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 $$\mathbb {Z}^d$$ for $$d\ge 2$$ . 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 difference 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 $$d\ge 2$$ . Afterward, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks. Consequently, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk.