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Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. (English) Zbl 1065.35047
Summary: We study the homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton-Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
76M50 Homogenization applied to problems in fluid mechanics
34K50 Stochastic functional-differential equations
93E03 Stochastic systems in control theory (general)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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