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Stochastic homogenization of Hamilton-Jacobi-Bellman equations. (English) Zbl 1111.60055
Summary: We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment.

##### MSC:
 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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