 (s)ds

and thus X is positive.

3.3.2 Anomalous Energy Dissipation Let us start to get an idea of the energy exchange between modes by looking at the signs (the analysis of this equation is often of algebraic nature instead of functional analytic one). Let us introduce the energy of first N modes N EN (t) := n=1 Xn2 (t) and recall from above that d 2 EN (t) = −2kN XN XN +1 . dt Thus, remembering that we assume kn > 0 for each n: • • •

If XN +1 > 0, the energy EN (t) decreases If all components XN are positive, there is a direct energy cascade: EN (t) decreases for each N If XN +1 < 0, the energy EN (t) increases; there is an inverse cascade at such N

The sign of components is thus very important. It is therefore relevant that positivity of each single component is preserved dynamically: •

if, for some n, Xn (0) ≥ 0 (resp. > 0) then Xn (t) ≥ 0 (resp. > 0) for all t ≥ 0.

The proof is the same of the positivity of the previous proposition. Putting together some of the previous facts, we see that the direct cascade is a robust phenomenon, while the inverse cascade may happens but is less stable. We denote by x = (xn )n≥1 the initial condition. Let us analyze more closely the case xn ≥ 0 for all n ≥ 1 which implies direct cascade. A direct cascade does not mean energy dissipation but just a shift of energy to the right (the right in n). To have energy dissipation we need a very fast shift. Does it occur? When 2 XN +1 = C > 0 lim inf kN XN N →∞

(3.2)


79

(possibly C = ∞), namely XN decreases not too fast, there is a constant rate of energy shift, the same for all blocks of modes, and this should lead to energy dissipation. For the aficionados of turbulence, we see intuitively from (3.2) that there 1/3 is a constant rate of shift of energy to high modes if XN ∼ kN , a condition related to Kolmogorov–Obukov 1941 theory and Onsager conjecture, Kolmogorov [133], Frisch [111], Gallavotti [114]. For our model it is not clear how to prove (if it is true) that solutions tend 1/3 to build up an inertial range XN ∼ kN which would allow for the energy shift (on the opposite, the forced deterministic model has an exponentially 1/3 attracting fixed point which is XN ∼ kN , see Cheskidov et al. [52, 53]). Our numerical simulations confirm that this happens but we do not have a proof. So, our proof of energy dissipation is not based on a property like (3.2). It is more abstract, in a sense. A proof with more physical content would be very interesting. Remark 3.1. Anomalous energy dissipation may happen only for solutions which are eventually positive: infinitely many negative component interrupts the flow. Intuitively it is clear, and can be easily proved by exercise. Let us prove energy dissipation. The idea is to prove first a control of the form En (τn ) ≤ En−1 (0) + δn with very small δn and not too large τn . Since En (0) can be larger than En−1 (0) + δn , this inequality is a quantitative control on the energy cascade (although not the first one that one would write down). Notice that E0 (0) = 0. Thus, when such an inequality is proved, one iterates it in n: if the series of ∞ τn ’s converges, we get a control on the global energy at time τ = n=1 τn . Lemma 3.1. Assume xn ≥ 0 for all n ≥ 1. Given any sequence of positive numbers (δn ), let τn be the first time we have En (τn ) ≤ En−1 (0) + δn if such a time exists, otherwise τn = +∞. Then τn is finite and τn ≤ Tn where Tn is defined by the identity 0

Tn

2

xn − δn E (0) . 1 − exp −kn E (0)s ds = 2 2kn δn

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3 Dyadic Models

Proof. Step 1. We shall use two basic relations: En (t) ≤ En−1 (0) + inf Xn2 [0,t]

En (t) = En−1 (0) + x2n − 2kn

t

0

Xn2 Xn+1 ds.

The first one is proved by (s ≤ t) En (t) ≤ En (s) = En−1 (s) + Xn2 (s) ≤ En−1 (0) + Xn2 (s) . The second one by

t

d En (s) ds ds 0 t 2 = xn + En−1 (0) − 2kn Xn2 Xn+1 ds.

En (t) = En (0) +

0

Step 2. Let us prove, by contradiction, that τn is finite. If it is not finite, from the first identity of step 1 we must have inf [0,∞) Xn2 ≥ δn , hence (from the second identity) En (t) ≤ En−1 (0) +

x2n

− 2kn δn

t 0

Xn+1 (s) ds.

At the same time, from the variation of constant formula above we have Xn+1 (t) ≥

t 0

t exp −kn Xn+2 (r) dr kn Xn2 (s) ds

≥ kn δn

t 0

s

exp −kn

E (0) (t − s) ds

δn 1 − exp −kn E (0)t =√ E0 2 where we have used the rough inequalities Xn+2 (r) ≤ En+2 (r) ≤ En+2 (0) ≤ E (0). Thus

2kn δ 2 t En (t) ≤ En−1 (0) + x2n − √ n 1 − exp −kn E (0)s ds. E0 0

t √ Since limt→∞ 0 1 − exp −kn E0 s ds = +∞, we get a contradiction. Step 3. Let Tn be defined as above (it exists and is unique). If inf [0,Tn ] Xn2 ≤ δn , then τn ≤ Tn . Otherwise, inf [0,Tn ] Xn2 > δn , thus, as


81

above, we get 2kn δ 2 En (Tn ) ≤ En−1 (0) + x2n − √ n E0

Tn

0

1 − exp −kn E (0)s ds.

But x2n

2kn δ 2 − √ n E0

Tn

0

1 − exp −kn E (0)s ds = δn

hence τn ≤ Tn . The previous equality is true since it is equivalent to the definition of Tn . The proof is complete. Theorem 3.1. For every positive solution, lim E (t) = 0.

t→∞

Proof. Step 1. Let us prove a more explicit estimate for τn . By definition of Tn we have

3/2 1 − exp −kn E (0)Tn E (0)

≤ Tn − . 2kn δn2 kn E (0) Hence

1 − exp −kn E (0)Tn 1

+

τn ≤ Tn ≤ Tn − kn E (0) kn E (0) 3/2 1 E (0) 1 . ≤ +

kn 2δn2 E (0) Step 2. Iterating the claim of the lemma we have En

n

τn

≤ E0 (0) +

k=1

n

δk =

k=1

n

δk .

k=1

Given α > 0, let us take (δk ) such that ∞ k=1

δk = α

and

1 < ∞. kn δn2

This is possible since kn ∼ λn , λ > 1. Set τ :=

∞

k=1 τn .

Thus

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3 Dyadic Models

En (τ ) ≤ En

n

τn

≤α

k=1

and thus E (τ ) ≤ α. The proof is complete.

We have proved energy dissipation for all positive solutions. The result remains true for a larger class of initial conditions. The proof however is quite tricky, we omit it.

3.3.3 Numerical Picture of the Anomalous Dissipation Assume we start with a bump-like, compact support profile as in picture (a) of Fig. 3.1. On the horizontal axis we have the indexes n ∈ N, on the vertical the values Xn , at a given time t; we draw a continuous graph for expository reasons, but it should be just the graph of a sequence. Then very slowly it modifies to a decreasing shape like the one of picture (b), with a little “head”; a sort of “snail”. Then the snail elongates, slowly at the beginning, faster and faster after a while, until it becomes, in finite time, an infinitely long snail, −1/3 see picture (c). The shape is ∼ kn . Finally, when such a spacial shape is built, dissipation starts, in a selfsimilar way: the shape goes down like −1/3

kn

t see picture (d). This beautiful movie is unfortunately unproved, mostly. Rigorously, we only know a few bits of it. First that anomalous dissipation occurs.

Fig. 3.1 (a) Initial configuration. (b) Preparation. (c) Dissipative shape. (d) Self-similar energy dissipation


83

Second, that the energy of dissipating solutions goes to zero roughly like 1 t2 (see Barbato et al. [20] for precise statements). Third, that self-similar solutions of the form an , t

an ∼ kn−1/3

exist, see [20] (the existence of a self-similar solution is particularly tricky).

3.3.4 Examples of Non-Uniqueness By time change, the previous result of energy dissipation easily implies the existence of negative solutions with increasing energy. Indeed, if (Xn (t))t∈[0,T ] is a solution, then Yn (t) = −Xn (T − t) ,

t ∈ [0, T ]

is again a solution, because 2 Yn (t) = Xn (T − t) = kn−1 Yn−1 − kn Yn Yn+1 ,

t ∈ [0, T ] .

But we have proved above the existence, for each x ∈ l 2 , of a solution with non-increasing energy. Thus we have proved: Theorem 3.2. There exist initial conditions x = (xn )n≥1 ∈ l2 , with negative components xn , with more than one corresponding solution of system (3.1). We may construct examples where one solution has increasing energy, the other decreasing energy. Figure 3.2 represents the increasing energy of a solution, with branching at every time of a decreasing energy one. On the horizontal axis we have time, on the vertical energy.

Fig. 3.2 Branching of increasing energy solutions

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3 Dyadic Models

This negative result is the starting point for the stochastic investigation we shall discuss below.

3.3.5 Uniqueness of Positive Solutions However, let us add more comments on uniqueness for (3.1). The result is taken from Barbato et al. [22]. Theorem 3.3. Assume

kn = 0. 2n Then, for all positive initial conditions, there exists only one Leray solution. lim

n→∞

Proof. Assume that X (1) and X (2) are two solutions. Let Zn := Xn(1) − Xn(2)

Yn := Xn(1) + Xn(2) .

Hence d d d Zn = Xn(1) − Xn(2) = dt dt dt 2 2 (1) (2) (1) (2) =kn−1 − kn Xn(1) Xn+1 − Xn(2) Xn+1 Xn−1 − Xn−1 Therefore, since

(1)

Xn−1 (1)

(2)

2

2 (2) − Xn−1 = Zn−1 Yn−1 (1)

(1)

(1)

(2)

Xn(1) Xn+1 − Xn(2) Xn+1 = Xn(1) Xn+1 − Xn(2) Xn+1 + Xn(2) Xn+1 − Xn(2) Xn+1 (1)

= Zn Xn+1 + Xn(2) Zn+1 we have

d dt Zn

(1) (2) = kn−1 Zn−1 Yn−1 − kn Zn Xn+1 + Xn Zn+1

Zn (0) = 0

∀n≥1 ∀n≥1

.

This implies 1 d 2 (1) Z = kn−1 Yn−1 Zn−1 Zn − kn Xn+1 Zn2 − kn Xn(2) Zn Zn+1 . 2 dt n The usual bounds, for equations of Navier–Stokes type, are based performed 2 Z , corresponding to the L2 norm, or energy norm. on the quantity ∞ n=1 n


85

For it we do not see any chance to prove a closed estimate, since kn−1 Yn−1 , (1) (2) kn Xn+1 and kn Xn may diverge. But notice that kn−1 Yn−1 Zn−1 Zn − kn Xn(2) Zn Zn+1 has almost a telescoping structure: it breaks by the small difference between Y and X (2) . If we could take advantage of a telescoping structure of this term, the other one (1) −kn Xn+1 Zn2 would be dissipative, for positive solutions. This is the idea. The difference between Y and X (2) can be eliminated with a little more algebra. Indeed, not only it is true that (1)

(2)

(1)

Xn(1) Xn+1 − Xn(2) Xn+1 = Zn Xn+1 + Xn(2) Zn+1 but also that (1)

(2)

(1)

(2)

(2)

(2)

Xn(1) Xn+1 − Xn(2) Xn+1 = Xn(1) Xn+1 − Xn(1) Xn+1 + Xn(1) Xn+1 − Xn(2) Xn+1 (2)

= Xn(1) Zn+1 + Zn Xn+1 and thus, summing these two identities, (1)

(2)

Xn(1) Xn+1 − Xn(2) Xn+1 =

1 (Zn Yn+1 + Yn Zn+1 ) . 2

Now we have d 1 Zn = kn−1 Zn−1 Yn−1 − kn (Zn Yn+1 + Yn Zn+1 ) dt 2 and thus 1 d 2 1 1 Z = kn−1 Yn−1 Zn−1 Zn − kn Zn2 Yn+1 − kn Yn Zn Zn+1 . 2 dt n 2 2 But now the telescoping feature is broken by the factor eliminated by multiplication by 2−n . Indeed,

1 2.

This can be

Zn−1 Zn Zn Zn+1 1 d Zn2 = kn−1 Yn−1 − 2−n−1 kn Zn2 Yn+1 − kn Yn n+1 . 2 dt 2n 2n 2 The telescoping character of kn−1 Yn−1

Zn−1 Zn Zn Zn+1 − kn Yn n+1 2n 2

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3 Dyadic Models

is restored! We get N N ZN ZN +1 1 d Zn2 = − 2−n−1 kn Zn2 Yn+1 − kN YN . n 2 dt n=1 2 2N +1 n=1

Being the solution positive, we simply throw away the first term and get t N kN 1 Zn2 (t) ≤ |YN (s) ZN (s) ZN +1 (s)| ds. 2 n=1 2n 2N +1 0 t

|YN (s)ZN (s)ZN +1 (s)|ds is bounded. N Z 2 (t) And 2N +1 → 0, by assumption, as N → ∞. Thus n=1 2nn → 0, but this N Z 2 (t) may happen only if n=1 2nn = 0. The proof is complete.

Since our solutions are Leray, the term kN

0

Remark 3.2. We have more general uniqueness results, but they do not change the picture. Remark 3.3. What we have done, in a sense, is to estimate the solution in ∞ Zn2 (t) a “distributional” topology, n=1 2n . More common estimates in the N 2 “energy” topology n=1 Zn (t) do not work.

3.3.6 Summary and Open Questions We have seen that uniqueness holds for all positive initial conditions, nonuniqueness holds for certain negative ones. These sets, positive and negative elements of l2 , are small sets in l 2 . In precise terms, they are meager sets, namely they are the countable union of nowhere dense sets. A set B is nowhere dense if, given an open set U , the set B is not dense in U (there exists an open set V ⊂ U such that B ∩ V = ∅). See Remark 3.4 below. Figure 3.3 shows l2

non unique

Fig. 3.3 Space l2 with uniqueness and non-uniqueness sets

unique

3.4 Random Perturbation

87

2 with two small sets in it: the set l+ of initial conditions with uniqueness, and the set of those negative initial conditions which have more than one solution. The question is thus: generically speaking (namely, on the complement of a meager set), what should we expect? Is non-uniqueness a rare phenomena or not? The intuition is that only well prepared initial conditions may lead to nonuniqueness. Non-uniqueness seems to require a great degree of organization. The only way we figure out how to prove non-uniqueness is by pumping energy from infinity. We have seen that positive solutions eventually dissipate energy, the latter moves to infinity in finite time. Reversing time, we have found solutions with increasing energy: they capture energy from infinity. They have to be the time-reversal of dissipating solutions and our intuition is that dissipation needs a rather precise structure to happen, xn ∼ αkn−1 , or at least we know it needs positive components except for a finite number 2 of them (the set l(+) of Remark 3.5 below). This happens in a meager set. Thus, by time-reversal, we may pump energy from infinity only if we live in a meager set; even more, presumably only if the shape is sufficiently close to −αkn−1 . If it were true that this is the only mechanism for non-uniqueness, we would have that non-uniqueness is an uncommon behavior. We cannot prove the previous claim, but, if it were true, the addition of a noise perturbation which changes continuously infinitely many signs and thus forbid the energy pumping, would restore uniqueness. This is exactly what happens, as we now explain. 2 Remark 3.4. Let l+ be the set of all positive x ∈ l2 . It is nowhere dense, hence meager. We leave the proof as an exercise. The main difficulty is to prove that any open set U in l2 , contains an element u with some un < 0. 2 Then the l 2 -distance of u from l+ is at least |un |. The non trivial claim can be proved by showing that any ball around any point must contain elements 2 with some negative component: given x0 ∈ l+ (otherwise is nothing to there 0 2 : there exists n prove), given any B x , ε , we cannot have B x0 , ε ⊂ l+ 2 such that x0n − ε < 0 and thus there exists x ∈ B (0, ε) such that x0 + x ∈ / l+ . 2 Remark 3.5. Similarly, let l(+) be the set of all x ∈ l 2 such that xn ≥ 0 for all n greater than some n0 . Prove it is meager.

3.4 Random Perturbation 3.4.1 Preliminary Remarks We want to investigate now whether noise improves the theory of this simple dyadic model. In the last section we argued that this could be the case concerning uniqueness, since non-uniqueness seems to be an exceptional

88

3 Dyadic Models

phenomenon related to energy injection from infinity, which may happen only for special configurations. The first, most straightforward idea, to improve the uniqueness properties of the dyadic model is to perturb it with additive noise, in the form 2 − kn Xn Xn+1 dt + σn dWn dXn = kn−1 Xn−1 where (Wn ) is a sequence of independent Brownian motions. This kind of random perturbation has two drawbacks: (1) it breaks the formal energy conservation property, (2) we do not know how to treat it! Point (1) is more a feature than a drawback. If we have reasons to claim that (formal) energy conservation is mandatory, in the nature of the problem, and any deviance from it is non physical, then (1) is a drawback; otherwise it is simply a remark. However, if we explicitly modify the energy of the system by an external mechanism, then we obscure the internal ones: how could we state that the addition of noise breaks injection of energy from infinity, if that noise itself injects energy? About point (2), recall that the methods we developed for additive noise work well when there is a viscous regularizing term. Indeed, very nondegenerate noise is needed, to exploit its regularizing properties (think to Girsanov, to have an idea); but a very non-degenerate noise needs a regularizing linear part to compensate it, at our present state of understanding of mathematical technologies. There is also a third reason, the comparison with (or inspiration coming from) the theory of diffusion of passive scalars and other similar models (also some forms of stochastic Navier–Stokes equations, see Mikulevicius and Rozovskii [161]). From the viewpoint of such theories it would be interesting to perturb Euler equations by a multiplicative, or random transport, term of the form du + [(u · ∇) u + ∇p] dt + (ξ · ∇) u = 0,

divu = 0

where ξ is a given random field, white noise in time, say of the form ξ (t, x) =

j

σj (x)

dW j (t) . dt

If we compare the terms (u · ∇) u ←→ (ξ · ∇) u we see that it is just natural to write the correspondence 2 kn−1 Xn−1 − kn Xn Xn+1 ←→ kn−1 Xn−1 ξn−1 − kn ξn Xn+1 .


89

There are also other minor variants which a priori could be of interest, but this, when multiplied by Xn , still produces as telescoping series, the basis of formal energy conservation. Let us finally mention two works on anomalous dissipation (very related to the issues treated here) in a stochastic setting similar to the one treated in this Chapter: Mattingly et al. [158], Barbato et al. [23]. Other properties for similar stochastic models can be found for instance in Barbato et al. [19], Bessaih and Millet [33]. Since we have mentioned the relation with Euler equations, we address to Brzezniak and Peszat [41] for the most advances results on 2D stochastic Euler equations.

