Celebratio Mathematica — Varadhan

Large deviations

Works connected to Fraydoun Rezakhanlou

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J. Quastel, F. Rezakhanlou, and S. R. S. Varadhan: “Large deviations for the symmetric simple exclusion process in dimensions \( d\geq 3 \),” Probab. Theory Related Fields 113 : 1 (1999), pp. 1–84. MR 1670733 Zbl 0928.60087 article

We consider symmetric simple exclusion processes with \( L=\overline{\rho}N^d \) particles in a periodic \( d \)-dimensional lattice of width \( N \). We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the \( d \)-dimensional torus. We are interested in the large deviations of the empirical process, \[ N^{-d}\Bigl[\sum_1^L\delta_{x_i(\,\cdot\,)}\Bigr] \] as random variables taking values in the space of measures on \( D[0,1] \). We prove a large deviation principle, with a rate function that is more or less universal, involving explicitly besides the initial profile, only such canonical objects as bulk and self diffusion coefficients.

Fraydoun Rezakhanlou	Related
Jeremy Quastel	Related

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