Exit Laws of Isotropic Diffusions in Random Environment from Large Domains

Benjamin Fehrman

Contact the author: Please use for correspondence this email.
Submission date: 08. Jul. 2016
Pages: 35
published in: Electronic journal of probability, 22 (2017), art-no. 63 
DOI number (of the published article): 10.1214/17-EJP79
Bibtex
MSC-Numbers: 35B27, 35J25, 60H25, 60J60, 60K37
Keywords and phrases: diffusion processes in random environment, stochastic homogenization, Dirichlet boundary-value problem
Download full preprint: PDF (448 kB)

Abstract:
This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [20]. Building upon their work, it is shown by analyzing the associated elliptic boundary-value problem that, almost surely, the smoothed exit law of the diffusion from large domains converges, as the domain's scale approaches infinity, to that of a Brownian motion. Furthermore, an algebraic rate for the convergence is established in terms of the modulus of the boundary condition.