3.4.2 Definitions and Main Results Thus, after all these intuitive remarks, we decide to consider the following model. On a filtered probability space (Ω, Ft , P ), let (Wn )n≥1 be a sequence of independent Brownian motions. We consider the infinite system of stochastic differential equations in Stratonovich form 2 dXn = kn−1 Xn−1 − kn Xn Xn+1 dt + σkn−1 Xn−1 ◦ dWn−1 − σkn Xn+1 ◦ dWn (3.3) for n ≥ 1, with X0 (t) = 0 and σ = 0. The choice of Stratonovich is simply motivated by Wong–Zakai principle (it is the natural model if we start with random perturbations more regular than white noise and take the white noise limit) and the fact that, due to the similarity between Stratonovich calculus and ordinary smooth calculus, energy is formally constant with this choice. Indeed, 1 2 Xn − kn Xn2 Xn+1 dt dXn2 = kn−1 Xn−1 2 + σkn−1 Xn−1 Xn ◦ dWn−1 − σkn Xn Xn+1 ◦ dWn which is telescoping both in the drift and the diffusion part. Remark 3.6. Concerning Stratonovich and Itˆ o integrals and joint quadratic variation, recall that 0

t

Xs ◦ dWs = lim

n→∞

0

ti ∈πn ,ti ≤t

t

Xs dWs = lim

n→∞

Xti+1 ∧t + Xti Wti+1 ∧t − Wti 2

ti ∈πn ,ti ≤t

Xti Wti+1 ∧t − Wti

90

3 Dyadic Models

[X· , W· ]t = lim

n→∞

Xti+1 ∧t − Xti Wti+1 ∧t − Wti

ti ∈πn ,ti ≤t

where πn is a sequence of finite partitions of [0, T ] with size |πn | → 0 and elements 0 = t0 < t1 < ... The limits are in probability, uniformly in time on compact intervals. From these facts we have the basic relation 0

t

Xs ◦ dWs =

0

t

Xs dWs +

1 [X· , W· ]t . 2

Details about these facts can be found in Kunita [143]; see also Russo and Vallois [182]. In order to give the definition of solution it is easier to translate to Itˆ o formulation. The result is the equation 2 − kn Xn Xn+1 dt + σkn−1 Xn−1 dWn−1 − σkn Xn+1 dWn dXn = kn−1 Xn−1 (3.4) σ 2 2 − Xn dt kn + kn−1 2 2

for all n ≥ 1, with k0 = 0 and X0 = 0. The following proposition explains us the equivalence. Proposition 3.2. If the processes (Xn (t))t≥0,n∈N are continuous semimartingales which satisfy either (3.4) or (3.3), then

t 0

kn−1 Xn−1 (s) ◦ dWn−1 (s) =

t

kn Xn+1 (s) ◦ dWn (s) =

0

t 0

t 0

kn−1 Xn−1 (s)dWn−1 (s) −

kn Xn+1 (s) dWn (s) +

σ 2

t 0

σ 2

t 0

2 kn−1 Xn (s)ds

2 kn Xn (s) ds

(the first identity holds true for n ≥ 2, the second one for n ≥ 1). Proof. From Remark 3.6 we have 0

t

Xn−1 (s) ◦ dWn−1 (s) =

t 0

Xn−1 (s) dWn−1 (s) +

1 [Xn−1 , Wn−1 ]t . 2

Moreover, 2 − kn−1 Xn−1 Xn dt dXn−1 = kn−2 Xn−2 + σkn−2 Xn−2 ◦ dWn−2 − σkn−1 Xn ◦ dWn−1 hence only the term −σkn−1 Xn ◦ dWn−1 contributes to the joint quadratic variation [Xn−1 , Wn−1 ]t (bounded variation terms vanish, term with


91

independent Brownian motion too). One has 0

·

σkn−1 Xn ◦ dWn−1 , Wn−1

=

t

0

t

σkn−1 Xn (s) ds

· · t (more generally 0 gdWn−1 , 0 hdWn−1 t = 0 g (s) h (s) ds). Thus we get the first identity of the proposition. Similarly, 0

t

Xn+1 (s) ◦ dWn (s) =

t 0

Xn+1 (s) dWn (s) +

and

[Xn+1 , Wn ]t =

0

1 [Xn+1 , Wn ]t 2

t

σkn Xn (s) ds.

The proof is complete.

All our rigorous analyses will be based on the Itˆ o form, the Stratonovich one serving mainly as an heuristic guideline. Let us introduce the concept of weak solution. Since our main emphasis is on uniqueness, we shall always restrict ourselves to a finite time horizon [0, T ]. Definition 3.2. Given x ∈ l 2 , a weak solution of (3.3) in l2 is a filtered probability space (Ω, Ft , P ), a sequence of independent Brownian motions (Wn )n≥1 on (Ω, Ft , P ), and an l2 -valued stochastic process (Xn )n≥1 on (Ω, Ft , P ), with continuous adapted components Xn , such that t 2 (s) − kn Xn (s) Xn+1 (s) ds kn−1 Xn−1 Xn (t) = xn + 0 t t + σkn−1 Xn−1 (s) dWn−1 (s) − σkn Xn+1 (s) dWn (s) 0 0 t 2 σ 2 2 Xn (s) ds − kn + kn−1 0 2 for each n ≥ 1, with k0 = 0 and X0 = 0. We denote this solution by (Ω, Ft , P, W, X) or simply by X. We say that a weak solution is of class L∞ if there is a ∞ constant C > 0 such that n=1 Xn2 (t) ≤ C for a.e. (ω, t) ∈ Ω × [0, T ].

92

3 Dyadic Models

Remark 3.7. The uniqueness result described in the sequel extends to the more general class of solutions such that ⎡ 1

E P ⎣e σ2

T 0

∞

n=1

2 Xn (t)dt

T

1+ 0

2 ⎤ Xi4 (t) dt

⎦ 0 such that n=1 Xn2 (t) ≤ C for a.e. (ω, t) ∈ Ω × [0, T ]. Our main result on the linear equation (3.5) is the following theorem. The idea of proof is related to Malliavin [153], Airault and Malliavin [1], Cruzeiro et al. [56]. Also the Girsanov transform in the “multiplicative” form used here is somewhat inspired by these works. Theorem 3.6. Given x ∈ l 2 , in the class of L∞ weak solutions of (3.5) on [0, T ], there is strong (in the probabilistic sense) uniqueness. Remark 3.8. The result extends to the class of solutions such that

T

0

for each n ≥ 1 and

E Q Xn4 (t) dt < ∞

T

lim

n→∞

0

E Q Xn2 (t) dt = 0.

(3.6)

(3.7)

The modifications in the following proof are obvious. It is easy to verify that L∞ weak solutions satisfy these two conditions. Proof. By linearity, it is sufficient to prove that an L∞ weak solution (Xn )n≥1 , with null initial condition is the zero solution. Assume thus x = 0. We have t t σkn−1 Xn−1 (s) dBn−1 (s) − σkn Xn+1 (s) dBn (s) Xn (t) = xn + 0 0 t 2 σ 2 2 Xn (s) ds − kn + kn−1 0 2 hence, from Itô formula, we have 1 1 dX 2 = Xn dXn + d [Xn ]t 2 n 2 2 σ2 2 σ2 2 2 2 2 =− + kn2 Xn+1 Xn dt + dMn + dt kn + kn−1 kn−1 Xn−1 2 2

94

3 Dyadic Models

where Mn (t) =

0

t

σkn−1 Xn−1 (s)Xn (s)dBn−1 (s) −

t

0

σkn Xn (s)Xn+1 (s)dBn (s).

for From the L∞ -bound (or from (3.6)), Mn (t) is a martingale, each n ≥ 1, hence E Q [Mn (t)] = 0. Moreover, for each n ≥ 1, E Q Xn2 (t) is finite and continuous in t: it follows easily from condition 2(3.6) and (3.5) itself. From Q Xn (0) = 0) we deduce that the previous equation (and the property E E Q Xn2 (t) satisfies t Q 2 2 E Xn (s) ds E Q Xn2 (t) = −σ 2 kn2 + kn−1 0 t 2 2 + σ2 kn−1 E Q Xn−1 (s) ds 0 t 2 2 2 E Q Xn+1 (s) ds + σ kn 0

for n ≥ 1, with u0 (t) = 0 for t ≥ 0. It follows 0

t

2 k2 Xn+1 (s) − Xn2 (s) ds ≥ n−1 E kn2 Q

t 0

EQ

2 2 Xn (s) − Xn−1 (s) ds.

Since X0 ≡ 0, we have

t

EQ

0

and thus

0

t

EQ

X12 (s) − X02 (s) ds ≥ 0

2 Xn+1 (s) − Xn2 (s) ds ≥ 0

for every n ≥ 1, by induction. This implies

T 0

E Q Xn2 (s) ds ≤

0

T

2 E Q Xn+1 (s) ds

for all n ≥ 1. Therefore, by property (3.7), which holds for L∞ solutions, for every n ≥ 1 we have T E Q Xn2 (s) ds = 0. 0

This implies Xn2 (s) = 0 a.s. in (ω, s), hence X is the null process. The proof is complete.


95

We complete this section with an existence result. Notice that this is a result of strong existence and strong (or pathwise) uniqueness. Theorem 3.7. Given x ∈ l2 , there exists a unique solution in L∞ (Ω × [0, T ]; l2), with continuous components. Proof. We have only to prove existence. For every positive integer N , consider the finite dimensional stochastic system (N )

(N )

dXn(N ) = σkn−1 Xn−1 dBn−1 − σkn Xn+1 dBn −

(N ) σ2 2 2 kn + kn−1 Xn dt 2

Xn(N ) (0) = xn (N )

(N )

for n = 1, ..., N , with k0 = kN = 0, X0 (t) = XN +1 (t) = 0. This linear finite dimensional equation has a unique global strong solution. By Itˆ o formula

1 (N ) 2 1 = Xn(N ) dXn(N ) + d Xn(N ) , Xn(N ) d Xn 2 2 t (N ) 2 σ2 2 (N ) (N ) 2 = σkn−1 Xn(N )Xn−1 dBn−1 − σkn Xn(N ) Xn+1 dBn − dt Xn kn + kn−1 2 2 2 σ2 (N ) (N ) 2 kn−1 + Xn−1 + kn2 Xn+1 2 hence N N N 1 (N ) 2 (N ) (N ) Xn d = σkn−1 Xn(N ) Xn−1 dBn−1 − σkn Xn(N ) Xn+1 dBn 2 n=1 n=1 n=1

−

N N σ 2 2 (N ) 2 σ 2 2 (N ) 2 Xn−1 kn Xn dt + k 2 n=1 2 n=1 n−1

−

N N σ 2 2 (N ) 2 σ 2 2 (N ) 2 kn−1 Xn dt + k Xn+1 . 2 n=1 2 n=1 n

2 2 (N ) 2 dt. Thus This is equal to − σ2 kN XN N n=1

Xn(N )

2

(t) ≤

N

x2n ,

Q-a.s.

n=1

In particular, this verystrong bound implies that there exists a subsequence (N ) converges weakly to some (Xn )n≥1 in Nk → ∞ such that Xn k n≥1 p 2 L Ω × [0, T ] ; l for every p > 1 and also weak star in L∞ Ω × [0, T ] ; l 2 .

96

3 Dyadic Models

Hence in particular (Xn )n≥1 belongs to L∞ (Ω×[0, T ]; l2 ). Now the proof proceeds by standard arguments typical of equations with monotone operators (which thus apply to linear equations), presented by Pardoux [171], Krylov and Rozovskii [139], Rozovskii [181], The subspace of Lp Ω × [0, T ] ; l2 of progressively measurable processes is strongly closed, hence weakly closed, hence (Xn )n≥1 is progressively measurable. The one-dimensional stochastic integrals which appear in each equation of system contin (3.5) are (strongly) uous linear operators from the subspace of L2 Ω × [0, T ] ; l 2 of progressively measurable processes to L2 (Ω), hence they are weakly continuous, a fact that allows us to pass to the limit in each one of the linear equations of system (3.5). A posteriori, from these integral equations, it follows that there is a modification such that all components are continuous. The proof of existence is complete.

3.4.4 Girsanov Transform The idea is that (3.4) written in the form

1 Xn−1 dt + dWn−1 dXn = σkn−1 Xn−1 σ σ2 2 2 Xn dt kn + kn−1 − 2

− σkn Xn+1

1 Xn dt + dWn σ

t becomes (3.5) because the processes Bn (t) := σ1 0 Xn (s) ds + Wn (t) are Brownian motions with respect to a new measure Q on (Ω, FT ); and conversely, so both weak existence and weak uniqueness statements transfer from (3.5) to (3.4). Equation (3.5) was also proved to be strongly well posed, but the same problem for the nonlinear model (3.4) is open. Let us give the details. We use results about Girsanov theorem that can be found in Revuz and Yor [177], Chap. 8, and an infinite dimensional version proved in Bensoussan [31], Kozlov [135], Da Prato and Zabczyk [65].

3.4.5 Proof of Theorem 3.4 Let us prepare the proof with a few remarks. that (X n )n≥1 is Assume T ∞ 2 X < ∞, an L∞ weak solution. Since in particular E 0 n=1 n (s)ds t ∞ the process Lt := − σ1 n=1 0 Xn (s)dWn (s) is well defined, is a mar t ∞ tingale and its quadratic variation [L, L]t is σ12 0 n=1 Xn2 (s)ds. Since

1

T 0

∞

2 Xn (t)dt

< ∞ (again because the solution is L∞ ), Novikov criterion applies, so exp Lt − 12 [L, L]t is a strictly positive martingale. Define E e 2σ2

n=1


97

the probability measure Q on FT by setting dQ 1 = exp LT − [L, L]T . dP 2

(3.8)

Notice also that Q and P are equivalent on FT , by the strict positivity and dP 1 = exp ZT − [Z, Z]T dQ 2 where

∞

(3.9)

t

1 Xn (s) dBn (s) σ n=1 0 t 1 Bn (t) = Wn (t) + Xn (s) ds. 0 σ = exp −LT + 12 [L, L]T and one can check that −LT + Indeed dP dQ 1 1 2 [L, L]T = ZT − 2 [Z, Z]T . Under Q, (Bn (t))n≥1,t∈[0,T ] is a sequence of independent Brownian motions. Since t t kn−1 Xn−1 (s) dBn−1 (s) = kn−1 Xn−1 (s) dWn−1 (s) 0 0 t kn−1 Xn−1 (s) Xn−1 (s) ds + Zt =

0

and similarly for

t 0

kn Xn+1 (s) dBn (s), we see that

t t kn−1 Xn−1 (s) dBn−1 (s) − kn Xn+1 (s) dBn (s) Xn (t) = Xn (0) + 0 0 t 1 2 2 Xn (s) ds. − kn + kn−1 0 2 This is (3.5). Moreover, the L∞ -condition is equivalent under P or under Q. We have proved the following lemma: Lemma 3.2. If (Ω, Ft , P, W, X) is an L∞ weak solution of the nonlinear equation (3.3), then it is an L∞ weak solution of the linear equation (3.5) where the processes Bn (t) = Wn (t) +

0

t

1 Xn (s) ds σ

are a sequence of independent Brownian motions on (Ω, FT , Q), Q defined by (3.8).

98

3 Dyadic Models

Remark 3.9. If a solution X of the nonlinear equation (3.3) satisfies only the integrability property of Remark 3.7, then it satisfies conditions (3.6) and (3.7) of Remark 3.8. Indeed, we have E

Q

T

0

Xn4

(t) dt = E

P

E (L)T

=E

P

T 0

Xn4

(t) dt

T 1 4 exp LT − [L, L]T + [L, L]T Xn (t) dt 2 0

⎡ 1/2 ≤ E P [exp (2LT − 2 [L, L]T )] E P ⎣

T

0

⎤1/2

2 Xn4 (t) dt

exp [L, L]T ⎦

.

The second factor is finite by the condition of Remark 3.7. and the term E P [exp (2LT − 2 [L, L]T )] is equal to one, by Girsanov theorem applied to the martingale 2Lt. This proves

As to condition (3.7), it condition (3.6). T ∞ 2 follows from the fact that E 0 n=1 Xn (s) ds < ∞, a consequence of the condition of Remark 3.7. One may also check that dXn = σkn−1 Xn−1 ◦ dBn−1 − σkn Xn+1 ◦ dBn so the previous computations could be described at the level of Stratonovich calculus. (the proof is now classical). Assume that Let us now prove weak uniqueness (i)

Ω(i) , Ft , P (i) , W (i) , X (i) , i = 1, 2, are two L∞ weak solutions of (3.3) with

the same initial condition x ∈ l 2 . Then (i)

(i)

(i)

dXn(i) = σkn−1 Xn−1 dBn−1 − σkn Xn+1 dBn(i) −

(i) σ2 2 2 Xn dt (3.10) kn + kn−1 2

where, for each i = 1, 2, Bn(i) (t) = Wn(i) (t) +

0

t

1 (i) X (s) ds σ n

(i) is a sequence of independent Brownian motions on Ω(i) , FT , Q(i) , Q(i) defined by (3.8) with respect to P (i) , W (i) , X (i) . We have proved in Theorem 3.6 that equation (3.5) has a unique strong (in the probabilistic sense) L∞ solution. Thus it has uniqueness in law on C ([0, T ] ; R)N , by Yamada–Watanabe theorem (see Revuz and Yor [177], Prévôt and R¨ ockner [176]), namely the laws of X (i) under Q(i) are the same.


99

The proof of Yamada–Watanabe theorem in this infinite dimensional conN text, with the laws on C ([0, T ] ; R) , is step by step identical to the finite dimensional proof, for instance of Revuz and Yor [177], Chap. 9, Lemma 1.6 and Theorem 1.7. We do not repeat it here. n ∈ N, t1 , ..., tn ∈ [0, T ] and a measurable bounded function f : 2Given n l → R, from (3.9) we have f X (i) (t1 ) , ..., X (i) (tn ) 1 (i) (i) (i) Q(i) (i) (i) exp Zt − f X (t1 ) , ..., X (tn ) =E Z ,Z 2 t

EP

(i)

∞ t (i) (i) (i) where Zt := n=1 0 σ1 Xn (s) dBn (s). Under Q(i) , the law of Z (i) , X (i) N N on C ([0, T ] ; R) × C ([0, T ] ; R) is independent of i = 1, 2. A way to explain this fact is to consider the enlarged system of stochastic equations made of (3.10) and equation ∞ 1 (i) (i) (i) Xn dBn . dZ = σ n=1 This enlarged system has strong uniqueness, for trivial reasons, and thus also weak uniqueness by Yamada–Watanabe theorem. Hence (1) (2) f X (1) (t1 ) , ..., X (1) (tn ) = E P f X (2) (t1 ) , ..., X (2) (tn ) . EP N

Thus we have uniqueness of the laws of X (i) on C ([0, T ] ; R) . The proof of uniqueness is complete.

3.4.6 Proof of Theorem 3.5 Let (Ω, Ft , Q, B, X) be a solution in L∞ Ω × [0, T ] ; l2 of the linear equation (3.5), provided by Theorem 3.7. Let us argue as in the previous subsection but from Q to P , namely by introducing the new measure P on (Ω, FT ) t 1 dP = exp ZT − 12 [Z, Z]T where Zt := ∞ defined as dQ n=1 0 σ Xn (s) dBn (s). Under P , the processes Wn (t) := Bn (t) −

0

t

1 Xn (s) ds σ

are a sequence of independent Brownian motions. We obtain that (Ω, Ft , P, W, X) is an L∞ -solution of the nonlinear equation (3.4). The L∞ property is preserved since P and Q are equivalent. The proof of existence is complete.

Chapter 4

Transport Equation

4.1 Introduction In Chap. 3 on the dyadic model, we have seen that a non well posed system can be made well posed by means of a special multiplicative noise. Translated in terms of nonlinear transport type equations ∂u + (b (u) · ∇) u = 0, ∂t

u|t=0 = u0

this means that there is hope that a stochastic analog of the form du + (b (u) · ∇) u dt +

(σk · ∇) u ◦ dW k = 0,

u|t=0 = u0

k

is well posed, even in cases when the deterministic case is not. Why should such a noise regularize? Just by analogy with the dyadic model? Maybe not, perhaps there is a structural reason (of which the dyadic model is an example). Consider the Itô formulation of the same equation du + (b (u) · ∇) u dt +

(σk · ∇) u dW k = Au dt

k

where order operator coming from the quadratic variation A is the second 1 k · D) u, W (σ . It is k k 2 Au =

1 2 (σk · ∇) u. 2 k

For instance, in the easy case when k

(σk · ∇) u ◦ dW k =

d

∂j u ◦ dW j

j=1


101

102

4 Transport Equation

we simply have 1 Δu. 2 The presence of the term Au does not mean anything in itself, from the regularity viewpoint. It does not mean that the equation became suddenly parabolic, regularizing. It is the same equation as the original one, which was of hyperbolic type, loosely speaking (for instance it can be solved forward and backward in time). When the coefficients are smooth, it is known (see Kunita [144, 145]) that the solution has the form u (t, x) = u0 ϕ−1 t (x) where ϕt (x) is a certain stochastic flow of diffeomorphisms, hence u (t, x) is not more o term. It regular than u0 . The term Au is perfectly compensated by the Itˆ is well know since the first studies on Zakai SPDEs of filtering theory, that an SPDE with a second order differential operator Au and a first order Itˆ o term like (σk · ∇) udW k is truly parabolic only when a condition of superellipticity is satisfied. The condition of super-ellipticity is precisely that the operator 1 2 Lu := Au − (σk · ∇) u 2 Au =

k

must be uniformly elliptic. But in our case Lu = 0. So, the term Au in the Itˆ o form does not mean parabolicity. But if we take expectation (we need the Itô term to be a martingale), we get ∂ E [u (t, x)] + E [(b (u) · ∇) u (t, x)] = AE [u (t, x)] . ∂t The linear part of this equation ∂ E [u (t, x)] = AE [u (t, x)] ∂t is now a true parabolic equation! From an SPDE of “hyperbolic” type we have found an equation with a parabolic part. Can we profit from this fact? There are two essential difficulties. One is the impossibility, a priori, to perform any closure of the expectation of the nonlinear term E [(b (u) · ∇) u (t, x)]. Another is that, even if we could prove that E [u (t, x)] has regularity properties similar to the solutions of parabolic equations, this would tell us not so much about the random field u (t, x). We do not know general ways to overcome these difficulties, and there cannot be: there are counterexamples, namely examples that formally fit with the previous description, but no improvement comes from the noise (Burgers equation for instance, see Chap. 5). Thus the idea described above is simply an indication that something regularizing could be around.

