Quantitative version of the Kipnis-Varadhan theorem and Monte Carlo approximation of homogenized coefficients

TL;DR

This paper develops a quantitative analysis of a Monte Carlo method for approximating homogenized coefficients in stochastic homogenization, providing sharp error estimates, fluctuation bounds, and limit theorems based on a refined Kipnis-Varadhan approach.

Contribution

It introduces a quantitative version of the Kipnis-Varadhan theorem for discrete elliptic equations, combining PDE and spectral theory to improve error and fluctuation estimates in homogenization.

Findings

Sharp error bounds for homogenized coefficient approximation

Quantitative fluctuation and deviation estimates

Optimal convergence rates with logarithmic correction in 2D

Abstract

This article is devoted to the analysis of a Monte Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time t>0, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions re-scaled by t of n independent random walks in n independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of PDE arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the re-scaled final position of the random walk in terms of t. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of n and t, and prove a large-deviation estimate, as well as a central limit theorem. Our estimates are optimal, up to a logarithmic correction in dimension 2.