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 ${\mathbb{Z}}^d$ only assuming ergodicity with respect to space-time 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.