4.1 Introduction

103

4.1.1 Linear Transport: Structural Approach The previous idea works at least in one interesting case: a class of linear transport equations with weakly differentiable fields. Assume b = b (t, x) does not depend on the solution u, but is given a priori. The Itô form is du + (b · ∇) u dt +

(σk · ∇) u dW k = Audt

k

and the expectation E [u (t, x)] really solves the parabolic equation ∂ E [u (t, x)] + (b · ∇) E [u (t, x)] = AE [u (t, x)] . ∂t Still this gives us only informations on E [u (t, x)]. To get more, let us use a common trick in the field of linear transport equations. Let β : R → R be a C 1 function. Formally speaking (some regularity is needed to be rigorous) the function β (u (t, x)) satisfies the equation dβ (u) + (b · ∇) β (u) dt +

(σk · D) β (u) ◦ dW k = 0

k

(Stratonovich calculus is like classical one). Now we move to Itô: dβ (u) + (b · ∇) β (u) dt +

(σk · D) β (u) dW k = Aβ (u)

k

and average ∂ E [β (u)] + (b · ∇) E [β (u)] = AE [β (u)] . ∂t This is much better: from this equation we get information on E [β (u)] which may tell us what we need, thanks to a proper choice of β. The same computation on the deterministic transport equation would give us (4.4) below; the stochastic case is better because the term AE [β (u)] regularizes and helps to prove results under weaker assumptions on b. The most obvious application is to uniqueness. Since the equation is linear, it is sufficient to prove u = 0 when u0 = 0. Choose β such that β (0) = 0 (hence β (u0 ) = 0), and strictly positive otherwise (so β (u) = 0 implies u = 0). Since E [β (u0 )] = 0, we can deduce E [β (u)] = 0 (and thus u = 0) for all t ≥ 0 under rather general assumptions on b, at least more general than those of the deterministic case, where no second order term AE [β (u)] comes to help. Another application is to understand the law of u (t, x), since we have some control on E [β (u)] for quite general β.

104


What we have just described is exactly what we have done for the linear dyadic model. We have computed the squares Xn2 (t)(it corresponds to the choice β (u) = u2 ), taken expectation and proved E Xn2 (t) = 0 when the initial condition is zero. Thus we see that what we have done in Chap. 3 is not only a very particular trick, but the application of a general idea (this was recognized after [21]). In this section we make rigorous the previous approach, for the linear transport equation, under suitable assumptions. But first it is necessary to get some more feeling about the deterministic case.

4.2 Deterministic Transport Equation Consider the linear equation ∂u + (b · ∇) u = 0, ∂t

u|t=0 = u0 .

(4.1)

The objects are: a given vector field b : [0, T ] × Rd → Rd , a given initial condition u0 : Rd → R, and the unknown scalar field u : [0, T ] × Rd → R.

4.2.1 Lipschitz Case The classical case is when b is Lipschitz with linear growth, uniformly in time: |b (t, x) − b (t, y)| ≤ C |x − y| ,

|b (t, x)| ≤ C (1 + |x|) .

One considers the associated ODE, called equation of the characteristics, dXtx = b (t, Xtx ) , dt

X0x = x

(4.2)

which has a unique solution Xtx for every x ∈ Rd . The map x → Xtx is bijective, since we can solve uniquely the same equation backward with any final time condition. The solution Xtx defines a map ϕt : Rd → Rd by ϕt (x) = Xtx . We denote by ϕ−1 its inverse. t If u is a sufficiently regular solution of (4.1), then ∂u d u (t, Xtx ) = (t, Xtx ) + ∇u (t, Xtx) · b (t, Xtx ) = 0. dt ∂t

4.2 Deterministic Transport Equation

Hence

105

u (t, Xtx ) = u0 (x) .

We have identified u (t, ·) in the range of the map x → Xtx , which is the full space Rd . Thus uniqueness holds for sufficiently regular solutions and one has the representation formula u (t, x) = u0 ϕ−1 t (x) . This proof works for regular solutions, but one can show, by means of the regularization method (and commutator lemma) used several times below, that we can extend the previous simple method to very weak solutions, let us say of class L∞ (or other similar spaces), and get uniqueness also in such spaces. There are other proofs of uniqueness, based only on PDE arguments. One of them is developed in the next section in a more general context.

4.2.2 Weakly Differentiable Case We call this way the very general class treated in the outstanding works of Di Perna and Lions [81], Ambrosio [8] and others. Let us consider only a particular case of the framework of [81]. Assume that |b| 1,1 d b ∈ L1 0, T ; Wloc , R ∈ L1 0, T ; L∞ Rd , 1 + |x| d 1 ∞ div b ∈ L 0, T ; L R

(4.3)

For d = 1, b = div b is bounded, so we go back to the Lipschitz case. But in dimension d > 1 this is far from (locally) Lipschitz. There is no nontrivial way to solve the ODE (1.2) and repeat the previous arguments. A posteriori of their results on the transport equation, existence and uniqueness of a regular Lagrangian flow (see the definition in Chap. 1) is proved. Let us remark that existence of weak solutions of the PDE (4.1) is not an issue: it can be proved in much larger generality on b by simple weak compactness arguments, which however are a little abstract and we omit them. The main problem is uniqueness. By linearity, it is sufficient to prove uniqueness when u|t=0 = 0. If u is a sufficiently regular solution of the PDE (4.1) and β : R → [0, ∞) is a C 1 -function with β (0) = 0 and β (x) > 0 for x = 0, then ∂β (u) + (b · ∇) β (u) = 0, ∂t

β (u) |t=0 = 0.

(4.4)

106


Let us interpret this equation in the weak sense over smooth compact support test functions ϕ:

t

β (u) (t, x) ϕ (x) dx =

β (u) (s, x) ϕ (x) div b (s, x) dxds

t

+ β (u) (s, x) b (s, x) · ∇ϕ (x) dxds. 0

0

Assume that β (u) ∈ L∞ 0, T ; L1 Rd , and, for a second, assume b ∈ L1 0, T ; L∞ Rd to simplify the integration by parts. Then, by a limit as ϕ goes to the constant 1, we get

t

β (u) (t, x) dx =

β (u) (s, x) div b (s, x) dx 0

and thus

β (u) (t, x) dx ≤

where, by assumption,

t 0

t 0

div b (s, ·)L∞ (Rd )

β (u) (s, x) dx

div b (s, ·)L∞ (Rd ) ds < ∞. By Gronwall lemma

β (u) (t, x) dx ≤ 0 · exp

t 0

div b (s, ·)L∞ (Rd ) ds

= 0.

Thus we have uniqueness, in a class of sufficiently regular solutions (some regularity is needed to pass from the equation for u tothe equation for β (u)). We shall remove the condition b ∈ L1 0, T ; L∞ Rd by a localization argument. But existence of regular solutions is an open problem. The general weak compactness argument which proves existence of weak solutions, mentioned above, only produces solutions of class L∞ , or similar (like Lp ). Regularity of the initial condition does not help to get more. Thus, the question is uniqueness of weak solutions. Let us give the precise definition of weak solution. To give a meaning to the last two integral terms of the next equation we need the condition b, div b ∈ L1 0, T ; L1loc Rd included in assumption (4.3). ∞ Definition 4.1. We calld L weak solution of the PDE (4.1) a function of ∞ class u ∈ L [0, T ] × R such that, for a.e. t ∈ [0, T ],


107

t

u (t, x) ϕ (x) dx =

u0 (x) ϕ (x) dx +

t

+ 0

u (s, x) ϕ (x) div b (s, x) dxds 0

u (s, x) b (s, x) · ∇ϕ (x) dxds

for all smooth compact support functions ϕ : Rd → R. The technical problem now is that, a priori, we cannot apply rigorous rules of calculus and prove that identity (4.4) holds true, in the weak sense. It may appear that the difficulty to go from the equation for u to the equation for β (u) is only a technical obstruction, removable by a suitable smoothing procedure. On the contrary, it is quite deep. A similar difficulty appears for Euler equations and is related to anomalous energy dissipation. It looks as a technical detail but it is an issue where we touch rather deep questions of fluid dynamic type. It deserves a definition. Definition 4.2. We say that an L∞ weak solution u of the PDE (4.1) is renormalized if, for every β ∈ C 1 (R), the function β (u) is again an L∞ weak solution of the PDE (4.1). Since we do not want to increase the regularity of u (because of the poor existence theorems), we need more regularity of b to renormalize L∞ weak It is here that the weak differentiability assumption solutions. 1,1 d 1 plays a role (otherwise, at the level of the a prib ∈ L 0, T ; Wloc R ori estimates needed to apply Gronwall lemma, we just need the other two conditions of assumption (4.3)). To renormalize u we regularize it by means of mollifiers. Given θ ∈ C0∞ Rd , non-negative, say equal to zero outside B (0, 1), such that θ (x) dx = 1, set θε (x) = ε1d θ xε , ε ∈ (0, 1], and define uε = θε ∗ u, the convolution in space

(θε ∗ u) (t, x) = θε (x − y) u (t, y) dy. Then (choose ϕ (x) = θε (x − x) in the weak formulation of Definition 4.1 and rename x by x) ∂uε + (b · ∇) uε = [b · ∇, θε ∗] u ∂t where we use the notation [b · ∇, θε ∗] for the commutator [b · ∇, θε ∗] u = (b · ∇) (θε ∗ u) − θε ∗ ((b · ∇) u) .

108


Remark 4.1. The rigorous formulation is a little more tricky. The function u is not differentiable, so when we write θε ∗ ((b · ∇) u) we mean the function θε ∗ ((b · ∇) u) (t, x ) = − −

u (t, x) θε (x − x) div b (t, x) dx u (t, x) b (t, x) · ∇x θε (x − x) dx

or the convolution of θε with the properly defined distribution (b · ∇) u. As a function of time, it is integrable. With this notation, from the weak formulation of Definition 4.1 we get

uε (t, x) +

t 0

θε ∗ ((b · ∇) u) (s, x) ds = uε (0, x)

which implies that uε (·, x) ∈ W 1,1 (0, T ) for each x ∈ Rd . In this weak sense we can write ∂uε + θε ∗ ((b · ∇) u) = 0 ∂t and thus ∂uε + (b · ∇) uε = [b · ∇, θε ∗] u. ∂t We deduce, in a weak sense (it can be justified by an approximation argument in t) ∂β (uε ) + (b · ∇) β (uε ) = β (uε ) [b · ∇, θε ∗] u ∂t namely

β (uε ) (t, x) ϕ (x) dx

t

β (uε ) (s, x) ϕ (x) div b (s, x) dx = β (uε ) (0, x) ϕ (x) dx + 0

t

β (uε ) (s, x) b (s, x) · ∇ϕ (x) dx + 0

t

+ β (uε ) (s, x) [b · ∇, θε ∗] u (s, x) ϕ (x) dx 0

for all ϕ ∈ C0∞ Rd . By definition of uε , it is not difficult to check that the first three terms converge to the corresponding ones with β (u) in place of β (uε ). Thus we get


109

β (u) (t, x) ϕ (x) dx

t

= β (u) (0, x) ϕ (x) dx + β (u) (s, x) ϕ (x) div b (s, x) dx 0

t

β (u) (s, x) b (s, x) · ∇ϕ (x) dx + 0

t

β (uε ) (s, x) [b · ∇, θε ∗] u (s, x) ϕ (x) dx. + lim ε→0

0

For every R > 0, let us denote by f L∞ and f L1 the norms of a function or R R vector field f in L∞ ([0, T ] × B (0, R)) and L1 ([0, T ] × B (0, R)) respectively. 1,1 d d Lemma 4.1. If b ∈ L1 0, T ; Wloc , u ∈ L∞ R loc [0, T ] × R , ϕ ∈ C0∞ Rd has support in B (0, R), then

T

|ϕ(x) [b · ∇, θε ∗] u(x)| dxdt ≤ Cθ uL∞

R+1

0

DbL1

R+1

+ div bL1

R+1

and ϕ (x) [b · ∇, θε ] u (s, x) converges to zero in L1 [0, T ] × Rd . Proof. Step 1. By definition (we drop t in the notation)

[b · ∇, θε ∗] u (x) = (b (x) · ∇x ) θε (x − y) u (y) dy

+ u (y) θε (x − y) div b (y) dy

+ u (y) b (y) · ∇y θε (x − y) dy

=

u (y) ((b (y) − b (x)) · ∇y ) θε (x − y) dy +

u (y) θε (x − y) div b (y) dy.

Step 2. Assume ϕ has support in B (0, R). Then, for x ∈ B (0, R),

|[b · ∇, θε ∗] u (x)| ≤ uL∞

R+1

+ uL∞

R+1

and with z = y − x

|((b (y) − b (x)) · ∇y ) θε (x − y)| dy |θε (x − y) div b (y)| dy

110


b (x + z) − b (x) 1 −z ≤ uL∞ ∇θ ε dz R+1 εd ε

+ uL∞ |θε (−z) div b (x + z)| dz R+1

≤ Cθ uL∞

R+1

+ Cθ uL∞

1 εd

R+1

1 0

1 εd

B(0,ε)

B(0,ε)

|Db (α (x + z) + (1 − α) x)| dzdα

|div b (x + z)| dz.

Step 3.

T

|ϕ (x) [b · ∇, θε ] u (x)| dxdt

1 1 T ≤ Cθ uL∞ ϕ (x) |Db (x + αz)| dx dzdtdα R+1 εd 0 0 B(0,ε)

T

1 + Cθ uL∞ ϕ (x) |div b (x + z)| dx dzdt R+1 εd 0 B(0,ε) 0

≤ Cθ uL∞ R+1

0

T

B(0,R+1)

|Db(x)| dxdt +

T 0

B(0,R+1)

|div b (x)| dxdt .

Step 4. We only outline this step, conceptually reasonable but long to be written down (see [13, 81]). If b is smooth, one can prove the convergence to zero of [b · ∇, θε ∗] u; then using the bound of the previous step and a smooth approximation of b, one proves convergence to zero of [b · ∇, θε ∗] u for general b. The proof is complete.

From this lemma and the previous computations, we get: 1,1 d , all L∞ weak solutions u of the R Corollary 4.1. If b ∈ L1 0, T ; Wloc PDE (4.1) are renormalized. We can now prove the following main theorem, proved by DiPerna and P.L. Lions [81]. Theorem 4.1. Under assumption (4.3), there is uniqueness of weak solution of the PDE (4.1) in the class L∞ 0, T ; L∞ Rd ∩ L1 Rd . Proof. Assume u0 = 0. Call u the difference of twosolutions, hence with zero initial condition. It is renormalized and lives in L∞ 0, T ; L∞ Rd ∩ L1 Rd . Let ϕ ∈ C0∞ be a test function with support in B (0, 2), equal to 1 in B (0, 1),


non negative. Let ϕR (x) = ϕ d dt

111

x 1 R . For every β ∈ C (R) we have

β (u) b · ∇ϕR dx +

β (u) ϕR dx =

β (u) ϕR div bdx.

We have t

) b · ∇ϕ dxds β (u s s R 0

t |bs | ≤ |β (us )| (1 + |x|) |∇ϕR | dxds 1 + |x| 0 ∞

T |bs |

1 |β (us )| dxds ≤ (1 + 2R) ∇ϕ∞ 1 + |x| R 0 ∞ |x|>2R

T f (s) |β (us )| dxds ≤C |x|>2R

0

with f ∈ L1 (0, T ). Take β such that

1 |u|2 ∧ |u| ≤ β (u) ≤ |u| . 2 Then

T

f (s) |x|>2R

0

|β (us )| dxds ≤

T

f (s) |x|>2R

0

|us | dxds.

Moreover we have t

t

β (u β (us ) ϕR dxds. ) ϕ div b dxds div b ≤ s R s s ∞ 0

0

Therefore d dt

β (us ) ϕR dx ≤

0

t

div bs ∞

β (us ) ϕR dxds

T

f (s)

+C 0

|x|>2R

|us | dxds.

By Gronwall lemma

β (ut ) ϕR dx ≤ e

t 0

div bs ∞ ds

T

C

f (s) 0

|x|>2R

|us | dxds.

112


We have f (s) |x|>2R |us | dx ≤ f (s) |us | dx, so we may apply Lebesgue dominated convergence theorem and get

f (s)

lim

R→∞

T 0

|x|>2R

|us | dxds = 0.

From 0 ≤ β (u) ≤ |u| and the boundedness of ϕ, we similarly may pass to the limit as R → ∞ in β (ut ) ϕR dx. Therefore we deduce

β (ut ) dx = 0. Since β(u) ≥ |u|2 ∧ 12 |u| , we deduce ut = 0. The proof is complete.

Remark 4.2. In the previous subsection, where we assumed Lipschitz b, we gave the proof of uniqueness only for regular solutions. However, with the same argument developed here, it can be rewritten for weak solutions. Of course, the Lipschitz case is a particular case of the weak differentiable one treated in this subsection. 4.2.2.1 Examples of Non-Uniqueness for Less Regular b The restrictions on b of the previous theorem are not optimal (see for instance Ambrosio [8]) but there is no hope to remove them too much. Let us discuss the condition div b ∈ L1 0, T ; L∞ Rd . In dimension d = 1 this is very restrictive: it is the Lipschitz condi|x| is weakly differentiable, in the sense tion. The case b (x) = 2sign (x) 1,1 d 1 (so in particular L∞ weak solutions would be renorb ∈ L 0, T ; Wloc R malized) but it does not satisfy div b ∈ L1 0, T ; L∞ Rd . And this is not an artificial restriction, since we have seen in Chap. 1, Example 1.6 and the associated figures, that non uniqueness holds in this case. So there is a barrier to the uniqueness theory in the deterministic case. We are going to show that noise allows us to overcome this barrier, to some extent. For completeness, let us expand Example 1.6 of Chap. 1. Consider the function 1 b(x) = sign (x)|x|β , β ∈ (0, 1) 1−β The deterministic transport equation is not well posed. Indeed, the Cauchy problem x (t) = b (x (t)) , t ≥ 0, x (0) = x0

4.3 Stochastic Case

113

has a unique solution for all x0 = 0, denoted by φt (x0 ). For x0 = 0 we have 1 1 two extremal solutions x+ (t) and x− (t), x+ (t) = t 1−β and x− (t) = −t 1−β for small t. In addition, for x0 = 0, we have the solution x (t) ≡ 0, and the solutions x(t) = x± (t − t0 )1t≥t0 for every t0 ≥ 0. Given t > 0 and x ∈ [x− (t), x+ (t)], there is a unique number t0 (t, x) ≥ 0 such that xsign(x) (t− t0 (t, x)) = x. The function φt maps R{0} one to one on (−∞, x− (t)) ∪ (x+ (t) , ∞); φ−1 will be its inverse, between these sets. With these notations, t given u0 ∈ L∞ and two bounded measurable functions γ+ , γ− : [0, ∞) → R, define the function ⎧ −1 ⎪ ⎪ u0 φt (x) for x > x+ (t) ⎨ γ+ (t0 (t, x)) for 0 ≤ x ≤ x+ (t) . (4.5) uγ± (t, x) = ⎪ γ (t (t, x)) for x− (t) ≤ x < 0 ⎪ ⎩ − 0−1 u0 φt (x) for x < x− (t) These are weak L∞ solutions, for every γ+ , γ− , of the deterministic transport equation with the same initial condition u0 . For instance, if u0 = 1x>0 and γ+ = γ− ≡ a for a constant value a, the shape of uγ± can be easily imaged; all these functions are solutions, corresponding to the same initial condition u0 .

