Random Fields and Stochastic Partial Differential Equations / Edition 1

by Y. Rozanov

Overview

This book considers some models described by means of partial dif­ ferential equations and boundary conditions with chaotic shastic disturbance. In a framework of shastic Partial Differential Equa­ tions an approach is suggested to generalize solutions of shastic Boundary Problems. The main topic concerns probabilistic aspects with applications to well-known Random Fields models which are representative for the corresponding shastic Sobolev spaces. {The term "shastic" 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 (shastic) 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.

Product Details

ISBN-13:	9780792349846
Publisher:	Springer Netherlands
Publication date:	03/31/1998
Series:	Mathematics and Its Applications , #438
Edition description:	1998
Pages:	232
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

I. Random Fields and Shastic Sobolev Spaces.- II. Equations for Generalized Random Functions.- III. Random Fields Associated with Partial Equations.- IV. Gaussian Random Fields.