Homogenization and Enhancement of theG‐Equation in Random Environments
Pierre Cardaliaguet, Panagiotis E. Souganidis · Communications on Pure and Applied Mathematics · 2013
Abstract We study the homogenization of aG‐equation that is advected by a divergence free “small mean” stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministicG‐equation, and we give necessary and sufficient conditions for enhancement. Since the problem is not assumed to be coercive, it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition that is necessary to apply the subadditive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence that has both a long‐time averaged limit (due to the subadditive ergodic theorem) and stays asymptotically close to the minimal time. © 2013 Wiley Periodicals, Inc.