4.3 Stochastic Case 4.3.1 Definitions and Preliminaries Consider the Stratonovich linear stochastic transport equation du + (b · ∇) u dt +

d

∂i u ◦ dW i = 0,

u|t=0 = u0

(4.6)

i=1 d d where the regularity of the vector field b : [0, T ] × R → R will be specified i and Wt t≥0 , i = 1, ..., d, are independent Brownian motions on a filtered probability space (Ω, Ft , P ). To shorten some notation and hint at more generality, let us define a few differential operators. For a.e. t ∈ [0, T ], denote by At , Bt , Ct,k the linear operators from C0∞ Rd to L1loc Rd defined as

(Bt f ) (x) = (b (t, x) · ∇) f (x) ,

(Ct,k f ) (x) = ∂k f (x)

(the operators Ct,k and At do not even depend on t here but we keep the general notation for completeness) (At f ) (x) =

1 Ct,k Ct,k f (x) , 2 k

f ∈ C0∞ Rd

114


∗ where, here, At f = 12 Δf . Then denote by A∗t , Bt∗ , Ct,k their formal adjoints, d d ∞ 1 again linear operators from C0 R to Lloc R , defined as

(Bt∗ ϕ) (x) = − (b (t, x) · ∇) ϕ (x) − ϕ (x) div b (t, x) ∗ Ct,k ϕ (x) = −∂k ϕ (x) (A∗t ϕ) (x) =

1 ∗ ∗ Ct,k Ct,k ϕ (x) , 2

ϕ ∈ C0∞ Rd .

k

d ∗ If ϕ ∈ ϕ ∈ L1loc [0, T ] × Rd . The next R we have A∗· ϕ, B·∗ ϕ, C·,k definition requires b, div b ∈ L1 0, T ; L1loc Rd . Definition 4.3. If u0 ∈ L∞ Rd , we say that a random field u (t, x) is a of equation (4.6) if u ∈ L∞ Ω × [0, T ] × Rd and, for all weak L∞-solution ∞ d ∗ ϕdx has a modification ϕ ∈ C0 R , the real valued process s → us Cs,k which is a continuous adapted semi-martingale and, for all t ∈ [0, T ], we have P -a.s.

t

t

∗ us Bs∗ ϕdx ds+ us Cs,k ϕdx ◦ dWsk = u0 ϕdx. ut ϕdx+ C0∞

0

k

0

A posteriori, form the equation itself, it follows that for all ϕ ∈ C0∞ Rd We shall the real valued process t → utϕdx has a continuous modification. ∗ always use it when we write ut ϕdx, ut Bt∗ ϕdx, ut Ct,k ϕdx. ∗ ϕdx is a continuous adapted The reason for the assumption that us Cs,k semi-martingale is that the Stratonovich integrals

t

0

∗ ϕdx us Cs,k

◦ dWsk

are thus well defined and equal to the corresponding Itˆ o integrals plus half of the joint quadratic variation:

t

= 0

1 ∗ ∗ us Cs,k u· C·,k ϕdx dWsk + ϕdx, W·k . 2 t

See also Remark 3.6 in Chap. 3. Proposition 4.1. A weak L∞ -solution in the previous Stratonovich sense satisfies the Itˆ o equation

4.3 Stochastic Case

115

t

ut ϕdx +

=

0

t

∗ ∗ us Bs ϕdx ds + us Cs,k ϕdx dWsk

t

u0 ϕdx +

0

us A∗s ϕdx ds

for all ϕ ∈ C0∞ Rd . Proof. We have only to compute equation itself,

k

0

∗ u· C·,k ϕdx, W·k . Notice that, by the t

t

∗ ∗ us Bs∗ Ct,k ϕdx + ϕdx ds utCt,k 0

t

∗ ∗ k ∗ ϕdx. + us Cs,k Ct,k ϕdx ◦ dWs = u0 Ct,k k

0

Thus, by the classical rules (see Kunita [143]) used also in the proof of Proposition 3.2, we have

∗ ϕdx, W·k u· C·,k

t

= t

0

∗ ∗ Ct,k ϕdx ds. us Cs,k

The proof is complete, recalling the definition of A∗t .

4.3.2 Renormalized Solutions Definition 4.4. We say that a weak L∞ -solution of (4.6) is renormalized if for every β ∈ C 2 (R) the process β (u (t, x)) is a weak L∞ -solution of the same (4.6). Definition 4.5. If v0 ∈ L∞ Rd , we say that v ∈ L∞ [0, T ] × Rd is a weak L∞ -solution of the PDE ∂v 1 + b · ∇v = Av, ∂t 2

v|t=0 = v0

if

t

vt ϕdx +

0

for all ϕ ∈ C0∞ Rd .

t

vs Bs∗ ϕdx ds = v0 ϕdx + vs A∗s ϕdx ds 0

116


Theorem 4.2. If b ∈ L1

1,1 d 0, T ; Wloc , all weak L∞ -solution are renorR

malized and, for any given β ∈ C 2 (R), the function v (t, x) = E [β (u (t, x))] is a weak L∞ -solution of the equation ∂v 1 + b · ∇v = Av, ∂t 2

v|t=0 = β (u0 ) .

Proof. Take ϕ (x) = θε (x − x) in the equation of Definition 4.3, where θε are the mollifiers introduced above in the deterministic section. We get (rename x with x)

uε (t, x) +

t

Bs uε (s, x) ds +

0

t 0

t

Cs,k uε (s, x) ◦ dWsk

0

k

= uε (0, x) +

[b · ∇, θε ∗] uε (s, x) ds.

Notice that Bs β (uε ) = β (uε ) Bs uε and Cs,k β (uε ) = β (uε ) Cs,k uε . Let us apply Itˆ o formula in Stratonovich form to β (uε ), which is the same as the classical chain rule for regular functions (see Kunita [143]). We get

β (uε ) (t, x) +

t

Bs β (uε ) (s, x) ds +

0

= β (uε ) (0, x) +

0

k t

0

t

Cs,k β (uε ) (s, x) ◦ dWsk

β (uε ) (s, x) [b · ∇, θε ∗] uε (s, x) ds.

By continuity properties of Stratonovich integrals that can be found in Kunita [143] (notice that Cs,k β (uε ) (s, x) are equibounded) and other simple arguments, one gets

β (u) (t, x) + 0

t

Bs β (u) (s, x) ds +

= β (u0 ) (x) + lim

ε→0

0

k

t

0

t

Cs,k β (u) (s, x) ◦ dWsk

β (uε ) (s, x) [b · ∇, θε ∗] uε (s, x) ds.

The last limit is zero, by Lemma 4.1, and thus we have proved that u is renormalized. Now, let us apply Proposition 4.1 to the weak solution β (u) and get

4.3 Stochastic Case

117

t

β(ut )ϕdx +

0

= β (u0 ) (x) +

t

∗ ∗ β(us )Bs ϕdx ds + β(us )Cs,k ϕdx dWsk

t

0

β (us ) A∗s ϕdx ds.

k

0

The integrands of the Itˆ o integrals are bounded, hence the integrals are martingales. Taking expectation we obtain

t

E [β (ut )] ϕdx +

E [β 0

t

= β (u0 ) (x) +

0

(us )] Bs∗ ϕdx

ds

E [β (us )] A∗s ϕdx ds.

This is the weak form of the parabolic equation above. The proof is complete.

4.3.3 Main Theorem In this section we shall use in essential way the strict ellipticity of the operator Δ, since we need parabolic regularization to prove uniqueness without the assumption div b (t, ·) ∈ L∞ Rd . Theorem 4.3. If b = b1 + b2 with −N Db ∈ L1 0, T ; L1 Rd , (1 + |x|) dx for some N > d,

b1 ∈ L2 0, T ; L∞ Rd

|b2 | ∈ L1 0, T ; L∞ Rd , 1 + |x|

div b2 ∈ L1 0, T ; L∞ Rd

then there exists a unique weak L∞ -solution of (4.6). When we say that two weak L∞ -solutions coincide, we mean they are ∞ d Ω × [0, T ] × R . It follows that, for in the same equivalence class of L d ∞ every ϕ ∈ C0 R , the continuous processes ut ϕdx of Definition 4.3 are indistinguishable. It is a pathwise uniqueness result. Proof. Step 1 (from the SPDE to a parabolic PDE). Call u the difference of two solutions. It is a weak L∞ -solution with zero initial condition. By Theorem 4.2, u is renormalized and, given β0 ∈ C 1 , the function v (t, x) = E [β0 (u (t, x))]

118


is a weak L∞ solution of the equation ∂v 1 + b · ∇v = Δv. ∂t 2 Choose β0 such that β0 (0) = 0, so v|t=0 = 0, and β0 (u) > 0 for u = 0. If we prove that vt = 0, we have proved ut = 0, P -a.s. This easily implies that u is the zero element of L∞ Ω × [0, T ] × Rd , which is our claim. Step 2. (uniqueness for the parabolic equation). Let θε be the mollifiers used in the previous sections and let vε (t, ·) = θε ∗ v (t, ·) (convolution in space). We have ∂vε 1 + b · ∇vε = Δvε + [b · ∇, θε ∗] v, ∂t 2 Consider the function ϕ (x) = (1 + |x|)

vε (0) = 0.

−N

where R > d is the number given in the assumptions of the theorem. We have −N −1

∇ϕ (x) = −N (1 + |x|)

x |x|

hence (1 + |x|) |∇ϕ (x)| ≤ N |ϕ (x)| (in fact they are equal). Using these facts, the boundedness of vε , ∇vε , Δvε and the integrability over Rd of ϕ (see also the next step for the finiteness of the term ϕvε [b · ∇, θε ∗] v) we have d dt

2

ϕ |vε | = −2

ϕvε b · ∇vε +

ϕvε Δvε + 2

ϕvε [b · ∇, θε ∗] v

ϕ|∇vε |2 − vε ∇ϕ · ∇vε ≤ − ϕ|∇vε |2 + N |vε ||ϕ||∇vε |

1 N2 2 2 ≤− ϕ |∇vε | + |vε | |ϕ| 2 2

ϕvε b · ∇vε = ϕvε b1 · ∇vε + ϕvε b2 · ∇vε

ϕvε Δvε = −

ϕvε b1 · ∇vε ≤ 2 b1 (t)L∞ (Rd ) ϕ |vε | |∇vε |

1 2 2 2 ϕ |∇vε | + C b1 (t)L∞ (Rd ) ϕ |vε | dx ≤ 4

−2

4.3 Stochastic Case

−2

119

ϕvε b2 · ∇vε = − ϕb2 · ∇vε2 = vε2 b2 · ∇ϕ + vε2 ϕ div b2

b2 (t) 2 vε2 (1 + |x|) |∇ϕ (x)| dx ≤ 1 + |x| L∞ (Rd )

+ div b2 L∞ (Rd ) vε2 ϕdx

b2 (t) 2 2 ≤ 1 + |x| ∞ d + div b2 L∞ (Rd ) N vε ϕdx. L (R )

Summarizing,

1 d 2 2 ϕ |vε | + ϕ |∇vε | ≤ CN α(t) |vε |2 ϕdx + CvL∞ ([0,T ]×Rd ) dt 4

× (1 + |x|)−N |[b · ∇, θε ∗] v| dx where α (t) :=

b1 (t)2L∞ (Rd )

b2 (t) 2 + 1 + |x| ∞ L

(Rd )

+ div b2 L∞ (Rd )

is integrable. By Gronwall lemma and the result of the next step we deduce

2 lim ϕ (x) |vε (t, x)| dx = 0 ε→0

for all t ∈ [0, T ], and thus v = 0.

−N dx ). To Step 3 (convergence of the commutator in L1 Rd , (1 + |x|) complete the previous proof we have to show that

0

T

(1 + |x|)

−N

|[b · ∇, θε ∗] v (t, x)| dxdt → 0.

We have [b · ∇, θε ∗] v (t, x) = b (x) · ∇vε (x) − (θε ∗ (b · ∇v)) (x)

= (b (x) − b (y)) · (∇θε ) (x − y) v (y) dy

+ θε (x − y) div b (y) v (y) dy

120


(1 + |x|)

−N

|[b · ∇, θε ∗] v (t, x)|

|b (x) − b (x + z)| −N 1 dz ≤ Cθ v∞ (1 + |x|) d ε B(0,ε) ε

−N 1 + Cθ v∞ (1 + |x|) |div b (x + z)| dz. εd B(0,ε)

This implies

T

−N

(1 + |x|)

0

≤C

T

0

|[b · ∇, θε ∗] v (t, x)| dxdt

(1 + |x|)−N |Db (x)| dx dt

(4.7)

because

0

T

−N

(1 + |x|)

1 εd

B(0,ε)

|div b (x + z)| dzdxdt

1 −N = dzdt (1 + |x − z|) |div b (x )| dx d B(0,ε) 0 ε

T

−N ≤C |div b (x )| dx dt (1 + |x |) T

0

and

T

|b (x) − b (x + z)| dzdxdt ε 0 B(0,ε) 1

T

|z| −N 1 (1 + |x|) ≤ Db (α (x + z) + (1 − α) x) dα dzdxdt d ε ε 0 0 B(0,ε)

1 T

1 −N (1 + |x|) ≤ |Db (x + αz)| dx dzdtdα d ε 0 0 B(0,ε)

T

−N (1 + |x |) |Db (x )| dx dt. ≤C (1 + |x|)

−N

1 εd

0

Since the left-hand-side of inequality (4.7) converges to zero for more regular fields b, it converges to zero in general. The proof is complete.

This theorem is taken from Attanasio and Flandoli [15], where the more general case of BV vector fields (instead of W 1,1 ) is treated. A basic point is a uniqueness result of weak solutions to parabolic equations with non regular drift; other results in this direction and their consequences for SDEs can be found in Figalli [94], Le Bris and P.L. Lions [147].

4.3 Stochastic Case

121

4.3.4 Representation of the Law of u (t, x) In this section we only outline a representation formula that may be useful to investigate properties of u (t, x). Let us work under the assumptions of Theorem 4.3. We know that, given β ∈ C 2 (R), v (t, x) := E [β (u (t, x))] satisfies the equation ∂v 1 + b · ∇v = Δv, ∂t 2

v|t=0 = β (u0 (x)) .

Set vT (t, x) := v (T − t, x), bT (t, x) := b (T − t, x), so that, for t ∈ [0, T ], 1 ∂vT − bT · ∇vT + ΔvT = 0, ∂t 2

vT |t=T = β (u0 (x)) .

On [t0 , T ] solve dXtt0 ,x0 = −bT t, Xtt0 ,x0 dt + dWt ,

Xtt00 ,x0 = x0 .

By some classical arguments that we omit (the idea is that, by Itˆ o formula, ∂vT 1 dvT t, Xtt0 ,x0 = dt + ∇vT · dXtt0 ,x0 + ΔvT dt ∂t 2 = ∇vT · dWt and then we integrate and take expected value), we have the representation formula . vT (t0 , x0 ) = E β u0 XTt0 ,x0 Thus we have E [β (u (t, x))] = v (t, x) = vT (T − t, x) = E β u0 XTT −t,x . We have proved (we omit the details about the backward SDE representation): Proposition 4.2. For all β ∈ C 2 (R), E [β (u (t, x))] = E β u0 XTT −t,x

122


where Xtt0 ,x0 is the solution of the SDE dXtt0 ,x0 = −bT t, Xtt0 ,x0 dt + dWt ,

Xtt00 ,x0 = x0 .

We have also the representation T,x E [β (u (t, x))] = E β u0 X t T,x is the solution of the backward SDE on [0, T ] where X t t , T,x = b t, X T,x dt + dW dX t t

T,x = x. X T

4.3.5 Analogy with Stabilization by Noise Let us finally mention that, a posteriori, we notice similarities with the theory of stabilization by noise developed by Arnold et al. [12], Arnold [11]. For a Stratonovich system written in abstract form as dXt = BXt dt +

Ck Xt ◦ dWtk

k

the Itô form is dXt =

B+

Ck2

Xt dt +

k

Ck Xt dWtk .

k

There are cases when Ck2 is a “negative” operator (in a sense), like when Ck∗ = −Ck and Ck Ck∗ is positive definite. This is, in a sense, the case of the first order differential operators Ck = ∂k . When Ck2 are “negative”, we may expect an increase of stability, because d E [Xt ] = dt

B+

Ck2

E [Xt ] .

k

At the PDE level, B + k Ck2 may be regularizing, when B is not. However, going in more details, stabilization can be proven only when the trace of B is negative, see Arnold et al. [12], [11], not in general as the operator B + k Ck2 would suggest. This tells us that the simple argument about regularization of E [Xt ] (or E [u] above) is only the signature of a possible but not sure regularization of the process itself.

4.4 Uniqueness by Stochastic Characteristics

123

4.4 Uniqueness by Stochastic Characteristics 4.4.1 Stochastic Flow of SDE with Non Regular Drift There is another way to prove uniqueness for a stochastic transport equation under poor regularity of the drift b: by stochastic characteristics. This method is somewhat more powerful than the structural one above, but maybe, due to its peculiarity, is less suitable for generalizations. We give an overview without all the details, that can be found in Flandoli et al. [100]. Consider the SDE (stochastic characteristics) dXtx = b (t, Xtx ) dt + dWt ,

X0x = x.

We have seen in Chap. 2 that under the assumption b ∈ C [0, T ] ; Cbα Rd for some α ∈ (0, 1), there exists a stochastic flow of diffeomorphisms. This means the existence of a measurable function ϕs,t (x, ω) defined for s, t ∈ [0, T ], s ≤ t, x ∈ Rd , ω ∈ Ω, with values in Rd , such that ϕs,s (x, ω) = x ϕr,t (ϕs,r (x, ω) , ω) = ϕs,t (x, ω)

t b (r, ϕs,r (x)) dr + Wt ϕs,t (x) = x + s

x → ϕs,t (x, ω) is a diffeomorphism of Rd (t, x) → ϕs,t (x, ω) is continuous for all s, t ∈ [0, T ], s ≤ t, x ∈ Rd , with probability one. Precisely, we have proved: Theorem 4.4. If b ∈ C [0, T ] ; Cbα Rd , the SDE has a stochastic flow of diffeomorphisms, with α -H¨ older continuous derivative for α < α.

4.4.2 Proof of Uniqueness Using the Flow: Introduction Let us describe the idea first. Assume u is a solution of the SPDE. It is a random field. There is a version of Itô formula, often called Itˆ o–Wentzell or Itô–Kunita–Wentzell formula (see Kunita [143]), which allows us to compute

124


du (t, ϕt (x)) when u is sufficiently regular (ϕt (x) denotes the stochastic flow ϕ0,t (x, ω) above). There is also a Stratonovich form of this formula, which looks like classical differential calculus, as usual (see Kunita [143]). If we could apply such a formula, we would have du (t, ϕt (x)) =

∂u ∂u dt + ∇u ◦ dϕt (x) = dt + ∇u · bdt + ∇u ◦ dW = 0 ∂t ∂t

hence u (t, ϕt (x)) = u0 (x) and thus

u (t, x) = u0 ϕ−1 t (x) .

This would proves uniqueness. The problem is similar to the one discussed above several times: we cannot perform computations on L∞ weak solutions u. If we mollify u, then we make computations but we have to control a commutator [b · ∇, θε ∗], which requires assumptions on b. Let us see the result of such a computation. From the equation duε + (b · ∇) uε dt + ∇uε ◦ dW = [b · ∇, θε ∗] u by Itˆ o–Kunita–Wentzell formula in Stratonovich form we get duε (t, ϕt (x)) = [b · ∇, θε ∗] u (t, ϕt (x)) . What is new here is that the commutator appears “free”, not multiplied by terms like β (uε ). This is a great advantage. It is sufficient that [b · ∇, θε ∗] u (t, ϕt (x)) converges in a distributional sense, in probability, namely

P − lim

ε→0

t

0

[b · ∇, θε ∗] u (s, ϕs (x)) ψ (x) dxds = 0

(4.8)

for all ψ ∈ C0∞ Rd , to get

u (t, ϕt (x)) ψ (x) dx =

u0 (x) ψ (x) dx

and thus uniqueness. This was not sufficient above, since distributional convergence of [b · ∇, θε ] does not imply any useful convergence of β (uε ) [b · ∇, θε ∗] since β (uε ) converges only in non-smooth topologies. Unfortunately, we have to prove distributional convergence of the composition [b · ∇, θε ∗] u (t, ϕt (x)), not only of the commutator [b · ∇, θε ∗] u (t, x).


