Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions, 1st Edition (Hardback) book cover

Stochastic Cauchy Problems in Infinite Dimensions

Generalized and Regularized Solutions, 1st Edition

By Irina V. Melnikova

Chapman and Hall/CRC

286 pages

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pub: 2016-02-19
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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.


"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

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

The Cauchy problem and strongly continuous semi-groups of solution operators

The Cauchy problem with generators of regularized semigroups: integrated, convoluted, and R-semi-groups

R-semi-groups and regularizing operators in the construction of approximated solutions to ill-posed problems

Distribution methods for construction of generalized solutions to ill-posed Cauchy problems

Solutions in spaces of abstract distributions

Solutions in spaces of abstract ultra-distributions

Solutions to the Cauchy problem for differential systems in Gelfand–Shilov spaces

Examples. Supplements

Examples of regularized semi-groups and their generators

Examples of solutions to Petrovsky correct, conditionally correct and incorrect systems

Definitions and properties of spaces of test functions

Generalized Fourier and Laplace transforms. Structure theorems

Infinite-Dimensional Stochastic Cauchy Problems

Weak, regularized, and mild solutions to Itô integrated stochastic Cauchy problems in Hilbert spaces

Hilbert space valued variables, processes, and stochastic integrals. Main properties and results

Solutions to Cauchy problems for equations with additive noise and generators of regularized semi-groups

Solutions to Cauchy problems for semi-linear equations with multiplicative noise

Extension of the Feynman–Kac theorem to the case of relations between stochastic equations and PDEs in Hilbert spaces

Infinite-dimensional stochastic Cauchy problems with white noise processes in spaces of distributions

Generalized solutions to linear stochastic Cauchy problems with generators of regularized semi-groups

Quasi-linear stochastic Cauchy problem in abstract Colombeau spaces

Infinite-dimensional extension of white noise calculus with application to stochastic problems

Spaces of Hilbert space valued generalized random variables: (S)−ρ(H). Basic examples

Analysis of (S)−ρ(H)-valued processes

S-transform and Wick product. Hitsuda–Skorohod integral. Main properties. Connection with Itô integral

Generalized solutions to stochastic Cauchy problems in spaces of abstract stochastic distributions

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.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis