Table Of ContentAdvanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matematica
Managing Editor:
Manuel Castellet
Giuseppe Da Prato
Kolmogorov Equations
for Stochastic PDEs
Springer Basel AG
Author:
Giuseppe Da Prato
Dipartimento di Matematica
Scuola Normale Superiore
Piazza dei Cava lieri 7
56126 Pisa
Italy
[email protected]
2000 Mathematical Subject Classification 35K57, 35030, 35Q53, 37A30, 60H15
A CIP catalogue record for this book is available from the
library of Congress, Washington D.C., USA
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ISBN 978-3-7643-7216-3 ISBN 978-3-0348-7909-5 (eBook)
DOI 10.1007/978-3-0348-7909-5
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© 2004Springer Basel AG
Originally published by Birkhauser Verlag Basel, Switzerland in 2004
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Printed on acid-free paper produced from chlorine-free pulp. TCFoo
ISBN 978-3-7643-7216-3
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Contents
Preface ix
1 Introduction and Preliminaries 1
1.1 Introduction........ 1
1. 2 Preliminaries . . . . . . . 5
1.2.1 Some functional spaces. 5
1.2.2 Exponential functions . 7
1.2.3 Gaussian measures . . . 8
1.2.4 Sobolev spaces Wl,2(H, p,) and W2,2(H, p,) 11
1.2.5 Markov semigroups. . . . . . . . . . . . . . 13
2 Stochastic Perturbations of Linear Equations 15
2.1 Introduction ........ . 15
2.2 The stochastic convolution. . . . . . 17
2.2.1 Continuity in time . . . . . . 20
2.2.2 Continuity in space and time 22
2.2.3 The law of the stochastic convolution 24
2.3 The Ornstein-Uhlenbeck semigroup Rt . 26
2.3.1 General properties . . . . . . . . 26
2.3.2 The infinitesimal generator of Rt 29
2.4 The case when Rt is strong Feller . . . . 32
2.5 Asymptotic behaviour of solutions, invariant measures 35
2.6 The transition semigroup in LP(H, p,) . 37
2.6.1 Symmetry of Rt ........ 42
2.7 Poincare and log-Sobolev inequalities. 45
2.7.1 Hypercontractivity of Rt . . . . 47
2.8 Some complements . . . . . . . . . . . 49
2.8.1 Further regularity results when Rt is strong Feller 49
2.8.2 The case when A and C commute ......... 49
2.8.3 The Ornstein-Uhlenbeck semigroup in the space of functions
of quadratic growth ...................... 51
vi Contents
3 Stochastic Differential Equations with Lipschitz Nonlinearities 59
3.1 Introduction and setting of the problem . . . . . . . . . . 59
3.2 Existence, uniqueness and approximation ......... 61
3.2.1 Derivative of the solution with respect to the initial datum 64
3.3 The transition semigroup .. 66
3.3.1 Strong Feller property 68
3.3.2 Irreducibility . . . . . 70
3.4 Invariant measure v ..... 73
3.5 The transition semigroup in L2(H, v) . 79
3.6 The integration by parts formula and its consequences 84
3.6.1 The Sobolev space W1,2(H, v) ........ . 85
3.6.2 Poincare and log-Sobolev inequalities, spectral gap 88
3.7 Comparison of v with a Gaussian measure . 91
3.7.1 First method ... 91
3.7.2 Second method .. 93
3.7.3 The adjoint of K2 97
4 Reaction-Diffusion Equations 99
4.1 Introduction and setting of the problem 99
4.2 Solution of the stochastic differential equation. 102
4.3 Feller and strong Feller properties. 109
4.4 Irreducibility . . . . . . . . . . 111
4.5 Existence of invariant measure . 114
4.5.1 The dissipative case ... 114
4.5.2 The non-dissipative case . 115
4.6 The transition semigroup in L2(H, v) . 117
4.7 The integration by parts formula and its consequences 122
4.7.1 The Sobolev space W1,2(H, v) ........ . 122
4.7.2 Poincare and log-Sobolev inequalities, spectral gap 123
4.8 Comparison of v with a Gaussian measure . . . . . . . 125
4.9 Compactness of the embedding W1,2(H, v) C L2(H, v) 127
4.10 Gradient systems .................... . 129
5 The Stochastic Burgers Equation 131
5.1 Introduction and preliminaries ......... 131
5.2 Solution of the stochastic differential equation. 135
5.3 Estimates for the solutions. . . . . . . . . . . . 138
5.4 Estimates for the derivative of the solution w.r.t. the initial datum 141
5.5 Strong Feller property and irreducibility ..... . 143
5.6 Invariant measure v ................ . 146
5.6.1 Estimate of some integral with respect to v 147
5.7 Kolmogorov equation. . . . . . . . . . . . . . . . . 150
Contents vii
6 The Stochastic 2D Navier-Stokes Equation 155
6.1 Introduction and preliminaries ... . 155
6.1.1 The abstract setting ..... . 157
6.1.2 Basic properties of the nonlinear term 158
6.1.3 Sobolev embedding and interpolatory estimates. 160
6.2 Solution of the stochastic equation 161
6.3 Estimates for the solution . . . . . 164
6.4 Invariant measure l/ ....... . 166
6.4.1 Estimates of some integral. 167
6.5 Kolmogorov equation. . . . . . . . 168
Bibliography 173
Index 181
Preface
This book is devoted to some basic stochastic partial differential equations, in
particular reaction-diffusion equations, Burgers and Navier-Stokes equations per
turbed by noise.