125

This requires assumptions and a lot of work, but at the end the conditions on b are quite general. In particular, no weak differentiability assumption on b is required any more. In the next two subsections we prove the distributional convergence (4.8).

4.4.3 Distributional Commutator Lemma If we only want distributional convergence of [b · ∇, θε ∗] u, we do not need 1,1 d . R the weak differentiability condition b ∈ L1 0, T ; Wloc

Denote by f Lp the norm of a function or vector field f in Lp ([0, T ] × R B(0, R)), for any p ∈ [1, ∞] and R > 0. We use the same notations for the norms in x only. There are several combinations of the integrability exponents of the next lemma that work. We choose the exponent 2 for sake of exposition. d 2 d , div b ∈ Lemma 4.2. If v ∈ L∞ loc [0, T ] × R , b ∈ Lloc [0, T ] × R L1loc [0, T ] × Rd , 1,2 d d ∩ L2 0, T ; Wloc R ψ ∈ L∞ loc [0, T ] × R

has support in [0, T ] × B (0, R), then

ψ (t, x) [b · ∇, θε ∗] v (t, x) dx dt 0 DψL2 + ψL2 bL2 ≤ Cθ vL∞

T

R+1

R

R

R+1

+ Cθ vL∞ ψL∞ div bL1 R+1

and

T

lim

ε→0

0

R

R+1

ψ (x) [b · ∇, θε ∗] v (t, x) dx dt = 0.

Proof. Step 1. Let us drop t. Assume ψ has support in B (0, R). By definition we have

ψ (x) [b · ∇, θε ∗] v (x) dx = I1 + I2 + I3

I1 :=

ψ (x) (b (x) · ∇x ) (θε ∗ v) (x) dx

ψ (x) v (y) b (y) · ∇y θε (x − y) dydx

I2 :=

I3 :=

ψ (x) v (y) θε (x − y) div b (y) dydx.

126


Moreover

I1 = − (θε ∗ v) (x) (b (x) · ∇x ) ψ (x) dx − ψ (x) (θε ∗ v) (x) div b (x) dx and thus |I1 | ≤ vL∞ DψL2 bL2 + vL∞ ψL∞ div bL1 . R+1

R

R

R+1

R

R

Similarly |I3 | ≤ vL∞ ψL∞ div bL1 R+1

R

R+1

.

Finally

I2 = − ψ (x) v (y) b (y) · ∇x θε (x − y) dydx

= v (y) θε (x − y) b (y) · ∇x ψ (x) dydx which yields |I2 | ≤ vL∞ ψL2 bL2 R+1

R

R+1

.

Summarizing these inequalities, we get the one of the statement of the lemma. Step 2. If b is smoother, just as in the commutator lemma of the previous sections, we have the convergence to zero of [b · ∇, θε ∗] v in L1loc . Then using the bound of the previous step and a smooth approximation of b, one proves convergence to zero of [b · ∇, θε ∗] v for general b. The proof is complete. It is interesting to notice that in d = 1 one can improve the result, with respect to ψ. The simple reason is that the assumption div b ∈ L1loc [0, T ] × Rd means Db ∈ L1loc [0, T ] × Rd , hence the classical commutator lemma applies! We state this for comparison with the previous lemma, but it is just Lemma 4.1 above. d 1 d Lemma 4.3. If d = 1, v ∈ L∞ loc [0, T ] × R , b, div b ∈ Lloc [0, T ] × R , d ψ ∈ L∞ loc [0, T ] × R has support in [0, T ] × B (0, R), then

0

T

ψ (t, x) [b · ∇, θε ∗] v (t, x) dx dt ≤ Cθ v ∞ ψ ∞ div b 1 LR+1 LR LR+1

and

lim

ε→0

0

T

ψ (x) [b · ∇, θε ∗] v (t, x) dx dt = 0.


127

4.4.4 Convergence of the Commutator Composed with the Flow We have to prove (4.8), P − lim

ε→0

t

0

[b · ∇, θε ∗] u (s, ϕs (x)) ψ (x) dxds = 0.

We have

t

0

[b · ∇, θε ∗] u (s, ϕs (x)) ψ (x) dxds

t

−1 = [b · ∇, θε ∗] u (s, y) ψ ϕ−1 s (y) Jϕs (y) dyds 0

where J denotes the Jacobian determinant; hence we have (4.8) if we prove that, with probability one, −1 (s, y) → ψ ϕ−1 s (y) Jϕs (y) satisfies the assumptions of Lemma 4.2, or Lemma 4.3 in dimension 1. The case d = 1 is trivial, we have only to prove boundedness and compact support [0, T ] × B (0, R (ω)) for some a.s. finite radius R (ω). The compact support property is true since the map (s, y) → ϕs (y, ω) is continuous, hence the image of [0, T ] × B (0, R) is bounded, where B (0, R) includes the support of ψ. Boundedness follows from the assumption on ψ and the continuity of the differential of ϕ−1 s . The case when we want to apply Lemma 4.2 is more have difficult. We to −1 −1 1,2 d 2 and prove that (s, y) → ψ ϕs (y) Jϕs (y) is of class L 0, T ; Wloc R has compact support. The compact support property is like in the previous 1,2 2 case d = 1. That (s, y) → ψ ϕ−1 0, T ; Wloc Rd is clear (y) is of class L s from ψ ∈ C0∞ Rd and the differentiability of the (inverse) flow. The only very delicate property is 1,2 d 2 (s, y) → Jϕ−1 0, T ; Wloc . R s (y) ∈ L

(4.9)

Indeed, a priori, the flow is only differentiable once, not twice. But Jϕ−1 s (y) is a very special combination of first derivatives so we can differentiate it again, just by a moderate integrability assumption on div b. Lemma 4.4. Assume b ∈ C [0, T ] ; Cbα Rd and div b ∈ Lp [0, T ] × Rd for some p > 2. Then (4.9) holds true.

128


Proof. The computations performed here require a little approximation procedure, not essential for the assumptions or the understanding, so we omit it, see Flandoli et al. [100], Theorem 11. Step 1. Write Yty := ϕ−1 s (y). It solves the equation dYty = −b (t, Yty ) dt + dWt ,

Y0y = y.

The Jacobian vty := Jϕ−1 s (y) satisfies dvty = − div b (t, Yty ) vty dt,

v0y = 1.

This is intuitive clear, having in mind the analogous result for ODEs, and considering the fact that the noise is independent of y. To prove it, introduce Vty := Yty − Wt . It solves dVty = −b (t, Vty ) , dt

Vty = y

where b (t, v) := b (t, v + Wt ). By one of the approximations mentioned above, one can apply the classical result on the determinant Jacobian of a deterministic flow and get d JV y = − div b (t, Vty ) JVty dt t which yields the above relation for vty . Step 2. Thus we have Jϕ−1 s

t

y (y) = exp − div b (s, Ys ) ds . 0

It is sufficient to prove 1,2 d 2 (y) ∈ L 0, T ; W R (s, y) → log Jϕ−1 s loc to prove (4.9). Indeed, −1 = exp log Jϕ−1 D log Jϕ−1 DJϕ−1 s (y) = D exp log Jϕs (y) s (y) s (y) and exp log Jϕ−1 = Jϕ−1 s (y) s (y) is locally bounded (the flow is made of diffeomorphisms). Step 3. Hence we need to show that

(t, y) →

t 0

1,2 d . R div b (s, Ysy ) ds ∈ L2 0, T ; Wloc

(4.10)


129

We see that this is a problem like those studied in Chap. 2: regularization of a function by noise. We invoke the following result on the backward Kolmogorov equation, which can be found in Krylov [137]: for all φ ∈ Lp [0, T ] × Rd , p > 1, the equation ∂Gφ 1 − b · ∇Gφ + ΔGφ = −φ, ∂t 2

u|t=T = 0

(it is sufficient that b is bounded; we work under the Hölder assumption for b) has a unique solution Gφ of class Gφ ∈ Lp 0, T ; W 2,p Rd ∩ W 1,p 0, T ; Lp Rd and the norm in these topologies is bounded by the norm of φ in Lp [0, T ] × Rd . Moreover, if p ≥ 2, then also Gφ ∈ L∞ 0, T ; W 1,p Rd again with norm bounded by the norm of φ in Lp [0, T ] × Rd . Modulo an approximation to apply Itˆ o formula, we get dGdiv b (t, Yty ) = − div b (t, Yty ) dt + ∇Gdiv b (t, Yty ) · dWt hence

t

t div b (s, Ysy ) ds = Gdiv b (0, y) − Gdiv b (t, Yty ) + ∇Gdiv b (s, Ysy ) · dWs . 0

0

Therefore, denoting by D the weak differential in y,

t

D 0

div b (s, Ysy ) ds = DGdiv b (0, y) − DGdiv b (t, Yty ) · DYty

t + D2 Gdiv b (s, Ysy ) DYsy · dWs . 0

We have to prove that each term on the right-hand-side is in L2 (0, T ; d 2 d 2 Lloc (R )). The first one, DGdiv b (0, y), is Lloc R , because Gφ is bounded in W 1,p Rd . The second term is left as an exercise after one has seen how to estimate the third one. Let us prove that

T

E 0

B(0,R)

t 2 2 y y D Gdiv b (s, Ys ) DYs · dWs dydt < ∞ 0

130


for every R > 0. This expression is bounded by

T

2 2 D Gdiv b (s, Y y ) DY y ds dy. s s

T

E 0

B(0,R)

By Holder inequality, it is sufficient to show that

B(0,R)

0

T

q

E [|DYsy | ] dsdy < ∞

for every q > 2 and

B(0,R)

T 0

p E D 2 Gdiv b (s, Ysy ) dsdy < ∞

for some p > 2. The first inequality is similar to those proved in Proposition 2.2 of Chap. 2 (we omit the proof). The second expression is equal to

T

T

E 0

B(0,R)

2 D Gdiv b (s, x)p Jϕs (x) dxds

= 0

≤

B(0,R)

2 D Gdiv b (s, x)p E [Jϕs (x)] dxds

sup t∈[0,T ],x∈B(0,R)

E [Jϕs (x)]

T 0

B(0,R)

2 D Gdiv b (s, x)p dxds.

Both factors are finite: the first one by computations like those of Chap. 2, the second one by the regularity theorem on the backward Kolmogorov equation. The proof is complete.

4.4.5 Final Statement From Lemma 4.2, or Lemma 4.3 in dimension 1 and the computations of the previous section we have: d α R and either (i) div b ∈ Theorem 4.5. Assume b ∈ C [0, T ] ; C b p d L [0, T ] × R for some p > 2, or (ii) d = 1 and div b ∈ L1loc [0, T ] × Rd . Then there exists a unique L∞ -solution of the SPDE.


131

This statement is remarkable for its simplicity and, in dimension d > 1, for its generality in terms of lack of weak differentiability of b. Thanks to the second case (ii), also this approach, like the structural one of the previous sections, covers the usual 1-dimensional examples b (x) = |x ∧ R|α for α ∈ (0, 1) and some R > 0.

Chapter 5

Other Models: Uniqueness and Singularities

This chapter contains a number of other examples, presented for different purposes. Not only the uniqueness problem but also emergence of singularities is discussed. First, we give a few examples where noise does not change the difficulties related to these two issues; a little bit improperly, we call them “negative” examples (in spite of the fact that they are very interesting). Then we show two examples where singularities are prevented by noise: continuity equation and vortex point motion. We call them “positive” examples. The next section on nonlinear Schr¨ odinger equation describes theoretical and numerical results both of positive and negative type. Finally, we summarize the attempts made on the 3D stochastic Navier–Stokes equations, in the direction of understanding uniqueness and singularities.

5.1 Negative Examples In Chaps. 3 and 4 we have seen that a suitable multiplicative bilinear noise may restore uniqueness. In this section we give three examples where a similar noise has no special effect on the equation. What makes the difference is not clear at present.

5.1.1 Linear Transport Equation with Random Coefficients It would be great if the methods of Chap. 4 apply to linear transport equation with random coefficient b (t, x, ω): we could approach nonlinear equation of the form du + (b (u) · ∇) udt + noise = 0 by the iteration dun+1 + (b (un ) · ∇) un+1 dt + noise = 0. F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Lecture Notes in Mathematics 2015, DOI 10.1007/978-3-642-18231-0 5, c Springer-Verlag Berlin Heidelberg 2011

133

134

5 Other Models: Uniqueness and Singularities

This is not possible, or at least not without special assumptions and a new insight, since there are easy counterexamples. One is the one-dimensional linear equation du (t, x) + (b (x − W (t)) · ∇) u (t, x) dt + ∇u (t, x) ◦ dW (t) = 0 with b (x) = 2sign (x)

|x|. The transformation v (t, x) = u (t, x + W (t))

leads to the equation (we heuristically use Itô–Wentzell–Kunita formula in Stratonovich form; a full justification by regular approximations looks possible but we do not give it since the final result is negative) dv (t, x) = (du) (t, x + W (t)) + (∇u) (t, x + W (t)) ◦ dW (t) = − (b (x) · ∇) u (t, x + W (t)) dt − ∇u (t, x + W (t)) ◦ dW (t) + (∇u) (t, x + W (t)) ◦ dW (t) = − (b (x) · ∇) v (t, x) dt which is our Example 1.6. The transformation is reversible: u (t, x) = v (t, x − W (t)). Thus the SPDE with random drift b (x − W (t)) is equivalent to the PDE with drift b. The SPDE is thus badly posed, in spite of the fact that, P -a.s., b (x − W (t)) satisfies the assumptions of the uniqueness theorems of Chap. 4. This example has been given in Flandoli et al. [100]. Remark 5.1. It is easy to see where the methods of Chap. 4 fail for this example. The structural method does not work since the expectation E [(b (x − W (t)) · ∇) u (t, x)] cannot be splitted. The method of characteristics is in trouble for more subtle reasons. Go back to Chap. 2 and see the analysis of characteristics done by the method based on the backward Kolmogorov equation. When the righthand-side of the backward Kolmogorov equation is random, the solution u (t, x, ω) is also random, but no more adapted to the forward filtration! Thus when we substitute it in the SDE of characteristics, we get an anticipating SDE and we cannot prove anymore the same results.

5.1.2 Euler Equation with Too Simple Noise Consider the Euler equations in vorticity form in R2 ∂ξ (t, x) + (u (t, x) · ∇ξ (t, x)) = 0 ∂t

5.1 Negative Examples

135

where ξ = ∂2 u1 − ∂1 u2 . It is a remarkable example of nonlinear transport equation, but with non-local nonlinearity: u is given in terms of ξ, but through a global space integral (Biot–Savart law, see Sect. 5.2.2). Add to this equation the same simple multiplicative noise used above for the linear transport equation: ∂t ξ (t, x) + (u (t, x) · ∇ξ (t, x)) dt +

2

∂i ξ (t, x) ◦ dW i (t) = 0

i=1

Following [152], this equation is (formally) equivalent to the family of stochastic ordinary equations depending on a parameter a ∈ R2 dXta =

R2

K(Xta − Xta )ξ0 (a )da dt + dWt

⊥

x with K (x) = |x| 2 , ξ0 being the initial condition of the vorticity equation. The equivalence is: given ξ (t, x), Xta is the solution of dXta = u (t, Xta) dt + dWt where u is found from ξ by Biot–Savart law; given Xta , we have ξ (t, Xta ) = ξ0 (a). The equation for (Xta ) is equivalent to

dYta

= R2

K(Yta

−

Yta )ξ0 (a )da

dt

by the change of variable Yta = Xta − Wt , and the equation for (Yta ) corresponds (by the same equivalence above) to the classical vorticity equation ∂t ξ (t, x) + (u (t, x) · Dξ (t, x)) dt = 0 ∂t

ξ = ∂2 u1 − ∂1 u2

with initial condition ξ0 . This means that the stochastic vorticity equation is (at least formally) equivalent to the deterministic one. Since the result would be a negative one and we are convinced that the argument is true, we do not develop rigorous details. This example has been given in Flandoli et al. [100]. The conclusion is that there is no advantage in introducing that kind of stochastic perturbation: we get an SPDE equivalent to the original PDE. Remark 5.2. The multiplicative noise 2i=1 ∂i ξ (t, x) ◦ dW i (t) corresponds to a random rigid translation in Lagrangian coordinates. It cannot change the dynamics. If we use a space-dependent noise, this symmetry is broken and we can hope to see something new. We shall see below that this indeed happens, with a noise that strongly separates nearby space points. It yields a form of regularization of the vortex point motion, a dynamic related to Euler evolution.

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5.1.3 Inviscid Burgers Equation: Non Uniqueness Recall Example 1.7 of Chap. 1: the inviscid Burgers equation has more than one solution (is we do not restrict ourselves to the concept of entropy solutions). So let us try to restore uniqueness by means of the multiplicative noise used above several times. Consider the one-dimensional stochastic Burgers equation dt u + u∂x udt + σ∂x u ◦ dWt = 0, u|t=0 = u0 (5.1) with x ∈ R, t ≥ 0. Its solution can be represented in the form (or better, it satisfies the identity) u (ϕt (x) , t) = u0 (x) (5.2) where ϕt (x) is the stochastic flow defined by (the characteristics) dϕt (x) = u (ϕt (x) , t) dt + σdWt . This is true where the flow is a bijection and under suitable regularity of solutions, but since the final result is clearly negative, we do no write a rigorous and detailed proof. To check the previous relation, assume that u is sufficiently regular and ϕt (x) is well-defined. Then, by Itˆ o–Wentzell–Kunita formula in Stratonovich form, du (ϕt (x) , t) = ∂t u + ∂x u ◦ dϕt = −u∂x udt − σ∂x u ◦ dWt + u∂x udt + σ∂x u ◦ dWt = 0. This proves (5.2). But the identity u (ϕt (x) , t) = u0 (x) implies that the equation for the flow in fact is dϕt (x) = u0 (x) dt + σdWt namely ϕt (x) = x + u0 (x) t + σWt . The drift is a direct function of the initial position. Now, consider the same initial condition as in Example 1.7 of Chap. 1: u0 = 1x>0 . In the deterministic case, σ = 0, the characteristics are ϕt (x) = x + u0 (x) t = x + t · 1x>0 so the relation u (ϕt (x) , t) = u0 (x) identifies the solution u (x , t), at a given time t > 0, only at points x of the form x + t · 1x>0 for some x thus for x ≤ 0 and x > t only. More solutions exist in the region 0 < x ≤ t. With the same initial condition u0 = 1x>0 , we observe the same result in the stochastic case: the relation u (ϕt (x) , t) = u0 (x) identifies the solution

5.1 Negative Examples

137

u (x , t) only at points x of the form x + t · 1x>0 + σWt for some x thus for x ≤ σWt and x > t + σWt only. More solutions exist in the random region σWt < x ≤ t + σWt . Of course this is similar to the previous subsection: the noise acts as a rigid translation of the background, no new phenomena may occur.

5.1.4 Inviscid Burgers Equation: Blow-Up Deterministic Burgers equation provides a simple example of blow-up of the derivative. Consider the Lipschitz continuous initial condition ⎧ ⎨ u0 (x) =

1 for x ≤ 0 1 − x for 0 ≤ x ≤ 1 . ⎩ 0 for x ≥ 1

Since the characteristics are ϕt (x) = x + u0 (x) t, a unique Lipschitz continuous solution exists until time t = 1, but then characteristics meet ϕ1 (0) = 1 = ϕ1 (1) and a discontinuity emerges (see Lax [146] for other details). The space derivative becomes infinite. One can also see this from the equation for the derivative (until it exists)

∂ 2 + ∂x ∂x u + (∂x u) = 0 ∂t

which gives us d ∂x u (ϕt (x) , t) = − (∂x u (ϕt (x) , t))2 dt that explodes in finite time when ∂x u0 (x) < 0. In the stochastic case (5.1), the flow is given by ϕt (x) = x + t + σWt for x ≤ 0 ϕt (x) = x + σWt for x ≥ 1 hence again ϕ1 (0) = ϕ1 (1), exactly as in the deterministic case. As we said above, the effect of the noise is just a background space translation. This kind of noise cannot improve the well-posedness and the regularity theory. Remark 5.3. The same result can be obtained by proving the blow-up of ∂x u (ϕt (x) , t).

