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.
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 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
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.
"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