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:	9781482210507
Publisher:	Taylor & Francis
Publication date:	02/19/2016
Series:	Chapman & Hall/CRC Monographs and Research Notes in Mathematics , #20
Pages:	306
Product dimensions:	6.12(w) x 9.19(h) x (d)

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.