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It seems that nothing changes if we consider more general 1-dimensional conservation laws, and we try with a more general noise ∂t u + ∂x f (u) +

∞

σk (x) ∂x u ◦ dWtk = 0,

u|t=0 = u0

k=1

(more vague computations on 2-dimensional systems of conservation laws, always 1-dimensional in space do not seem to give different results). Space dependent coefficients σk (x) of the noise will be the basic ingredient to regularize the vortex point motion, opposite to the absence of regularization shown above for 2D Euler equations when σk are constant. But here it does not seem they matter. Let us see, heuristically, a few details. For the deterministic equation ∂t u + ∂x f (u) = 0,

u|t=0 = u0

set a = f , ∂t u + a (u) ∂x u = 0,

u|t=0 = u0

which is a nonlinear transport equation. The equation of characteristics is d ϕt (x) = a (u (ϕt (x) , t)) , dt

ϕ0 (x) = x.

d If u and ϕ exist and are sufficiently regular, than dt u (ϕt (x) , t) = 0 and thus, as above for Burgers, relation (5.2) holds. Therefore a (u (ϕt (x) , t)) is constant characteristics are always straight lines

ϕt (x) = x + a (u0 (x)) t. This is true until everything is regular. Depending on data, precisely when a ◦ u0 is decreasing somewhere, these lines may coalesce and a discontinuity may arise. The class of discontinuous solutions is needed to deal with the equation (this is also another motivation for the investigation of weak solutions of linear transport equations). In the stochastic case, the stochastic characteristics are dϕt (x) = a (u (ϕt (x) , t)) dt +

∞

σk (ϕt (x)) dWtk ,

ϕ0 (x) = x

k=1

since again du(ϕt (x), t) = 0 is u and ϕ are regular. Hence again u(ϕt (x), t) = u0 (x). This implies that characteristics are martingale perturbations of straight lines:

5.2 Positive Examples

139

ϕt (x) = x + a (u0 (x)) t +

∞ k=1

t 0

σk (ϕs (x)) dWsk

until everything is regular. A question now is: could it be that these characteristics do not meet (discontinuities do not arise from regular initial data)? If, for instance, a (u0 (x1 )) = 1,

a (u0 (x2 )) = −1

at two points x1 < x2 we have ϕt (x1 ) = x1 + t + M1 (t) ϕt (x2 ) = x2 − t + M2 (t) where Mi (t) =

∞

t 0

k=1

σk (ϕs (x)) dWsk ,

i = 1, 2

are two martingales. Under very general assumptions we have a.s. lim

t→+∞

Mi (t) =0 t

and thus coalescence happens again, with probability one. It seems that there is no hope to regularize inviscid Burgers equations by the kind of multiplicative noise used successfully for other equations. However, maybe other kind of noise behave differently. See Albeverio and Rozanova [5] for an intriguing result.

5.2 Positive Examples Both the following two examples show that certain singularities in PDEs may be prevented by noise. This is a very open research direction which hopefully will receive more attention.

5.2.1 Stochastic Continuity Equation Example 1.8 of Chap. 1 shows a kind of singularity that we may expect in deterministic continuity equations with poor drift coefficients. Let us give

140


more details. Given a vector field b (x, t) on Rd ×[0, T ], the continuity equation is the following linear PDE ∂t p + div (bp) = 0,

p|t=0 = p0 .

We use the letter p since we concentrate on the case when the solution is a probability density. The equation can be interpreted in a distributional way and it has a meaning even for measure valued functions t → pt . It is dual to the backward transport equation on Rd × [0, T ] ∂t u + b · ∇u = 0,

u|t=T = uT .

The duality comes from the fact, true in certain classes of regularity, that ∂t up dx = − pb · ∇u dx − u div (bp) dx = 0 namely

uT p (T, x) dx =

u (0, x) p0 dx

(and generalizations to smaller time intervals). Thus, if we do not have uniqueness for the transport equation, namely we have non-zero solutions

from uT = 0, we get u (0, x) p0 dx = 0 which is absurd for all p0 non orthogonal to u (0, x). This may be intuitively interpreted as the indication that the existence of a solution p, sufficiently regular to perform the previous computations, is wrong. Non-uniqueness for u corresponds to some kind of non-existence for p. With this general idea in mind, one takes the nonuniqueness Example 1.6 of Chap. 1 for the transport equation, write it in backward form (this is the origin of a change of sign in b), and guess an example of continuity equation with existence troubles. This is Example 1.8 of Chap. 1. In that example, given a probability density p0 , unless it is very symmetric, solutions such that p (t, ·) is a probability density for every t do not exist. This pathology is not the rule, it depends on the regularity of b. Also quite weak drift b do not produce troubles. An is advancedaccount 1 d 0, T ; BV R with given by Ambrosio [8] in the case when b is only L loc div b ∈ L1 0, T ; L∞ Rd . Under these assumptions the continuity equation is well posed in the class of weak L∞ -solutions, a Lagrangian flow ϕt (x) exists for the equation dϕt (x) = b (ϕt (x)) , dt

ϕ0 (x) = x

(5.3)

and, given a probability measure p0 , the image law pt := ϕt p0 is the solution of the continuity equation.


141

However, if b is only H¨ older continuous, we may have the blow-up phenomenon of Example 1.8. Solutions ϕt (x) of (5.3) coalesce as t increase: for instance, ϕ1 (1) = ϕ1 (−1) = 0. The only mass preserving generalized solution pt of the continuity equation is not a function but a probability measure with a concentrated mass at x = 0, equal to the portion of mass of p0 given to those points that at time t have coalesced at zero. The situation is different under random perturbations. Consider the stochastic continuity equation dp + div (bp) dt +

d

div (ei p) ◦ dW i = 0,

p|t=0 = p0 .

(5.4)

i=1

Denote by L1+ (Rd ) the space of all probability densities on Rd . The following definition requires boundedness of b (assumed below in the theorem). Definition 5.1. Given p0 ∈ L1+ (Rd ), a weak L1+ -solution of the Cauchy problem (5.4) is a non-negative measurable function p : Rd × [0, T ] × Ω → R, such that p (., t, ω) ∈ L1+(Rd ), for all t ∈ [0, T ] and P -a.s. ω ∈ Ω, t →

Rd p(x, t)g(x, t)dx is integrable for P -a.s. ω ∈ Ω for every bounded cond ∞ d tinuous

function g : R × [0, T ] → R, for all f ∈ C0 (R ) the process t → Rd p(x, t)f (x)dx is a continuous adapted semimartingale and

p(x, t)f (x)dx − p0 (x)f (x)dx Rd t = ds p(x, s)b(x, s) · ∇f (x)dx Rd

Rd

0

+

d t i=1

If

Rd

0

Rd

p(x, s)Di f (x)dx ◦ dW i (s) .

p(x, t)dx = 1 for a.e. t, we call it a mass preserving solution.

The following theorem (taken from Flandoli et al. [101]) shows that the blow-up of Example 1.8 of Chap. 1 disappears in the stochastic case. Due to the boundedness restriction we have to impose, think to the variant b (x) = −2sign (x) |x| ∧ 1 of Example 1.8. Theorem 5.1. Assume that b ∈ C [0, T ] ; Cbα Rd , Rd for some α ∈ (0, 1), with div b ∈ Lp [0, T ] × Rd for some p > 2 (or just div b ∈ L1loc ([0, T ] × R) in dimension 1). Let p0 ∈ L1+ (Rd ) be given. Let ϕt (x) be the stochastic flow of diffeomorphisms associated to the SDE (1.1) (Chap. 2) and let μt be the image (random) measure of μ0 (dx) := p0 (x) dx under ϕt . Then μt is absolutely continuous with respect to Lebesgue measure and its density p(x, t) is a masspreserving weak L1+ -solution of the continuity equation (5.4). Proof. By definition, μt is a random probability measure. See Crauel [55] for general facts about random measures and Flandoli [96] for some detail related

142


to their use for random continuity equations. Since ϕt is differentiable and invertible, μt is absolutely continuous with respect to Lebesgue measure and its density p(x, t) is given by the formula −1 p(x, t) = | det Dϕ−1 t (x) | · p0 ϕt (x) and satisfies the identity p(x, t)g(x)dx = Rd

Rd

p0 (x) g(ϕt (x))dx

(5.5)

(5.6)

for all bounded continuous function g : Rd → R. Let us prove that p(x, t) is a weak L1+ -solution. The measurability in all arguments comes from (5.5). The property p (., t, ω) ∈ L1+(Rd ), for all t ∈ [0, T ] and P -a.s. ω ∈ Ω is true by definition of p. Identity (5.6) implies one of the conditions of Definition 5.1 and, restricted to g = f ∈ C0∞ (Rd ), implies the continuous semimartingale property. Finally, again from (5.6) and Itˆ o formula in Stratonovich form (see Kunita [143]) we have p(x, t)f (x)dx = p0 (x) f (ϕt (x))dx

Rd

Rd

= Rd

+

p0 (x) f (x)dx +

d k=1

t

Rd

p0 (x) dx

t

∇f (ϕs (x)) · b(ϕs (x) , s)ds

0

k

Dk f (ϕs (x)) ◦ dW (s)

0

and now, in the last expression, the first term is equal to 0

t

ds p0 (x) ∇f (ϕs (x)) · b(ϕs (x) , s)dx Rd t ds p (x, s) ∇f (x) · b(x, s)dx; = Rd

0

and the second one is equal to d t k=1

=

0

Rd

p0 (x) Dk f (ϕt (x))dx ◦ dW k (s)

d t k=1

0

Rd

p (x, s) Dk f (x)dx ◦ dW k (s) .


143

We have used Fubini theorem both in the classical and stochastic version, taking advantage of the boundedness of all terms except the L1 function p0 (x). Thus the equation in Definition 5.1 is satisfied. The proof is complete.

The stochastic continuity equation is related to the SPDE studied by Le Jan and Raimond [148]. However, in our case we insist on an irregular drift, while the main examples of generalized flows of [148] are related to irregular diffusion terms. Remark 5.4. Depending on the level of intuition, it may appear strange that the coalescence associated to the drift b (x) = −2sign (x) |x| ∧ 1 disappears under noise. Indeed, let us think to what happens when randomness is superimposed to a pair of coalescing trajectories: the first intuition could be that these trajectiories fluctuate but still coalesce, after a random time. If we think so (this happened to the author), we make a mistake: b is not affected by noise, so the singular region where coalescence may take place is always the neighborhood of x = 0. Only when two trajectories lie in such region, they approach each other very fast, otherwise, they approach each other like in any Lipschitz contracting environment (which does not lead to coalescence in finite time). And, as we understood in Chap. 2 with the occupation measure, the time spent by trajectories in the neighborhood of x = 0 is small; at the end of the story it is not sufficient for coalescence. Remark 5.5. This argument applies only to the linear case when b does not depend on the solution. Otherwise, the region of collapse may change with the solution and noise may loose its effect. In a sense, this is what happened above for stochastic conservation laws.

5.2.2 Point Vortex Motion At the beginning of Chap. 3 we have already seen that Euler equations ∂u + u · ∇u + ∇p = 0, ∂t

divu = 0,

u|t=0 = u0

(5.7)

are, especially in dimension d = 3, a rich example of basic problems, like uniqueness, singularities and anomalous energy dissipation. The 2D case is a little bit better: see, for instance, Majda and Bertozzi [152] and P.L. Lions [150] for reviews of several results. A classical one is existence of solutions when u0 is in the Sobolev space W 1,2 (namely L2 -vorticity ξ = ∇⊥ · u = ∂2 u1 − ∂1 u2 ) and uniqueness when the vorticity is bounded. When we go essentially below this regularity, we meet troubles, also in d = 2. The case when the vorticity is a signed measure is intriguing and received much attention, due to the interest in the evolution of vortex

144


structures like sheets or points of vorticity concentration. See Majda and Bertozzi [152] for a review. Deep existence and stability results for distributional vorticity fields of distinguished (constant) sign have been proved, first for a class of distributions which includes vortex sheets but not vortex points, then also for point vortices, see among others Delort [80], Poupaud [172]. Uniqueness is an open problem in all such cases. When the vorticity has variable sign, even existence is not clear. Even more, when the vorticity is pointwise distributed (point vortices) and of variable sign, a reasonable formulation of the Euler equations is missing and the Eulerian viewpoint have to be replaced by a Lagrangian one. We address to Marchioro and Pulvirenti [155] for an extensive discussion of this formulation and its motivations. Here we take it for granted. The Lagrangian formulation of point vortex motion is given by a finite dimensional system of ordinary differential equations for the positions of the vortices. If we have n point vortices, with intensities ω1 , ..., ωn ∈ R, which occupy the positions x1t , ..., xnt ∈ R2 at time t, then dxit ωj K(xit − xjt ), = dt

i = 1, ..., n

(5.8)

j=i

where, in full space, K(x) =

x⊥ |x|

2

with the notation x⊥ = (−x2 , x1 ) if x = (x1 , x2 ). It is a closed system: no other variable is needed to describe the evolution (the intensity of point vortices is constant, corresponding to the fact that vorticity is transported in 2D). However, this Lagrangian description contains a problem, partially described in Example 1.9 of Chap. 1. The system (5.8) is well posed only for almost all initial configurations with respect to Lebesgue measure and one can give explicit examples of initial configurations with vortices that coalesce in finite time, see Marchioro and Pulvirenti [155]. The Lagrangian equations loose meaning at the coalescing time and it is not clear how to continue the evolution afterwards; the natural idea is that we have a single new point vortex with intensity equal to the algebraic sum of the coalescing ones, but then the number of equations is changed. ODEs with variable number of equations is not an usual concept. Perhaps a proper Eulerian description could be meaningful also after the coalescence time, but a rigorous formulation of this fact is not known. We consider this coalescing phenomenon as an emergence of singularity. It is certainly such for the ODE; maybe it is not for a proper Eulerian formulation. Notice however that Euler equations are reversible. If it could be possible to introduce a concept of solution such that more point vortices coalesce in finite time and become a single vortex, by time reversal we immediately have


145

an example of non-uniqueness: a point vortex may remain a point vortex for ever, or may split in more point vortices at any time. So, in case an Eulerian formulation exists and the example is no more of blow-up in nature, then an example of non-uniqueness immediately emerges. In Sect. 5.1.2 above we have already discussed the vorticity field of the 2D Euler equations and we have seen that it is not reasonable to hope for 2 a regularizationi by noise if we use the simple multiplicative noise i=1 ∂i ξ (t, x) ◦ dWt . It produces only a rigid random translation of the environment. Thus, if a set of point vortices coalesce for the deterministic dynamics, it does the same under such simple noise. But what happens if the noise is highly uncorrelated in space at small distances? When two (or more) point vortices get very close to each other, they are subject to quite uncorrelated impulses. This could prevent coalescence. We can rigorously prove that this happens, under suitable assumptions on the noise. Let us describe in more detail the stochastic model we are going to study and its motivation. Euler equations can be recast in terms of vorticity as the system ∂ξ + u · ∇ξ = 0, ξ|t=0 = ξ0 (5.9) ∂t u = −∇⊥ −1 ξ

(5.10)

where the second relation is the so called Biot–Savart law. Consider the limit case when ξ is the sum of finite number of delta Dirac masses (see Fig. 5.1) ξ(., t) =

n i=1

ωi δxit .

(5.11)

Such field is too singular for an Eulerian description: the velocity field is not square integrable (so even the weakest form of (5.7) is not meaningful) and its singularity coincides with the delta Dirac points of the vorticity (so also (5.9) is not meaningful). In spite of this, there are good arguments, based on the limit of regular solutions supported around the ideal point vortices (Marchioro and Pulvirenti [155]), to accept that a certain finite dimensional differential equation for the position of the point vortices is the correct physical description of the evolution of fields like (5.11). The equations for the evolution of the positions of point vortices have the form (5.8) above.

Fig. 5.1 Point vortices in the plane

146


Now let us investigate the effect of a multiplicative noise on the Euler equations. As we said above, let us consider a noise with a potentially rich correlation structure at small distances, thus the vorticity equation may be dξ + u · ∇ξdt +

N

σk (x) · ∇ξ ◦ dβtk = 0,

ξ|t=0 = ξ0

(5.12)

k=1

where σk (x) are suitable 2d vector fields and {βtk }k=1,...,N are independent Brownian motions and u is again reconstructed from ξ by means of Biot– Savart law (5.10). We need nontrivial fields σk and presumably a high value of N (maybe infinite) to hope for a regularization. The stochastic point vortex dynamics which corresponds to (5.12) has the form dxit =

j=i

ωj K(xit − xjt )dt +

N

σk (xit ) ◦ dβtk ,

xi0 = xi

(5.13)

k=1

for i = 1, ..., n, with each single xit in R2 . We have denoted by x1 , ..., xn ∈ R2 the initial positions of the point vortices. It can be proven that, under suitable hypoellipticity assumptions on the fields σk (x), this stochastic point vortex dynamics is globally well posed (in particular coalescence of point vortices disappear) for all initial conditions. The hypoellipticity condition, moreover, is generically satisfied. All the details can be found in Flandoli et al. [102]. Here we outline the framework and the proofs. Remark 5.6. A very important remark is that the noise in system (5.13) is the same for all particles. Thus this is very similar to the so called n-point motion of a single SDE. The regularizing effect of the noise at the level of the n-point motion is a very non-trivial fact. It would be trivial to improve the regularity of the deterministic system (5.8) by adding independent Brownian motions to each component, but this would not correspond to a Lagrangian point vortex formulation of stochastic Euler equations. To avoid certain difficulties related to the infinite space, let us consider the artificial environment of the 2D Torus T = R2 /(2πZ2 ). Our point vortices x⊥ will move on it. The kernel K (x) is no more simply given by |x| 2 as in the full space, but its regularity properties are similar (the Fourier transform of 1 ik⊥ 2 K is k x , ..., xn ∈ Tn 2 , k = (k1 , k2 ) ∈ Z \{0}). Let Γ be the set of all

such that xi = xj for some i = j (Γ is the union of the generalized diagonals of Tn ). Let {σk }k=1,...,N be a finite number of smooth vector fields on T. Introduce the associated vector fields on Tn : Aσk (x1 , . . . , xn ) = Ak (x1 , . . . , xn ) = σk (x1 ), . . . , σk (xn )

(5.14)


147

Recall that given vector fields A, B in Rm , their Lie bracket [A, B] is the vector fields in Rm defined by [A, B] = (A · ∇)B − (B · ∇)A. We assume that {σk }k=1,...,N satisfies: 1. σk are periodic, infinitely differentiable and div σk = 0 2. the vector space spanned by the vector fields A1 , ..., AN ,

[Ai , Aj ], 1 ≤ i, j ≤ N,

[Ai , [Aj , Ak ]], 1 ≤ i, j, k ≤ N, ...

at every point x ∈ Γ c is R2n . The second assumption is a form of H¨ ormander’s condition, and it is proved by Flandoli et al. [102] that it is generic when N is large enough; see also Dolgopyat et al. [83]. It is the way we make rigorous the idea that the noise must act quite independently at small distances, everywhere. Technically speaking, this conditions will ensure that the law at any time t > 0 of the solution of a regularized stochastic equation is absolutely continuous with respect to Lebesgue measure if we start outside the diagonal Γ . Under these hypotheses we are able to prove the following result of wellposedness of the dynamics for all initial n-point configurations. Theorem 5.2. For all X0 = (x10 , ..., xn0 ) ∈ Tn \Γ (5.13) has one and only one global strong solution. The notion of strong solution (Xt )t≥0 to (5.13) is the classical one for SDEs with the additional condition that P (Xt ∈ Γ c for all t ≥ 0) = 1. The proof of Theorem 5.2 is based on two ingredients. The first one is the analog of the deterministic result: well posedness for almost every initial configuration. The proof is long and somewhere heavier than the deterministic case due to Itô formula. We omit it here, see [102]. Lemma 5.1. For Lebesgue almost every X0 = (x10 , ..., xn0 ) ∈ Tn , (5.13) has one and only one global strong solution. The second ingredient is the fact that N is not visible by the dynamics (5.13) because of the Hörmander’s condition above. To put rigorously in practice this simple argument we need to work with a regularized problem where the singular Biot–Savart kernel K is replaced by a smooth one K δ . The equations read dxi,δ t =

j=i

j,δ ωj K δ (xi,δ t − xt )dt +

N k=1

k σk (xi,δ t ) ◦ dβt .

