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Stochastic cauchy problems in infinite dimensions: generalized and regularized solution

By: Melnikova, Irina V.
Series: Monographs and research notes in mathematics. Publisher: Boca Raton CRC Press 2016Description: xix, 286 p.ISBN: 9781482210507.Subject(s): Cauchy problem | Stochastic processesDDC classification: 515.35 Summary: Features Describes deterministic techniques and results of stochastic equations Includes examples involving generators of integrated, convoluted, and R-semi-groups Discusses basic properties of regularized semi-groups and illustrates them with numerous examples, paying special attention to differential systems in Gelfand–Shilov spaces Incorporates novel material that extends white noise analysis to Hilbert spaces and allows you to obtain new types of solutions to stochastic problems Summary 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. (https://www.crcpress.com/Stochastic-Cauchy-Problems-in-Infinite-Dimensions-Generalized-and-Regularized/Melnikova/p/book/9781482210507)
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Table of Contents:

Part – I: Well-Posed and Ill-Posed Abstract Cauchy Problems. The Concept of Regularization
1. Semi-group methods for construction of exact, approximated, and regularized solutions

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

3. Examples. Supplements

Part – II: Infinite-Dimensional Stochastic Cauchy Problems
4. Weak, regularized, and mild solutions to Itô integrated stochastic Cauchy problems in Hilbert spaces

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

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

Features

Describes deterministic techniques and results of stochastic equations
Includes examples involving generators of integrated, convoluted, and R-semi-groups
Discusses basic properties of regularized semi-groups and illustrates them with numerous examples, paying special attention to differential systems in Gelfand–Shilov spaces
Incorporates novel material that extends white noise analysis to Hilbert spaces and allows you to obtain new types of solutions to stochastic problems

Summary

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.


(https://www.crcpress.com/Stochastic-Cauchy-Problems-in-Infinite-Dimensions-Generalized-and-Regularized/Melnikova/p/book/9781482210507)

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