Optimal convergence rates in stochastic homogenization in a balanced random environment

Abstract

We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\ge 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for $d\ge 3$ and nearly optimal when $d=2$. Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk.

The quantitative stochastic homogenization of the non-divergence form operators are improved: optimal (and nearly optimal) rates are obtained for dimensions $d\ge 3$ (and $d=2$ resp.). Correctors are constructed for all dimensions using "local correctors"

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