(5.15)

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The smooth function K δ is taken equal to K up to the closed δ-neighborhood Γδ of Γ ; hence solutions of system (5.15) are solutions of (5.13) and viceversa, as far as they stay outside Γδ . For system (5.15) we immediately have strong well posedness and also the existence of a smooth stochastic flow ϕδt on Tn , see Kunita [143]. Denote by δ τX (ω) the first instant when ϕδt (X0 ) ∈ Γδ and set it equal to +∞ if this 0 δ > 0 = 1 by continuity of trajectories. fact never happens. We have P τX 0 δ Assume X0 ∈ Γ c . The solution ϕδt (X0 ), on the random interval 0, τX , is 0 δ (ω) is also the first instant also the unique solution (Xt ) of (1.1). Thus τX 0 when Xt ∈ Γδ . Set δ τX0 (ω) = sup τX (ω). 0 δ∈(0,1)

By localization, we have a unique solution of (1.1) on [0, τX0 ). Let us add a point Δ to Tn and set ϕt (X0 ) = Δ for t ≥ τX0 , where τX0 < ∞. The family of processes ϕt (X0 ), X0 ∈ Γ c , so defined, lives in Γ c ∪ Δ for positive times and is Markov. Then P (ϕ[ε,T ] (X0 ) ∈ Γ c ) = P (ϕ[0,T −ε] (Y ) ∈ Γ c )μϕε (X0 ) (dY ) Γ c ∪{Δ}

where {ϕ[ε,T ] (X0 ) ∈ Γ c } = {ω ∈ Ω : ϕt (X0 )(ω) ∈ Γ c , for any t ∈ [ε, T ]} and μϕε (X0 ) is the law of ϕε (X0 ). Recall the definition of N . We have P (ϕ[0,T −ε] (Y ) ∈ Γ c ) = 1 for all Y ∈ N c . Then c

P (ϕ[ε,T ] (X0 ) ∈ Γ ) ≥

Nc

P (ϕ[0,T −ε] (Y ) ∈ Γ c )μϕε (X0 ) (dY )

= 1 − μϕε (X0 ) (N ). Now, assume X0 ∈ Γδc∗ for some δ ∗ > 0. We have, for all δ ∈ (0, δ ∗ ), μϕε (X0 ) (N ) = P (ϕε (X0 ) ∈ N ) δ δ > ε) + P (ϕε (X0 ) ∈ N, τX ≤ ε) = P (ϕε (X0 ) ∈ N, τX 0 0 δ δ ≤ P (ϕδε (X0 ) ∈ N, τX > ε) + P (τX ≤ ε) 0 0 δ ≤ ε) ≤ P (ϕδε (X0 ) ∈ N ) + P (τX 0 δ = P (τX ≤ ε). 0

To say that P (ϕδε (X0 ) ∈ N ) = 0 we have used two facts: N is Lebesguenegligible, the law of ϕδt (X0 ) on Tn is absolutely continuous with respect to Lebesgue measure, for each X0 ∈ Γ c , δ > 0, t > 0. The latter property

5.3 Singularities of Stochastic Schr¨ odinger Equation

149

is a consequence of the H¨ ormander’s condition above, due to the following lemma, taken from Nualart [169], Theorem 2.3.2. Lemma 5.2. Consider the stochastic equation in Stratonovich form in Rm Xt = x0 +

N j=1

t

Aj (Xs ) ◦

0

dWsj

t

+

A0 (Xs )ds 0

with infinitely differentiable coefficients with bounded derivatives of all order. Assume the following H¨ ormander’s condition at point x0 : the vector space spanned by the vector fields A1 , ..., AN ,

[Ai , Aj ], 0 ≤ i, j ≤ N,

[Ai , [Aj , Ak ]], 0 ≤ i, j, k ≤ N, ...

at point x0 is Rm . Then, for every t > 0, the law of Xt is absolutely continuous with respect to the Lebesgue measure. Let us complete the proof of the theorem. Just by continuity of trajectories, δ we have limε→0 P (τX ≤ ε) = 0. Hence 0 lim P (ϕ[ε,T ] (X0 ) ∈ Γ c ) = 1.

ε→0

The family of events (ϕ[ n1 ,T ] (X0 ) ∈ Γ c ) is decreasing in n, hence P (ϕ[ n1 ,T ] (X0 ) ∈ Γ c ) is also decreasing. This implies P (ϕ[ε,T ] (X0 ) ∈ Γ c ) = 1 for every ε giving P (ϕ[0,T ] (X0 ) ∈ Γ c ) = 1.

5.3 Singularities of Stochastic Schr¨ odinger Equation The nonlinear Schr¨ odinger (NLS) equation ∂u = iΔu + i |u|2σ u, ∂t

u|t=0 = u0

in Rd or a bounded domain in the space variable, is an interesting example of PDE where, for some values of σ, certain solutions develop singularities. The unknown u (t, x) is complex valued. In the space H 1 Rd , the equation is well posed for σd < 2. However, 2 if d > 2), if u0 ∈ H 1 Rd , there is a unique when σd ≥ 2 (and σ < d−2 1 d local solution d in H R , but it blows-up in finite time if H (u0 ) < 0 and 2 xu0 ∈ L R , where the energy H (u) is defined as H (u) =

1 2

2

Rd

∇u (x) dx −

1 2 (σ + 1)

2(σ+1)

Rd

|u (x)|

dx.

150


The papers by de Bouard and Debussche [69–72], Debussche and Di Menza [74, 75], Barton-Smith et al. [29], investigate the effect of noise on this equation. Two kind of noises have been considered: additive noise and bilinear multiplicative one of Stratonovich type, in the form iεdW 2σ du = iΔu + i |u| u dt + iεu ◦ dW where W is a real valued infinite dimensional (or space-dependent) Brownian motion (possibly delta-correlated in space). This bilinear multiplicative noise has some feature similar to the one considered above for transport type equations, but it is not the same. Notice that u (t, x) takes values in C, where the inner product ξ, ηC := Re (ξη) is used. Then, formally speaking, iu ◦ dW, uC = 0. This means that the noise iu ◦ dW acts as a fast random rotation in C, somewhat similarly to a noise of the form σ (x) · ∇u ◦ dW when div σ = 0, which is a random rotation with respect to the L2 -scalar product. The analogy is admittedly weak and requires more understanding. 2 if d > 2) and additive noise, Consider the case σd ≥ 2 (and σ < d−2 with a certain trace class assumption on the covariance. Local (on a random time interval) well posedness in H 1 Rd continues to hold. But, under a suitablenon degeneracy assumption of the noise, if u0 ∈ H 1 Rd satisfies xu0 ∈ L2 Rd , then for every t > 0 the probability of blow-up before time t is positive. Remarkable is that the result holds for all initial condition, without the assumption H (u0 ) < 0. The intuitive reason is that, with positive probability, the non degenerate noise moves the system in the region H < 0, where blow-up starts and go on if we turn off the noise. A similar result is true for the multiplicative noise case (with important differences in the proof), in the regime σd > 2 (with other conditions) and suitable nuclear covariance of the noise. There are also results for σd = 2. Summarizing, in the supercritical cases σd > 2 blow-up occurs for “any” initial condition, and also in the critical case σd = 2 for additive noise. These and other results can be found in de Bouard and Debussche [69–72]. In the language (improperly) used in this Chapter, this is a very “negative” result (not in the sense that it is not interesting!): noise collaborates to blowup. The blow-up region of the deterministic problem is fat (includes the region defined by the constraints H < 0 and xu0 ∈ L2 Rd ). It is not the same as in certain examples seen above, where pathologies happen for the deterministic problem only for a thin class of initial data. Noise moves away from such special pathological conditions and restores well posedness. On the contrary,

5.3 Singularities of Stochastic Schr¨ odinger Equation

151

in the case of a fat bad region, the “transitivity” due to non-degenerate noise yields blow-up even if we start in the good region. Numerical simulations show that, in the same regimes where the blow-up results above have been proved, solutions blow-up with probability one (at due random time). Moreover, comparing blow-up time of the deterministic and stochastic case, a remarkable numerical observation is that multiplicative noise (regular in space) delays blow-up. This is the origin of the next step of numerical investigation. For these results see Debussche and Di Menza [74, 75], Barton-Smith et al. [29]. If the noise W is delta-correlated in space, the NLS equation with multiplicative noise cannot be studied rigorously, with known methods; the force is too singular. However, numerical simulations have been performed, with a very accurate adaptive procedure of space grid refinement, which improves both the accuracy of observation of the blow-up region and the point-by-point independence of the noise input. The result is surprising: blow-up disappears. Certain solutions develop the beginning of a singularity but after a while the pronounced pre-singular shape decreases. This is an extremely intriguing a “positive” numerical result. More recently, Debussche and Tsutsumi [77] have studied the following stochastic nonlinear Schrödinger (NLS) equation with quintic nonlinearity and white noise dispersion on the real line idu + Δu ◦ dβ + |u|4 u dt = 0 u|t=0 = u0 with x ∈ R, t > 0. The process β is a real valued Brownian motion. The product is a Stratonovich product and the Itˆ o form is i idu + Δ2 u dt + Δu dβ + |u|4 u dt = 0. 2 It seems as if the principal part of (2.2) were the double Laplacian, which does not appear to be degenerate. But this is not true. Indeed, the explicit formula of the fundamental solution for the linear equation shows the high degeneracy of the principal part:

ut (x) =

1 d/2

(4iπ (β(t) − β(s)))

Rd

exp i

|x − y|2 4(β(t) − β(s))

us (y)dy

The main tool to solve NLS type equation is to use Strichartz estimates. They give ultra integrability properties for the linear equation. Is not obvious whether the Strichartz type estimate holds or not unlike the deterministic case. Indeed the dispersion coefficient is highly degenerate and has infinitely many zeros.

152


Strichartz estimates for white noise dispersion were first obtained by de Bouard and Debussche [73] but only subcritical nonlinearity could be treated. In space dimension one, these can be improved and are in fact more powerful than in the deterministic case. It is possible to prove global well posedness in L2 (R) or H 1 (R) for any initial data. Thus the white noise dispersion restores well posedness. As we have seen several times in past sections, regularization is obtained by a multiplicative noise in Stratonovich form. The common root of these examples (if any) is, however, still unclear.

5.4 Additive Noise in 3D Navier–Stokes Equations The long time project where the topics of this note take their origin is the attempt to improve the theory of fluid dynamics by means of random perturbations in the governing fluid equations. One of the millennium prize problems (see the presentation of Fefferman [91]) is concerned with the 3D Navier–Stokes equations in a domain D ⊂ R3 ∂u + u · ∇u + ∇p = ν u + f ∂t divu = 0, u|t=0 = u0

(5.16)

which are a rather good model for incompressible, viscous, constant-density fluids; here u : D × [0, ∞) → R3 is the velocity field of the fluid, p : D × [0, ∞) → R is the pressure field, both unknown, f : D × [0, ∞) → R3 is the body force and ν > 0 is the kinematic viscosity. Without details, let us just recall that, chosen suitable boundary conditions, if u0 ∈ H, where H is a suitable space of divergence free square integrable field, and f satisfies natural integrability conditions, there is at least one weak Leray–Hopf solution (u, p), a notion which includes the property

2

sup t∈[0,T ]

D

|u (x, t)| dx + 0

T

2

D

∇u (x, t) dxdt < ∞

and the energy in equality 1 2

t 2 2 |u (x, t)| dx + ν ∇u dxds 0 D D t 1 2 ≤ |u (x, 0)| dx + u · f dxds. 2 D 0 D

Additional properties are known at least for simple domains, like

5.4 Additive Noise in 3D Navier–Stokes Equations

T

153

0

D

|p (x, t)|3/2 dxdt < ∞

sup t∈[0,T ]

D

∇u (x, t) dx < ∞

but they do not improve sufficiently the theory. Uniqueness of these weak solutions is open. If the initial condition is sufficiently regular (for instance in the Sobolev class W α,2 for some α > 1/2), then a unique local solution exists with the same space-regularity; but globality, or on the contrary the emergence of singularities, is open. A natural question then is whether a suitable stochastic perturbations, (bilinear) multiplicative or additive, could improve the theory. The equations take the form du + (u · ∇u + ∇p − ν u − f ) dt = bk · ∇u ◦ dW k + σj ej dβ j k

j

where bk : D × [0, ∞) → R3 are given, {ej } is a complete orthonormal system of H, σj are real numbers, W k , β j are independent one-dimensional Brownian motions on some probability space (Ω, F, P ). Existence of weak Leray–Hopf solutions of the martingale problem for these equations have been proved under various generality, see for instance Flandoli and Gatarek [98], Mikulevicius and Rozovskii [161]; the 2-dimensional case is well posed also in the stochastic case, see Schmalfuss [186] for an interesting proof of uniqueness. Uniqueness of weak solutions in dimension 3 remains an open problem as well as the existence or absence of singularities, but a few progresses have been made, that we are going to summarize. Most of the attempts until now have been concerned with a non degenerate additive noise. But tools like those of Chap. 2 are restricted to much simpler nonlinearities. Let us mention, nevertheless, two progresses made on the additive noise case.

5.4.1 About Singularities A time-space point (t, x) ∈ D is called regular when the solution u is bounded on some neighborhood of (t, x). By booth strap arguments, usually boundedness implies more regularity, locally. A non-regular point is called singular. Denote by S the set of all singular points, S ⊂ (0, T ) × D. A famous result of Caffarelli et al. [43], following previous results of Scheffer ([183] and previous works), states that S has Hausdorff dimension at most one, and the one-dimensional Hausdorff measure of S is zero. The result is true for the so called suitable weak solutions.

154


A generalization of this result to the stochastic Navier–Stokes equations with additive noise by Flandoli and Romito [105] allowed to prove the following intriguing fact (later on a simplified proof was given by Da Prato and Debussche [61], without Caffarelli–Kohn–Nirenberg theory). Denote by St (ω) the set of all singular points at time t (namely of the form (t, x) for some x ∈ D) of the function u (t, x, ω), suitable weak solution of the stochastic Navier–Stokes equations. The result is that, if u is time stationary, we have P (St = ∅) = 1 for every given t ≥ 0. At any given time, there is no chance to observe singularities. Unfortunately this does not mean that there are no singularities: for a.e. ω, the function (t, x) → u (t, x, ω) may have singular points, but it cannot happen that for a positive measure set of ω they appear at the same time t. The singular time could be like an exponential random variable, to have an idea. This result was already known in the deterministic case, if we interpret “time-stationary” in the sense of “constant in time”. Solutions of Navier– Stokes equations which do not depend on time cannot have singularities, even in dimension 4. But in the stochastic case, a time-stationary solution can be extremely complex, chaotic in a sense. Thus the stochastic result applies to very non-trivial solutions. Moreover, when the noise is non-degenerate, if μ = μt denotes the law of u (t, ·) on H (it is independent of t), then μ has full support; thus u has wild variability in time.

5.4.2 Markov Selections and Strong Feller Property Consider again the stochastic Navier–Stokes equations with additive noise. The main break-through on them is due to Da Prato and Debussche [61], followed by contributions of Debussche and Odasso [76], Odasso [170], Da Prato and Debussche [63], Flandoli and Romito [106, 107], Romito [180], Goldys et al. [117], Blömker et al. [35]. They have obtained a direct solution of the Kolmogorov equation (we have seen in Chap. 2 its importance for uniqueness). The solution has several differentiability properties. Unfortunately the derivatives are not sufficiently regular (as functions on infinite dimensional spaces) to apply known arguments for proving uniqueness. Nevertheless, a striking positive consequence of that theory and subsequent development (see in particular Flandoli and Romito [107]) is the following theorem. Theorem 5.3. There exists Markov selections in H associated to the stochastic Navier–Stokes equations with additive noise. Moreover, under proper non-degeneracy conditions on the noise, all of them are strong Feller in a suitable space W ⊂ H.

5.5 Conclusions and Open Problems

155

The statement has been given in a necessarily vague form; see Da Prato and Debussche [61] and Flandoli and Romito [107] for details. The continuous dependence on initial conditions in W, of the elements of a Markov selection, is a property without anything similar in the deterministic case, where lack of (proof of) uniqueness goes parallel to lack of (proof of) any continuous dependence. Using these tools one can also construct new criteria of uniqueness; one of the most appealing ones is given by Romito [180], which can be summarized, roughly speaking, as follows: Theorem 5.4. Assume the non-degeneracy conditions on the noise of the previous theorem, which in particular imply existence and uniqueness of an invariant measure, for each Markov selection. If these invariant measures, among the Markov selections, coincide, then uniqueness of weak solutions holds for the stochastic Navier–Stokes equations. Whether the unique invariant measures of different Markov selections coincide or not, it is an open problem. However, Romito [180] proves that these invariant measures are equivalent. The transition probabilities of the different Markov selections are also equivalent, see Flandoli and Romito [106].

5.5 Conclusions and Open Problems 5.5.1 SPDEs: The Effects of Noise These notes are devoted to a very particular research direction in the area of SPDEs. But clearly this research would not be possible without the preliminary and contemporary wide development of the general theory of SPDEs and in particular the part related to fluid dynamics. A number of books and fundamental works exist on these topics: see Albeverio and Ferrario [4], Blömker [34], Cerrai [46], Chow [50], Da Prato [59], Da Prato and Zabczyk [65–67], Flandoli [95], Hairer [125], Holden et al. [128], Kotelenez [134], Krylov and Rozovskii [139], Kuksin [140], Kuksin and Shirikyan [142], Malliavin [153], Métivier [160], Mueller [164], Nualart [169], Pardoux [171], Prévôt and Röckner [176], Rozovskii [181], Visik and Fursikov [194], Walsh [196] and others. The philosophy of these notes, namely that noise may improve results on PDEs, applies to other questions than uniqueness and singularities. Let us mention a few of them. Randomizing initial conditions by an invariant measure may allow to study very difficult equations; see Albeverio and Cruzeiro [2] and several related works with application to Euler equations; somewhat related is renormalization of 2D Navier–Stokes equations and their analysis under space-time white noise, see Albeverio and Ferrario [3, 4], Brze´zniak and Ferrario [39], Da Prato and Debussche [60]. Noise greatly

156


simplifies ergodic theory of PDEs, see for instance Da Prato and Zabczyk [66], Flandoli and Maslowski [104], but taking advantage of cascade mechanisms of fluid equations and use just a little amount of noise is not easy; the literature in this difficult direction (degenerate noise) is wide, but see in particular Hairer and Mattingly [126, 127], Kuksin and Shirikyan [142] and related facts about Malliavin calculus and control theory in Mattingly and Pardoux [157], Shirikyan [185]; support theorems and smoothness of the law of solutions are also related, see for instance Millet and Sanz-Solé [162, 163]. Averging theory is much improved by noise, see Kuksin and Piatnitski [141], Cerrai and Friedlin [47]. The diffusion of passive scalars and passive vectors is better investigated when the driving velocity field is random, see for instance Brze´zniak and Neklyudov [40], Celani et al. [44], Cranston et al. [54], E and Vanden-Eijnden [84], Le Jan and Raymond [148], Lototskii and Rozovskii [149] and more classical works quoted in these references. Randomness can be included in fluid dynamic models in other ways, see Cruzeiro et al. [56], Bessaih et al. [32], Flandoli et al. [103] and its effect is only partially understood. Special properties related to noise input are investigated, see for instance Dalang and Walsh [57], Dalang et al. [58], Mueller and Tribe [167]. This list of problems and references is extremely incomplete and its purpose is only to mention very interesting questions not treated here.

