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May, 1994 A Limit Theorem for Linear Boundary Value Problems in Random Media
Jean-Pierre Fouque, Ely Merzbach
Ann. Appl. Probab. 4(2): 549-569 (May, 1994). DOI: 10.1214/aoap/1177005072


The asymptotic behavior of the solutions of linear equations with random coefficients, random external forces and with affine boundary conditions is studied, motivated by a transmission-reflection problem for a one-dimensional wave equation in a random slab. The fluctuations of the coefficients are on a small scale in such a way that our problem is a diffusion-approximation problem except that we impose boundary conditions which force the solution to be anticipating. In the limit we obtain linear stochastic differential equations with affine boundary conditions, studied by Ocone and Pardoux. Our main tools are diffusion approximation results (Papanicolaou, Stroock and Varadhan or Ethier and Kurtz) and the properties of the limiting equations involving generalized Stratonovich integrals (Ocone and Pardoux). As an application, the transmission-reflection problem is discussed. We prove that the solution has a density with respect to the Lebesgue measure and satisfies the Markov field property.


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Jean-Pierre Fouque. Ely Merzbach. "A Limit Theorem for Linear Boundary Value Problems in Random Media." Ann. Appl. Probab. 4 (2) 549 - 569, May, 1994.


Published: May, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0802.60058
MathSciNet: MR1272739
Digital Object Identifier: 10.1214/aoap/1177005072

Primary: 60H30
Secondary: 73D70

Keywords: boundary value problem , Density , diffusion approximation , generalized Stratonovich integral , Markov field , random coefficients , Waves in random media

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 2 • May, 1994
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