Celebratio Mathematica — Varadhan

Large deviation and homogenization

Works connected to Fraydoun Rezakhanlou

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E. Kosygina, F. Rezakhanlou, and S. R. S. Varadhan: “Stochastic homogenization of Hamilton–Jacobi–Bellman equations,” Comm. Pure Appl. Math. 59 : 10 (2006), pp. 1489–1521. MR 2248897 Zbl 1111.60055 article

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.

Fraydoun Rezakhanlou	Related
Elena Kosygina	Related