# Random Fields and Stochastic Partial Differential Equations

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

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

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.

Probability theory Sobolev space partial differential equation random function stochastic processes

- 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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