Lower bounds for transition probabilities on graphs

TL;DR

This paper establishes lower bounds for transition probabilities and exit times of random walks on weighted graphs, linking these bounds to the elliptic Harnack inequality and providing new off-diagonal estimates.

Contribution

It introduces conditions for lower bounds on transition probabilities and exit times, demonstrating their connection to the elliptic Harnack inequality on weighted graphs.

Findings

Lower bounds for exit times derived from elliptic Harnack inequality

Off-diagonal lower bounds for transition probabilities established

Conditions for upper and lower estimates of distribution of exit times

Abstract

The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.