Journal
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
Volume 97, Issue 5, Pages 460-504Publisher
GAUTHIER-VILLARS/EDITIONS ELSEVIER
DOI: 10.1016/j.matpur.2011.09.009
Keywords
Stochastic homogenization; Viscous Hamilton-Jacobi equation; Poissonian potential
Categories
Funding
- NSF [DMS-1004645, DMS-0901802]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0901802] Funding Source: National Science Foundation
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We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools. (C) 2011 Elsevier Masson SAS. All rights reserved.
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