Gaussian upper bounds for heat kernels of continuous time simple random walks

TL;DR

This paper establishes Gaussian upper bounds for heat kernels of continuous time simple random walks on weighted graphs, introducing a new adapted metric and extending bounds to long-range non-Gaussian cases, with applications to random environments.

Contribution

The paper derives Gaussian upper bounds for heat kernels using a novel metric adapted to the random walk, extending classical results to more general settings.

Findings

Gaussian upper bounds for heat kernels are established.

A new metric adapted to the random walk is introduced.

Long-range non-Gaussian bounds are also derived.

Abstract

We consider continuous time simple random walks with arbitrary speed measure $\theta$ on infinite weighted graphs. Write $p_t(x,y)$ for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points $x_1,x_2$, we obtain a Gaussian upper bound for $p_t(x_1,x_2)$. The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.