ISBN: 3-540-66545-5
TITLE: Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions
AUTHOR: Krylov, N.V.; Röckner, M.; Zabczyk, J.
TOC:
Preface V
On Kolmogorov's equations for finite dimensional diffusions
N.V. Krylov 1
1. Solvability of It's stochastic equations 1
2. Markov property of solutions 8
2.1 Regular equations 8
2.2 Some properties of Euler's approximations 11
2.3 Markov property 15
3. Conditional version of Kolmogorov's equation 16
4. Differentiability of solutions of stochastic equations with respect to initial data 21
4.1 Estimating moments of solutions of It's equations 22
4.2 Smoothness of solutions depending on a parameter 26
4.3 Estimating moments of derivatives of solutions 31
4.4 The notions of L-continuity and L-differentiability 33
4.5 Differentiability of certain expectations depending on a parameter 36
5. Kolmogorov's equation in the whole space 42
5.1 Stratified equations 43
5.2 Sufficient conditions for regularity 46
5.3 Kolmogorov's equation 48
6. Some integral approximations of differential operators 53
7. Kolmogorov's equations in domains 58
L^panalysis of finite and infinite dimensional diffusion operators
Michael Röckner 65
1. Introduction 65
2. Solution of Kolmogorov equations via sectorial forms 66
2.1 Preliminaries 66
2.2 Sectorial forms 68
2.3 Sectorial forms on L2(E; m) 70
2.4 Examples and Applications 72
Table of Contents VII
3. Symmetrizing measures 78
3.1 The classical finite dimensional case 78
3.2 Representation of symmetric diffusion operators 80
3.3 OrnsteinUhlenbeck type operators81
3.4 Operators with nonlinear drift83
4. Non-sectorial cases: perturbations by divergence free vector fields 86
4.1 Diffusion operators on L^p(E; m) 86
4.2 Solution of Kolmogorov equations on L^1(E; m) 88
4.3 Uniqueness problem. 92
4.4 Concluding remarks. 95
5. Invariant measures: regularity, existence and uniqueness 96
5.1 Sectorial case 96
5.2 Nonsectorial cases 99
6. Corresponding diffusions and relation to Martingale problems 103
6.1 Existence of associated diffusions 103
6.2 Solution of the martingale problem 105
6.3 Uniqueness 105
7. Appendix 106
7.1 Kolmogorov equations in L^2(E; mu) for infinite dimensional manifolds E: a case study from continuum statistical mechanics 106
7.2 Ergodicity 110
Parabolic equations on Hilbert spaces
J. Zabczyk 117
1. Preface 117
2. Preliminaries 119
2.1 Linear operators 119
2.2 Measures and random variables 123
2.3 Wiener process and stochastic equations 127
3. Heat Equation 130
3.1 Introduction 131
3.2 Regular initial functions 135
3.3 Gross Laplacian 137
3.4 Heat equation with general initial functions 139
3.5 Generators of the heat semigroups 143
3.6 Nonparabolicity 147
4. Transition semigroups 149
4.1 Transition semigroups in the space of continuous functions 150
4.2 Transition semigroups in spaces of square summable functions 154
5. Heat equation with a first order term 157
5.1 Introduction 158
5.2 Regular initial functions 159
5.3 General initial functions 163
5.4 Range condition and examples 170
6. General parabolic equations. Regularity 174
6.1 Convolution type and evaluation maps 174
6.2 Solutions of stochastic equations 178
6.3 Space and time regularity of generalized solutions 179
6.4 Strong Feller property 181
7. General parabolic equations - Uniqueness - 186
7.1 Uniqueness for the heat equation 186
7.2 Uniqueness in the general case 187
8. Parabolic equations in open sets 191
8.1 Introduction 191
8.2 Main theorem 192
8.3 Estimates of the exit probabilities 195
9. Applications 198
9.1 HJB equation of stochastic control 198
9.2 Solvability of HJB equation 202
9.3 Kolmogorov's equation in mathematical finance 204
10. Appendix 206
10.1 Implicit function theorems 206
END