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" Random Fields and Stochastic Partial Differential Equations "
by Yu. A. Rozanov.
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BL
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| Record Number
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579168
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b408387
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| Main Entry
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Rozanov, Yu. A.
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| Title & Author
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Random Fields and Stochastic Partial Differential Equations\ by Yu. A. Rozanov.
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| Publication Statement
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Dordrecht :: Springer Netherlands :: Imprint: Springer,, 1998.
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| Series Statement
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Mathematics and Its Applications ;; 438
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| ISBN
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9789401728386
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: 9789048150090
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| Contents
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I. Random Fields and Stochastic Sobolev Spaces -- II. Equations for Generalized Random Functions -- III. Random Fields Associated with Partial Equations -- IV. Gaussian Random Fields.
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| Abstract
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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.
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Mathematics.
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Differential equations, Partial.
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Distribution (Probability theory).
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SpringerLink (Online service)
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