Stochastic Equations in Infinite Dimensions
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The aim of this book is to give a systematic and self-contained presentation of the basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikhman that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measures on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof.
Reviews & endorsements
"...very fine...provides the first comprehensive synthesis of the semigroup approach to SPDE....The exposition is excellent and readable throughout, and should help bring the theory to a wider audience." Daniel L. Ocone, Stochastics and Stochastics ReportsSee more reviews
"...this is an excellent book which covers a large part of stochastic evolution equations with clear proofs and a very interesting analysis of their properties...In my opinion this book will become an indispensable tool for everyone working on stochastic evolution equations and related areas." P. Kotelenez, Mathematical Reviews
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- Date Published: April 2011
- format: Adobe eBook Reader
- isbn: 9780511877223
- availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
Part I. Foundations:
1. Random variables
2. Probability measures
3. Stochastic processes
4. The stochastic integral
Part II. Existence and Uniqueness:
5. Linear equations with additive noise
6. Linear equations with multiplicative noise
7. Existence and uniqueness for nonlinear equations
8. Martingale solutions
Part III. Properties of Solutions:
9. Markov properties and Kolmogorov equations
10. Absolute continuity and Girsanov's theorem
11. Large time behaviour of solutions
12. Small noise asymptotic.
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