Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

Abstract

We establish a quenched local central limit theorem for the dynamic random conductance model on ${\mathbb{Z}}^d$ only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi's iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.

arXiv comment33 pages, accepted version, to appear in Probab. Theory Relat. Fields

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Cited by in corpus (5)

• Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla \varphi$ Interface Model
• A quenched local limit theorem for stochastic flows
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Homogenization theory of random walks among deterministic conductances