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＜電子ブック＞
Random Fields and Stochastic Partial Differential Equations / by Y. Rozanov
(Mathematics and Its Applications ; 438)

版	1st ed. 1998.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	1998
本文言語	英語
大きさ	VII, 232 p : online resource
著者標目	*Rozanov, Y author SpringerLink (Online service)
件　名	LCSH:Probabilities LCSH:Differential equations FREE:Probability Theory FREE:Differential Equations
一般注記	I. Random Fields and Stochastic Sobolev Spaces -- II. Equations for Generalized Random Functions -- III. Random Fields Associated with Partial Equations -- IV. Gaussian Random Fields 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 HTTP:URL=https://doi.org/10.1007/978-94-017-2838-6

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