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

Conlon, Joseph

doi:10.1090/S0002-9947-04-03467-1

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: 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. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
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. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245–287 (English, with French summary). MR 898496
4. Joseph G. Conlon and Ali Naddaf, Green’s functions for elliptic and parabolic equations with random coefficients, New York J. Math. 6 (2000), 153–225. MR 1781430
5. E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239
6. T. Delmotte and J. Deuschel,
On estimating the derivatives of symmetric diffusions in stationary random environment,
preprint 2003.
7. Giambattista Giacomin, Stefano Olla, and Herbert Spohn, Equilibrium fluctuations for ∇𝜙 interface model, Ann. Probab. 29 (2001), no. 3, 1138–1172. MR 1872740, https://doi.org/10.1214/aop/1015345600
8. C. Landim, S. Olla, and H. T. Yau, Convection-diffusion equation with space-time ergodic random flow, Probab. Theory Related Fields 112 (1998), no. 2, 203–220. MR 1653837, https://doi.org/10.1007/s004400050187
9. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
10. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
11. V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242