2017/06/28 by Andres, Sebastian, Neukamm, Stefan
#35B27 #35K65 #60F05 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1706.09493
We study the random conductance model on the lattice ℤd, 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≥ 3 quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed t-\frac 1 5+ε for d≥ 4 and t-(1)/(10)+ε 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≥ 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.