Stochastic Equations in Infinite Dimensions

by Guiseppe Da Prato , Jerzy Zabczyk

• ISBN13: 9780521059800
• ISBN10: 0521059801
• Edition: 1st
• Format: Paperback
• Copyright: 2008-02-04
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Foundations

Random variables

Probability measures

Stochastic processes

The stochastic integral

Existence and Uniqueness

Linear equations with additive noise

Linear equations with multiplicative noise

Existence and uniqueness for nonlinear equations

Martingale solutions

Properties of Solutions

Markov properties and Kolmogorov equations

Absolute continuity and Girsanov's theorem

Large time behaviour of solutions

Small noise asymptotic

Table of Contents provided by Publisher. All Rights Reserved.