The volume and time comparison principle and transition probability estimates for random walks

TL;DR

This paper establishes conditions for transition probability estimates of random walks on weighted graphs, focusing on volume, mean exit time, and their implications for uniformity and estimates.

Contribution

It provides necessary and sufficient conditions for transition probability estimates, extending understanding to non-uniform cases on various graph structures.

Findings

Upper estimates without uniformity assumptions

Two-sided estimates under mean exit time uniformity

Application to integer lattices and fractal graphs

Abstract

This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball is independent of the centre, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if uniformity in the space assumed only for the mean exit time.