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Random Fields and Stochastic Partial Differential Equations

  • Yu. A. Rozanov

Part of the Mathematics and Its Applications book series (MAIA, volume 438)

Table of contents

  1. Front Matter
    Pages i-4
  2. Yu. A. Rozanov
    Pages 193-229
  3. Back Matter
    Pages 231-232

About this book

Introduction

This book considers some models described by means of partial dif­ ferential equations and boundary conditions with chaotic stochastic disturbance. In a framework of stochastic Partial Differential Equa­ tions an approach is suggested to generalize solutions of stochastic Boundary Problems. The main topic concerns probabilistic aspects with applications to well-known Random Fields models which are representative for the corresponding stochastic Sobolev spaces. {The term "stochastic" in general indicates involvement of appropriate random elements. ) It assumes certain knowledge in general Analysis and Probability {Hilbert space methods, Schwartz distributions, Fourier transform) . I A very general description of the main problems considered can be given as follows. Suppose, we are considering a random field ~ in a region T ~ Rd which is associated with a chaotic (stochastic) source"' by means of the differential equation (*) in T. A typical chaotic source can be represented by an appropri­ ate random field"' with independent values, i. e. , generalized random function"' = ( cp, 'TJ), cp E C~(T), with independent random variables ( cp, 'fJ) for any test functions cp with disjoint supports. The property of having independent values implies a certain "roughness" of the ran­ dom field "' which can only be treated functionally as a very irregular Schwarz distribution. With the lack of a proper development of non­ linear analyses for generalized functions, let us limit ourselves to the 1 For related material see, for example, J. L. Lions, E.

Keywords

Probability theory Sobolev space partial differential equation random function stochastic processes

Authors and affiliations

  • Yu. A. Rozanov
    • 1
    • 2
  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.CNR-IAMIMilanItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-2838-6
  • Copyright Information Springer Science+Business Media B.V. 1998
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5009-0
  • Online ISBN 978-94-017-2838-6
  • Buy this book on publisher's site
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