Stochastic Equations in Infinite Dimensions (Encyclopedia of Mathematics and its Applications, Series Number 45)

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. The Aim Of This Book Is To Give A Systematic And Self-contained Presentation Of 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 Gikham 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 Measure 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. The Book Ends With A Comprehensive Bibliography That Will Contribute To The Book's Value For All Working In Stochastic Differential Equations. Lifts Of Diffusion Processes -- 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 Nehaviour Of Solutions -- Small Noise Noise Asymptotic -- A Linear Deterministic Equations -- Some Results On Control Theory -- Nuclear And Hilbert, Schimidt Operators -- Dissipative Mappings. Giuseppe Da Prato, Jerzy Zabczyk. Includes Bibliographical References (p. 427-449) And Index. "The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalisation of stochastic differential equations as introduced by Ito 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."--BOOK JACKET. "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. The book ends with a comprehensive bibliography that will contribute to the book's value for all working in stochastic differential equations."--Jacket As we have said in the Preface, stochastic evolution equations in infinite dimensions are natural generalizations of stochastic ordinary differential equations and their theory has motivations coming both from mathematics and the natural sciences: physics, chemistry and biology.