Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations

TL;DR

This paper demonstrates that ergodic constants in stochastic homogenization of certain nonlinear PDEs can be approximated by periodic versions, providing error estimates when convergence rates are available.

Contribution

It introduces a method to approximate stochastic ergodic constants using periodic equations and establishes error bounds based on convergence rates.

Findings

Ergodic constants can be approximated via periodic equations.

Error estimates depend on the convergence rate of homogenization.

Applicable to Hamilton-Jacobi and nonlinear elliptic PDEs.

Abstract

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, "viscous" Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate "periodizations" of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.