Random Fields and Stochastic Partial Differential Equations [Paperback]

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.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 stlÓ+