Celebratio Mathematica — Varadhan

Large deviation and homogenization

Works connected to Elena Kosygina

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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

E. Kosygina and S. R. S. Varadhan: “Homogenization of Hamilton–Jacobi–Bellman equations with respect to time-space shifts in a stationary ergodic medium,” Comm. Pure Appl. Math. 61 : 6 (2008), pp. 816–847. MR 2400607 Zbl 1144.35008 article

We consider a family \( \{u_{\varepsilon}(t,x,\omega)\} \), \( \varepsilon < 0 \), of solutions to the equation \[ \frac{\partial u_{\varepsilon}}{\partial t} + \frac{\varepsilon}{2}\,\Delta u_{\varepsilon} + H\Bigl(\frac{t}{\varepsilon}, \frac{x}{\varepsilon}, \nabla u_{\varepsilon},\,\omega\Bigr) = 0 \] with the terminal data \( u_{\varepsilon}(T,x,\omega) = U(x) \). Assuming that the dependence of the Hamiltonian \( H(t,x,p,\omega) \) on time and space is realized through shifts in a stationary ergodic random medium, and that \( H \) is convex in \( p \) and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of \( u_{\varepsilon}(t,x,\omega) \) as \( \varepsilon\to 0 \) to the solution \( u(t,x) \) of a deterministic averaged equation \[ \frac{\partial u}{\partial t} + \overline{H}(\nabla u) = 0, \qquad u(T,x) = U(x) .\] The “effective” Hamiltonian \( \overline{H} \) is given by a variational formula.