Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances
Sebastián Andrés, Stefan Neukamm · Stochastic Partial Differential Equations Analysis and Computations · 2018
We study the random conductance model on the lattice $${\mathbb {Z}}^d$$ , i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension $$d\ge 3$$ quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed $$t^{-\frac{1}{5}+\varepsilon }$$ for $$d\ge 4$$ and $$t^{-\frac{1}{10}+\varepsilon }$$ for $$d=3$$ . Additionally, in the uniformly elliptic case in low dimensions $$d=2,3$$ we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for $$d\ge 3$$ we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.