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.
Highlights
One of the most studied models for random walks in random environments is the random conductance model (RCM)
Objectives of particular interest are homogenisation results such as invariance principles or stronger local limit theorems for the associated heat kernel
In [5] a local limit theorem has been proven for random walks under general ergodic conductances satisfying a certain moment condition
Summary
One of the most studied models for random walks in random environments is the random conductance model (RCM). In this paper we significantly relax these assumptions and show a quenched local limit theorem for the dynamic RCM with degenerate spacetime ergodic conductances that only need to satisfy a moment condition. It turns out that the De Giorgi’s iteration method performs far more efficiently for proving Hölder regularity of time-space harmonic functions. On one hand, it avoids the need for a parabolic Harnack inequality in contrast to the arguments in [5,24], and it makes the proof significantly simpler and shorter
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
