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Tables of Contents for Stochastic Pde's and Kolmogorov Equations in Infinite Dimensions
Chapter/Section Title
Page #
Page Count
Preface
v
 
On Kolmogourv's equatorns for finite dimensional diffusions
1
64
N.V.Krylov
Solvability of Ito's stochastic equations
1
7
Markov property of solutions
8
8
Regular equations
8
3
Some Properties of Euler's approximations
11
4
Markov property
15
1
Conditional version of Kolmogorov's equation
16
5
Differentiability of solutions of sochastic equations with respect to initial data
21
21
Estimating moments of soulations of Ito's equations
22
4
Smoothness of solutions depending on a parameter
26
5
Estimating moments of derivatives of solutions
31
2
The nations of L-Continuity and L-differentiability
33
3
Differentiability of certain expectations depending on parameter
36
6
Kolmogorov's equation in the whole space
42
11
Stratified equations
43
3
Sufficient conditions for regularity
46
2
Kolmogorov's equation
48
5
Some integral approximations of differential operators
53
5
Kolmogorov's equations in domanis
58
7
Lp-analysis of finite and infinite dimensional diffusion operators
65
31
Micheal Rochner
Introduction
65
1
Solution of Kolmogorov equations via sectorial forms
66
12
Preliminaties
66
2
Sectorial forms
68
2
Sectorial forms on L2 (E;m)
70
2
Examples and Applications
72
6
Symmetrizing measures
78
8
The classical finite dimensional case
78
2
Representation of symmetric diffusion operators
80
1
Ornstein-Uhlenbeck type operators.
81
2
Operators with non-linear drift.
83
3
Non-sectorial cases: perturbations by divergence free vector fields
86
10
Diffusion operators on Lp(E;m)
86
2
Solution of Kolmogorov equations on L1 (E;m)
88
4
Uniquenees Problem
92
3
Concluding remarks
95
1
Invariant mesures: regularity, existence and uniqueness
96
21
Sectorial case
96
3
Non-sectorial cases
99
4
Corresponding diffusions and relation to Martingale Problems
103
3
Existence of associated diffusions
103
2
Solution of the martingle Problem
105
1
Uniqueness
105
1
Appendix
106
11
Kolmogorov equations in L2(E;μ) for infinite demensional manifolds E: a case study from continuum statistical mechanics
106
4
Ergodicity
110
7
Parabolic equations on Hilbert spaces
117
 
J. Zabczyk
Preface
117
 
Preliminaries
119
 
Linear operators
119
 
Measures and random variables
123
 
Wiener process and stochastic equations
127
 
Heat Equation
130
 
Introduction
131
 
Regular initial functions
135
 
Gross Laplacian
137
 
Heat equation with general initial functions
139
 
Generators of the heat semigroups
143
 
Nonparabolicity
147
 
Transition semigroups
149
 
Transition semigroups in the space of continuous functions
150
 
Transition semigroups in spaces of square summable functions
154
 
Heat equation with a first order term
157
 
Introduction
158
 
Regular initial functions
159
 
General initial functions
163
 
Range codition and examples
170
 
General parabolic equations. Regularity
174
 
Convolution type and evaluation maps
174
 
Solutions of stochastic equations
178
 
Space and time regularity of generalized solutions
179
 
Strong Feller property
181
 
General parabolic equations. Uniqueness
186
 
Uniquensess for the heat equation
186
 
Uniquensess in the general case
187
 
Parabolic equations in open sets
191
 
Introduction
191
 
Main theorm
192
 
Estimates of the exit probabilites
195
 
Applications
198
 
HJB equation of stochastic control
198
 
Solvability of HJB equation
202
 
Kolmogorve's equation in mathematical finance
204
 
Appendix
206
 
Implicit function theorems
206