Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions

by Irina V. Melnikova

Overview

Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions presents stochastic differential equations for random processes with values in Hilbert spaces. Accessible to non-specialists, the book explores how modern semi-group and distribution methods relate to the methods of infinite-dimensional stochastic analysis. It also shows how the idea of regularization in a broad sense pervades all these methods and is useful for numerical realization and applications of the theory.

The book presents generalized solutions to the Cauchy problem in its initial form with white noise processes in spaces of distributions. It also covers the "classical" approach to stochastic problems involving the solution of corresponding integral equations. The first part of the text gives a self-contained introduction to modern semi-group and abstract distribution methods for solving the homogeneous (deterministic) Cauchy problem. In the second part, the author solves stochastic problems using semi-group and distribution methods as well as the methods of infinite-dimensional stochastic analysis.

Product Details

ISBN-13:	9781315360263
Publisher:	CRC Press
Publication date:	09/03/2018
Series:	Chapman & Hall/CRC Monographs and Research Notes in Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	306
File size:	33 MB
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About the Author

Irina V. Melnikova is a professor in the Institute of Mathematics and Computer Sciences at Ural Federal University. Her research interests include analysis, applied mathematics, and probability theory.

Table of Contents

Well-Posed and Ill-Posed Abstract Cauchy Problems. The Concept of Regularization: Semi-group methods for construction of exact, approximated, and regularized solutions. Distribution methods for construction of generalized solutions to ill-posed Cauchy problems. Examples. Supplements. Infinite-Dimensional Stochastic Cauchy Problems: Weak, regularized, and mild solutions to Itô integrated stochastic Cauchy problems in Hilbert spaces. Infinite-dimensional stochastic Cauchy problems with white noise processes in spaces of distributions. Infinite-dimensional extension of white noise calculus with application to stochastic problems.

Editorial Reviews

"Written by a distinguished expert in the field of generalized functions and semigroups of operators, the book represents an excellent introduction to a theory of increasing power and relevance in contemporary stochastic analysis. […] Due to its clear, systematic and comprehensive style of exposition, it will make the subject accessible to a broad mathematical audience. [...] The book is designed to be most appealing for graduate students, postgraduates and experienced scientists who work in the field of stochastic partial differential equations, but it should be welcome in the library of any researcher who has a broad mathematical interest."

- Dora Seleši, Mathematical Reviews, March 2017

From the Publisher