5.5.2 What We Have Learned on the Effect of Noise on Uniqueness Two main questions have driven in the last 10 years or more the researches reported here: initially and above all, try to prove the well posedness of the 3D Navier–Stokes equations under suitable random perturbations; then, more realistically, try to identify some class or just some example of PDE that is regularized by noise, and extract some general principle if possible. The state of the art about stochastic 3D Navier–Stokes equations has been roughly summarized in Sect. 5.4: no final answer, a few intermediate results, some technique and some criteria developed for future research. The problem is open. About the more realistic purpose of compiling a list of easier examples and detect some general idea, maybe we can now write some provisory conclusion. First, what kind of noise is expected to regularize? It is natural to start from additive non degenerate noise, because of its success to regularize ODEs. This is not wrong, but we have learned that it is also promising to try with another noise, bilinear multiplicative, thus strongly degenerate in a sense. The latter at present has been more successful for problems of fluid dynamic nature. The area of potential success of these two noises (additive and bilinear multiplicative) are different. To give a meaning to the next sentences assume


157

we deal with an evolutionary PDE with first order space derivatives (typically a transport term) and maybe a second order elliptic operator, let us say a Laplacian for simplicity; other models, like those of Chap. 3, are not PDEs, but there are objects in the equation which correspond to the differential operators just described. Similarly, an analogy can be found with the two examples of multiplicative noise of Sect. 5.3. The difference is: viscous or inviscid. If there is a Laplacian, namely a viscous term of the form νΔu with ν > 0, the most natural attempt is still additive noise, although at present the examples are limited. Bilinear multiplicative noise of the kind used successfully in Chaps. 3–5, which involved first order space derivatives, is useless when there is already a Laplacian in the equation: as explained at the beginning of Chap. 4, the reason of its success is that it works as a Laplacian at the level of certain mean values of the solution. If the equation contains a Laplacian from the beginning, there is only an improvement of the viscosity, in a sense, but not a qualitative change. If the PDE is inviscid (a sort of degenerate parabolic problem), then some kind of parabolicity is restored, for mean values, by Stratonovich multiplicative noise of the form above. In a number of examples we have shown that this restores uniqueness and prevents blow-up. On the contrary, additive noise cannot be studied, for inviscid problems, with the available techniques. Mathematical fluid dynamics is not only 3D Navier–Stokes equations. A large portion of the former is devoted to inviscid problems, first of all Euler equations. They present a lot of phenomena of non uniqueness and potential singularities (blow-up is not proved for Euler equations themselves but for related models, like the evolution of vortex sheets). Certainly a research direction that is worth to be tried is the investigation of the effect of bilinear multiplicative noise on Euler equations and the related models. Examples of results on related models (not directly Euler equations) are the results of Chap. 3 on the dyadic model and the vortex point problem in Chap. 5. More difficult is to claim that the results on the transport equation are relevant for Euler equations, but it is not excluded. Maybe they could be more useful for other degenerate parabolic equations. Finally, for the 3D Navier–Stokes equations, which are viscous, what should we expect? Still additive noise is the only reasonable one to try, for the reasons just described, but the distance between the framework of Chap. 2 and this example is large. The assumptions to apply Girsanov transform or to prove strong uniqueness are certainly too restrictive. The approach to weak uniqueness by Kolmogorov equation or Fokker–Planck equation looks more promising but still restricted to much easier nonlinearities. Maybe the ideas of Chap. 2 could be explored not immediately for uniqueness but to understand something of the following side but appealing problem. There exist integral (in time) quantities, associated to the velocity field u (t, x) or the vorticity field ω (t, x) of a fluid, which contain information of interest and sometimes are related to uniqueness or blow up. One of them

T∗ is 0 ω (t, ·)∞ dt: if finite, no blow-up of regular solutions may appear

158


until time T ∗ (Beale–Kato–Majda criterium, see Majda and Bertozzi [152]). However, let us exclude those which are norm-like, positive definite, since the following argument looks less promising for them. Another interesting quantity is the vortex stretching over [0, T ]

T

T

ξ (t, xt ) S (t, xt ) ξ (t, xt ) dt 0 ω(t,x) is the direction of vorticity, S (t, x) is the symmetwhere ξ (t, x) = |ω(t,x)| ric part of Du (t, x), thus the deformation tensor, and xt denoted either a fixed space point x or a fluid particle trajectory. This quantity captures the amount of stretching of the vortex field (at a given point or along the motion) due to the space deformation produced by the fluid motion. It is not impossible to hope that quantities like this one, very difficult at present in the deterministic case (but see Chae [48] for a review of several results), may be approached by the occupation measure technique of Chap. 2, in the stochastic case. In other words, if general functional analysis estimates (or geometric or harmonic analysis arguments) do not allow to control sufficiently well such quantities, the regularity of wide fluctuations of ξ (t, x) and S (t, x) could be such that statistically only a small portion of time there is too large stretching. Maybe an excessive stretching leading to blow-up requires a great degree of geometrical organization, which is broken by fluctuations. A first attempt should be made in the simpler case of Gaussian fields, solutions of some linear SPDE.

5.5.3 Other Open Questions Several open problems are related to these lectures. We have always used noise which is white in time. What happens for other noises, like the derivative of fractional Brownian motion? And, perhaps more interesting, for smooth approximations of white noise, close to the limit? For Levy noise, see Priola [174]. Even more interesting for general mathematics (namely not only for stochastic analysis), would be to investigate regularization produced by single deterministic curves in place of noise, in the spirit of the last remarks to Sect. 1.3.3; curves which are sufficiently rough. Maybe it is possible to apply occupation-measure ideas to uniqueness or selection directly for ODEs. Another direction is the generality of the (white) noise. What happens for degenerate additive noise, in cases where non degenerate one regularizes? Similarly, the regularization performed by the special multiplicative noise of Chaps. 3–5 persists if we modify the noise up to some extent? This group of questions is not in the spirit of these lectures, where we cared to find at least one suitable noise that regularizes (a sort of conceptual information on


159

the relation between random perturbations and difficult problems of PDEs). Nevertheless, to claim that physical models are better behaved when noise is incorporate, it is relevant to know that this happens not only for one special noise but for a sufficiently large class. Related to the last issue, it is necessary to point out that a very advanced literature on uniqueness for SDEs and SPDEs has the irregularity or degeneracy of the diffusion coefficient as its main objective. The motivations come from applications in Physics, Biology, Finance, and internal to stochastic analysis. It is impossible to list here the contributions in this deep direction. Let us mention only Mytnik [165], Mueller and Tribe [168], Mytnik et al. [166], Burdzy et al. [42] and references therein. Apparently, at present, the methods discussed in these notes are disjoint from those useful for such problems; a more unified picture could be interesting.

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List of Participants

40th Probability Summer School, Saint-Flour, France July 4–17, 2010 Lecturers Franco FLANDOLI

Universit` a di Pisa, Italy

Giambattista GIACOMIN

Université Paris Diderot, France

Takashi KUMAGAI

Kyoto University, Japan

Participants Sergio ALMADA

Georgia Inst. Technology, Atlanta, USA

Marek ARENDARCZYK

Univ. Wroclaw, Poland

David BARBATO

Univ. Padova, Italy

David BELIUS

ETH Zurich, Switzerland

Pierre BERTIN

Univ. Paris 6 et 7, F

Luigi Amedeo BIANCHI

Scuola Normale Superiore, Pisa, Italy

Thomas BOUILLOC

Univ. Nice Sophia Antipolis, F

Omar BOUKHADRA

U. Provence, F & U. Constantine, Algeria

Charles-Edouard BREHIER

ENS Cachan, Antenne Bretagne, Rennes, F

Elisabetta CANDELLERO

Graz Univ. Technology, Austria

Francesco CARAVENNA

Univ. Padova, Italy

Reda CHHAIBI

Univ. Pierre et Marie Curie, Paris, F

Mirko D’OVIDIO

Sapienza Univ. Roma, Italy

Latifa DEBBI

Univ. Setif, Algeria

Fran¸cois DELARUE


Francisco DELGADO

Univ. Barcelona, Spain

Aurélien DEYA

Univ. Henri Poincaré, Nancy, F

Hacène DJELLOUT

Univ. Blaise Pascal, Clermont-Ferrand, F

Aurélien EBERHARDT

Univ. Strasbourg, F

172


Fran¸cois EZANNO

Univ. de Provence, Marseille, F

Ennio FEDRIZZI

Univ. Paris Diderot, F

Matthieu FELSINGER

Univ. Bielefeld, Germany

Robert FITZNER

Eindhoven Univ. Technology, NL

Elena ISSOGLIO

Friedrich Schiller Univ., Jena, Germany

Shuai JING

Univ. Bretagne Occidentale, Brest, F

Yasmina KHELOUFI

Univ. Setif, Algeria

Konrad KOLESKO

Univ. Wroclaw, Poland

Noemi KURT

TU Berlin, Germany

Kazumasa KUWADA

Ochanomizu Univ., Tokyo, Japan

Mateusz KWASNICKI

Wroclaw Univ. Technology, Poland

Clément LAURENT


Qian LIN

Univ. Bretagne Occidentale, Brest, F

Arnaud LIONNET

Univ. Oxford, UK

J.-A. LOPEZ-MIMBELA

CIMAT, Guanajuato, Mexico

Eric LUC ¸ ON


Camille MALE

ENS Lyon, F

Mario MAURELLI

Univ. Pisa, Italy

Francesco MORANDIN

Univ. Parma, Italy

Jean-Christophe MOURRAT


Mikhail NEKLYUDOV

Univ. York, UK

Eyal NEUMANN

Technion Inst. Technology, Haifa, Israel

Harald OBERHAUSER

TU Berlin, Germany

Jean PICARD


K. PIETRUSKA-PALUBA

Univ. Warsaw, Poland

Marco ROMITO

Univ. Firenze, Italy

´ Univ. Blaise Pascal, Clermont-Ferrand, F Erwan SAINT LOUBERT BIE Martin SAUER

TU Darmstadt, Germany

Georg SCHOECHTEL

TU Darmstadt, Germany

Laurent SERLET


Yuhao SHEN


Mykhaylo SHKOLNIKOV

Stanford Univ., USA

Damien SIMON


Julien SOHIER

Univ. Paris Diderot, F


173

Philippe SOSOE

Princeton Univ., USA

Andrzej STOS


E. TODOROVA KOLKOVSKA

CIMAT, Guanajuato, Mexico

Dario VINCENZI


Jing WANG

Purdue Univ., West Lafayette, USA

Frédérique WATBLED

Univ. Bretagne-Sud, Vannes, F

Hendrik WEBER

Univ. Warwick, UK

Lihu XU

Eindhoven Univ. Technology, NL

Ramon XULVI-BRUNET

Harvard Univ., Cambridge, MA, USA

Danyu YANG

Oxford Univ., UK

Lorenzo ZAMBOTTI


174


Programme of the School

Main lectures Franco Flandoli

Random perturbations of PDEs and fluid dynamic models

Giambattista Giacomin

Disorder and critical phenomena through basic probability models

Takashi Kumagai

Random walks on disordered media and their scaling limits

Short lectures Sergio Almada Monter

Scaling limit for the diffusion exit problem

Marek Arendarczyk

Asymptotics of supremum distribution of a Gaussian process over a random time interval

David Barbato

Girsanov transform of SPDEs with multiplicative noise

David Belius

Cover levels and random interlacements

Pierre Bertin

Linear stochastic evolutions

Omar Boukhadra

Return probabilities of random walks among random conductances

Charles-Edouard Brehier

Averaging for some SPDEs: strong and weak order

Elisabetta Candellero

Phase transitions for random walks, asymptotics on free products of groups

Latifa Debbi

Numerical approximation of the solutions of fractional stochastic Burgers equations

Fran¸cois Delarue

Density estimates for a random noise propagating through a chain of differential equations

Fran¸cois Ezanno

Markov models of crystal growth

Ennio Fedrizzi

Uniqueness and flow theorems for solutions of SDEs with low regularity of the drift

Robert Fitzner

Lace expansion for dummies

Elena Issoglio

On the solution of a stochastic PDE with fractal noise


175

Shuai Jing

Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in (0, 1/2)

Kazumasa Kuwada

Duality results on gradient estimates and Wasserstein controls

Mateusz Kwa´snicki

On exit distribution of Markov processes

Clément Laurent

Large deviations for self-intersection local times of stable random walks

José Alfredo López-Mimbela

Finite-time blowup and existence of global positive solutions of a semi-linear SPDE

Eric Lu¸con

Quenched and averaged convergence and fluctuations of the empirical measure in the Kuramoto model

Camille Male

Random matrix theory and free probability

Francesco Morandin

Anomalous dissipation in a birth and death process and in two stochastic dyadic models

Jean-Christophe Mourrat

Subdiffusive scaling limit for the random walk among random traps

Misha Neklyudov

Ergodicity for infinite particle systems with locally conserved quantities

Eyal Neumann

Sample path properties of Volterra processes

Harald Oberhauser

Towards a theory of rough viscosity solutions

Katarzyna Pietruska-Paluba

Function spaces arising in connection with stochastic processes on metric spaces

Marco Romito

Non-uniqueness and uncertainty: untold stories about the Navier-Stokes equation

Mykhaylo Shkolnikov

On competing particle systems and their applications

Julien Sohier

Scaling limits of a stripe wetting model

Ekaterina Todorova Kolkovska The expected Gerber-Shiu penalty function for a perturbed classical risk process Dario Vincenzi

Dispersion and collapse in turbulent transport

Hendrik Weber

Tightness for a stochastic Allen-Cahn equation

176


Lihu Xu

Harnack and log-Harnack inequalities of SPDEs

Danyu Yang

Rough path, p-variation of a path from its lifted rough path

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´ Vol. 1974: F. den Hollander, Random Polymers: Ecole ´ e de Probabilités de Saint-Flour XXXVII – 2007 d’Et´ (2009) Vol. 1975: J.C. Rohde, Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication (2009) Vol. 1976: N. Ginoux, The Dirac Spectrum (2009) Vol. 1977: M.J. Gursky, E. Lanconelli, A. Malchiodi, G. Tarantello, X.-J. Wang, P.C. Yang, Geometric Analysis and PDEs. Cetraro, Italy 2001. Editors: A. Ambrosetti, S.-Y.A. Chang, A. Malchiodi (2009) Vol. 1978: M. Qian, J.-S. Xie, S. Zhu, Smooth Ergodic Theory for Endomorphisms (2009) ´ Vol. 1979: C. Donati-Martin, M. Emery, A. Rouault, ´ C. Stricker (Eds.), Seminaire de Probablitiés XLII (2009) Vol. 1980: P. Graczyk, A. Stos (Eds.), Potential Analysis of Stable Processes and its Extensions (2009) Vol. 1981: M. Chlouveraki, Blocks and Families for Cyclotomic Hecke Algebras (2009) Vol. 1982: N. Privault, Stochastic Analysis in Discrete and Continuous Settings. With Normal Martingales (2009) Vol. 1983: H. Ammari (Ed.), Mathematical Modeling in Biomedical Imaging I. Electrical and Ultrasound Tomographies, Anomaly Detection, and Brain Imaging (2009) Vol. 1984: V. Caselles, P. Monasse, Geometric Description of Images as Topographic Maps (2010) Vol. 1985: T. Linß, Layer-Adapted Meshes for ReactionConvection-Diffusion Problems (2010) Vol. 1986: J.-P. Antoine, C. Trapani, Partial Inner Product Spaces. Theory and Applications (2009) Vol. 1987: J.-P. Brasselet, J. Seade, T. Suwa, Vector Fields on Singular Varieties (2010) Vol. 1988: M. Broué, Introduction to Complex Reflection Groups and Their Braid Groups (2010) Vol. 1989: I.M. Bomze, V. Demyanov, Nonlinear Optimization. Cetraro, Italy 2007. Editors: G. di Pillo, F. Schoen (2010) Vol. 1990: S. Bouc, Biset Functors for Finite Groups (2010) Vol. 1991: F. Gazzola, H.-C. Grunau, G. Sweers, Polyharmonic Boundary Value Problems (2010) Vol. 1992: A. Parmeggiani, Spectral Theory of NonCommutative Harmonic Oscillators: An Introduction (2010) Vol. 1993: P. Dodos, Banach Spaces and Descriptive Set Theory: Selected Topics (2010) Vol. 1994: A. Baricz, Generalized Bessel Functions of the First Kind (2010) Vol. 1995: A.Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls (2010) Vol. 1996: T. Lorenz, Mutational Analysis. A Joint Framework for Cauchy Problems In and Beyond Vector Spaces (2010) Vol. 1997: M. Banagl, Intersection Spaces, Spatial Homology Truncation, and String Theory (2010) Vol. 1998: M. Abate, E. Bedford, M. Brunella, T.-C. Dinh, D. Schleicher, N. Sibony, Holomorphic Dynamical Systems. Cetraro, Italy 2008. Editors: G. Gentili, J. Guenot, G. Patrizio (2010) Vol. 1999: H. Schoutens, The Use of Ultraproducts in Commutative Algebra (2010) Vol. 2000: H. Yserentant, Regularity and Approximability of Electronic Wave Functions (2010) Vol. 2001: T. Duquesne, O. Reichmann, K.-i. Sato, C. Schwab, Lévy Matters I. Editors: O.E. BarndorffNielson, J. Bertoin, J. Jacod, C. Klüppelberg (2010)

Vol. 2002: C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems (2010) Vol. 2003: A. Cousin, S. Crépey, O. Guéant, D. Hobson, M. Jeanblanc, J.-M. Lasry, J.-P. Laurent, P.-L. Lions, P. Tankov, Paris-Princeton Lectures on Mathematical Finance 2010. Editors: R.A. Carmona, E. Cinlar, I. Ekeland, E. Jouini, J.A. Scheinkman, N. Touzi (2010) Vol. 2004: K. Diethelm, The Analysis of Fractional Differential Equations (2010) Vol. 2005: W. Yuan, W. Sickel, D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel (2011) Vol. 2006: C. Donati-Martin, A. Lejay, W. Rouault (Eds.), Séminaire de Probabilités XLIII (2011) Vol. 2007: E. Bujalance, F.J. Cirre, J.M. Gamboa, G. Gromadzki, Symmetries of Compact Riemann Surfaces (2010) Vol. 2008: P.F. Baum, G. Cortiñas, R. Meyer, R. SnchezGarca, M. Schlichting, B. Toën, Topics in Algebraic and Topological K-Theory. Editor: G. Cortiñas (2011) Vol. 2009: J.-L. Colliot-Thélène, P.S. Dyer, P. Vojta, Arithmetic Geometry. Cetraro, Italy 2007. Editors: P. Corvaja, C. Gasbarri (2011) Vol. 2010: A. Farina, A. Klar, R.M.M. Mattheij, A. Mikelić, N. Siedow, Mathematical Models in the Manufacturing of Glass. Cetraro, Italy 2008. Editor: A. Fasano (2011) Vol. 2011: B. Andrews, C. Hopper, The Ricci Flow in Riemannian Geometry (2011) Vol. 2012: A. Etheridge, Some Mathematical Models ´ ´ e de Probabilités de from Population Genetics. Ecole d’Et´ Saint-Flour XXXIX – 2009 (2011) Vol. 2013: A. Bobenko, C.Klein, Computational Approach to Riemann Surfaces (2011) Vol. 2014: M. Audin, Fatou, Julia, Montel. The Great Prize of Mathematical Sciences of 1918, and beyond (2011) Vol. 2015: F. Flandoli, Random Perturbation of PDEs and ´ ´ e de Probabilités de d’Et´ Fluid Dynamic Models. Ecole Saint-Flour XL – 2010 (2011)

Recent Reprints and New Editions Vol. 1629: J.D. Moore, Lectures on Seiberg-Witten Invariants. 1997 – 2nd edition (2001) Vol. 1638: P. Vanhaecke, Integrable Systems in the realm of Algebraic Geometry. 1996 – 2nd edition (2001) Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999 – Corr. 3rd printing (2007) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 – 2nd corr. and augmented edition (2007) Vol. 1693: S. Simons, From Hahn-Banach to Monotonicity (Minimax and Monotonicity 1998) – 2nd exp. edition (2008) Vol. 470: R.E. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. With a preface by D. Ruelle. Edited by J.-R. Chazottes. 1975 – 2nd rev. edition (2008) Vol. 523: S.A. Albeverio, R.J. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integral. 1976 – 2nd corr. and enlarged edition (2008) Vol. 1764: A. Cannas da Silva, Lectures on Symplectic Geometry 2001 – Corr. 2nd printing (2008)

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