Particular attention is paid to the corresponding Kolmogorov equations
which are elliptic or parabolic equations with infinitely many variables.
The aim of the book is to present the basic elements of stochastic PDEs in
a simple and self-contained way in order to cover the program of one year PhD
course both in Mathematics and in Physics.
The needed prerequisites are some basic knowledge of probability, functional
analysis (including fundamental properties of Gaussian measures) and partial dif
ferential equations.
This book is an expansion of a course given by the author in 1997 at the
"Center de Recerca Matematica" in Barcelona (see [30]), which I thank for the
warm hospitality.
I wish also to thank B. Goldys for reading the manuscript and making several
useful comments.
This work was also supported by the research program "Analisi e controllo di
equazioni di evoluzione deterministiche e stocastiche" from the Italian "Ministero
della Ricerca Scientifica e Tecnologica" .
Pisa, October 2004 Giuseppe Da Prato
Chapter 1
Introduction and Preliminaries
1.1 Introduction
We are here concerned with a stochastic differential equation in a separable Hilbert
space H,
dX(t,x) = (AX(t, x) + F(X(t, x)))dt + B dW(t), t > 0, x E H,
{ (1.1)
X(O,x) = x, x E H.
Here A: D(A) c H ---+ H is the infinitesimal generator of a strongly continuous
semigroup etA in H, B is a bounded operator from another Hilbert space U and
H, F: D(F) c H ---+ H is a nonlinear mapping and W(t), t ;:::: 0, is a cylindrical
Wiener process in U defined in some probability space (n, §, IF), see Chapter 2
for a precise definition.
In applications equation (1.1) describes the evolution of an infinite dimen
sional dynamical system perturbed by noise (the system being considered "iso
lated" when F = 0).
In this book we shall consider several stochastic partial differential equations
which can be written in the form (1.1). In each case we shall first prove existence
and uniqueness of a mild solution. A mild solution of equation (1.1) is a mean
square conti°nuo us stochastic process, adapted to W(t), such that X(t,x) E D(F)
for any t ;:::: and
where WA(t) is the stochastic convolution defined by
(1.3)
G. Prato Kolmogorov Equations for Stochastic PDEs
© Birkhauser Verlag 2004
2 Chapter 1. Introduction and Preliminaries
Moreover, we shall study several properties of the transition semigroup Pt defined
by (1)
Ptip(X) = JE[ip(X(t, x))], ip E Bb(H), t ~ 0, x E H, (1.4)
as Feller and strong Feller properties and irreducibility. We recall that P is Feller
t
if Ptip is continuous for all t ~ 0 and any continuous and bounded function ip,
strong Feller if Ptip is continuous for all t > 0 and all ip E Bb(H). Moreover, Pt
is irreducible if Pt1J(x) > 0 for all x E H and all open sets I, where 1J is the
characteristic function of I (2).
To study asymptotic properties of the transition semigroup P an important
t
tool is provided by invariant measures. A Borel probability measure v in H is said
to be invariant for Pt if L L
Ptipdv = ipdv (1.5)
for all continuous and bounded functions ip: H ....... R
If Pt is irreducible, then any invariant measure v is full, that is we have
v(B(x, r)) > 0 for any ball B(x, r) of center x E H and radius r. In fact from (1.5)
it follows that L
II(B(x,r)) = Pt1J(x)v(dx) > O.
If Pt is at the same time irreducible and strong Feller, then there is at most one
invariant measure in view of the Doob theorem, see Theorem 1.12 (3) .
We shall prove, under suitable assumptions, existence (and in some cases
uniqueness) of an invariant measure v. As it is well known, this allows us to
extend uniquely P to a strongly continuous semigroup of contractions in L2(H, v)
t
(still denoted Pt). We shall denote by K2 its infinitesimal generator.
Particular attention will be paid to describing the relationship between K2
and the concrete differential operator Ko defined by
1
Koip(x) = "2 Tr [CD2ip(X)] + (Ax + F(x), Dip(x)), ip E iffA(H), (1.6)
where Tr denotes the trace, C = BB* (B* being the adjoint of B), and D denotes
the derivative with respect to x. Moreover, iffA(H) is the linear span of all real and
imaginary parts of exponential functions ip h,
iph(X) := ei(h,x), x E H, hE D(A*),
where A* is the adjoint of A. It is easy to see that the space iffA(H) is dense in
L2(H, v). The reason for taking h E D(A*) is that this fact is necessary in order
(1) Bb(H) is the space of all bounded and Borel real functions in H.
(2)lJ(x) = 1 if x E I, IJ(x) = 0 if x fJ. I.
(3) Another powerful method to prove the uniqueness of an invariant measure is based on cou
pling, see [54], [74], [73], [68], [40].
