Quenched Local Limit Theorem for Random Walks Among Time-Dependent Ergodic Degenerate Weights

We establish a quenched local central limit theorem for the dynamic random conductance model on Zd only assuming ergodicity with respect to spacetime shifts and a moment condition. As a key analytic ingredient we show H¨older 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.