Random Fields and Stochastic Partial Differential Equations

Random Fields and Stochastic Partial Differential Equations

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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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Product details

  • Hardback | 232 pages
  • 152.4 x 241.3 x 20.3mm | 544.32g
  • Dordrecht, Netherlands
  • English
  • 1998 ed.
  • VII, 232 p.
  • 0792349849
  • 9780792349846

Table of contents

Foreword. 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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