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Transactions of the American Mathematical Society

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Green's functions for elliptic and parabolic equations with random coefficients II

Author: Joseph G. Conlon
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4085-4142
MSC (2000): Primary 81T08, 82B20, 35R60, 60J75
Published electronically: January 16, 2004
MathSciNet review: 2058840
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Abstract: This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

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Additional Information

Joseph G. Conlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Keywords: pde with random coefficients, homogenization
Received by editor(s): July 23, 2002
Received by editor(s) in revised form: July 15, 2003
Published electronically: January 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society