The systematic study of existence, uniqueness, and properties of solutions to stochastic differential equations in infinite dimensions arising from practical problems characterizes this volume that is intended for graduate students and for pure and applied mathematicians, physicists, engineers, professionals working with mathematical models of finance. Major methods include compactness, coercivity, monotonicity, in a variety of set-ups. The authors emphasize the fundamental work of Gikhman and Skorokhod on the existence and uniqueness of solutions to stochastic differential equations and present its extension to infinite dimension. They also generalize the work of Khasminskii on stability and stationary distributions of solutions. New results, applications, and examples of stochastic partial differential equations are included. This clear and detailed presentation gives the basics of the infinite dimensional version of the classic books of Gikhman and Skorokhod and of Khasminskii in one concise volume that covers the main topics in infinite dimensional stochastic PDE’s. By appropriate selection of material, the volume can be adapted for a 1- or 2-semester course, and can prepare the reader for research in this rapidly expanding area.
Author(s): Leszek Gawarecki, Vidyadhar Mandrekar
Series: Probability and Its Applications
Edition: 1st Edition.
Publisher: Springer
Year: 2010
Language: English
Pages: 308
Cover......Page 1
Probability and Its Applications......Page 2
Stochastic Differential Equations in Infinite Dimensions: with Applications to Stochastic Partial Differential Equations......Page 4
9783642161933......Page 5
Preface......Page 8
Acknowledgements......Page 12
Contents......Page 14
Acronyms......Page 16
Part I - Stochastic Differential Equations in Infinite Dimensions......Page 18
1.1 The Heat Equation as an Abstract Cauchy Problem ......Page 20
1.2 Elements of Semigroup Theory ......Page 22
1.3 Commonly Used Function Spaces ......Page 27
1.4 The Abstract Cauchy Problem ......Page 28
1.5 The Variational Method ......Page 32
2.1.1 Cylindrical and Hilbert-Space-Valued Gaussian Random Variables ......Page 34
2.1.2 Cylindrical and Q-Wiener Processes ......Page 36
2.1.3 Martingales in a Hilbert Space ......Page 38
2.2 Stochastic Integral with Respect to a Wiener Process ......Page 40
2.2.1 Elementary Processes ......Page 42
2.2.2 Stochastic Itô Integral for Elementary Processes ......Page 43
2.2.3 Stochastic Itô Integral with Respect to a Q-Wiener Process ......Page 51
2.2.4 Stochastic Itô Integral with Respect to Cylindrical Wiener Process ......Page 61
2.2.5 The Martingale Representation Theorem ......Page 66
2.2.6 Stochastic Fubini Theorem ......Page 74
2.3.1 The Case of a Q-Wiener Process ......Page 78
2.3.2 The Case of a Cylindrical Wiener Process ......Page 86
3.1 Stochastic Differential Equations and Their Solutions ......Page 90
3.2 Solutions Under Lipschitz Conditions ......Page 101
3.3 A Special Case ......Page 120
3.4 Markov Property and Uniqueness ......Page 124
3.5 Dependence of the Solution on the Initial Value ......Page 128
3.6 Kolmogorov’s Backward Equation ......Page 134
3.7 Lipschitz-Type Approximation of Continuous Coefficients ......Page 142
3.8 Existence of Weak Solutions Under Continuity Assumption ......Page 145
3.9 Compact Semigroups and Existence of Martingale Solutions ......Page 152
3.10 Mild Solutions to SSDEs Driven by Cylindrical Wiener Process ......Page 158
Appendix: Compactness and Tightness of Measures in C([0,T], \mathscr{M})......Page 165
4.1 Introduction ......Page 168
4.2 Existence of Weak Solutions Under Compact Embedding ......Page 169
4.3 Strong Variational Solutions ......Page 191
4.4 Markov and Strong Markov Properties ......Page 198
5.2 Unbounded Spin Systems, Solutions in C([0,T],H^w)......Page 202
5.3 Locally Interacting Particle Systems, Solutions in C([0,T], \mathbb{R}^{\mathbb{Z}^d})......Page 211
Part II - Stability, Boundedness, and Invariant Measures......Page 218
6.1 Introduction ......Page 220
6.2 Exponential Stability for Stochastic Differential Equations ......Page 228
6.3 Stability in the Variational Method ......Page 242
Appendix: Stochastic Analogue of the Datko Theorem ......Page 247
7.1 Exponential Ultimate Boundedness in the m.s.s ......Page 250
7.2 Exponential Ultimate Boundedness in Variational Method ......Page 254
7.3 Abstract Cauchy Problem, Stability and Exponential Ultimate Boundedness ......Page 263
7.4 Ultimate Boundedness and Invariant Measure ......Page 272
7.4.1 Variational Equations ......Page 278
7.4.2 Semilinear Equations Driven by a Q-Wiener Process ......Page 283
7.4.3 Semilinear Equations Driven by a Cylindrical Wiener Process ......Page 288
7.5 Ultimate Boundedness and Weak Recurrence of the Solutions ......Page 291
References ......Page 302
Index ......Page 306
