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

DOI:
https://doi.org/10.1090/S0002-9947-04-03467-1

Published electronically:
January 16, 2004

MathSciNet review:
2058840

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Abstract | References | Similar Articles | Additional Information

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.

**1.**D. G. Aronson,*Non-negative solutions of linear parabolic equations*,

Ann. Sci. Norm. Sup. Pisa (3)**22**(1968) , 607-94. MR**55:8553****2.**Joseph G. Conlon,*Homogenization of random walk in asymmetric random environment*, New York J. Math.**8**(2002), 31–61. MR**1887697****3.**E. A. Carlen, S. Kusuoka and D. Stroock,*Upper bounds for symmetric Markov transition functions*,

Ann. Inst. H. Poincaré**23**(1987), 245-87. MR**88i:35066****4.**J. Conlon and A. Naddaf,*Green's functions for elliptic and parabolic equations with random coefficients*,

New York J. Math.**6**(2000), 153-225. MR**2001j:35282****5.**E. B. Davies,*Heat Kernels and Spectral Theory*,

Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, New York, 1989. MR**90e:35123****6.**T. Delmotte and J. Deuschel,*On estimating the derivatives of symmetric diffusions in stationary random environment*,

preprint 2003.**7.**G. Giacomin, S. Olla and H. Spohn,*Equilibrium fluctuations for**interface model*,

Ann. Probab.**29**(2001), 1138-1172. MR**2003c:60161****8.**C. Landim, S. Olla and H. T. Yau,*Convection-diffusion equation with space-time ergodic random flow*,

Probab. Theory Relat. Fields**112**(1998), 203-220. MR**99j:35084****9.**M. Reed and B. Simon,*Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness*,

Academic Press, New York, London, 1975. MR**58:12429b****10.**E. Stein and G. Weiss,*Introduction to Fourier Analysis on Euclidean Spaces*,

Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR**46:4102****11.**V. Zhikov, S. Kozlov and O. Oleinik,*Homogenization of Differential Operators and Integral Functionals*,

Springer-Verlag, Berlin, 1994. MR**96h:35003b**

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Additional Information

**Joseph G. Conlon**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
conlon@umich.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03467-1

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