Heat kernel fluctuations for a resistance form with non-uniform volume growth

TL;DR

This paper investigates heat kernel behavior on resistance form spaces with non-uniform volume growth, revealing how volume fluctuations influence heat kernel estimates, applicable to fractals and dendrites.

Contribution

It provides new global and local heat kernel estimates accounting for non-uniform volume fluctuations on resistance form spaces.

Findings

Heat kernel fluctuations are linked to volume growth irregularities.

Non-trivial volume fluctuations cause corresponding heat kernel fluctuations.

Bounds are established for both on-diagonal and off-diagonal heat kernel parts.

Abstract

In this article, we consider the problem of estimating the heat kernel on measure-metric spaces equipped with a resistance form. Such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set. In this situation, it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on-diagonal part of the heat kernel is completely determined by this rate of volume growth. However, recent results have shown that for certain random fractal sets, there are global and local (point-wise) fluctuations in the volume as $r\rightarrow 0$ and so these uniform results do not apply. Motivated by these examples, we present global and local on-diagonal heat kernel estimates when the volume growth is not uniform, and demonstrate that when the volume fluctuations are non-trivial, there will be nontrivial fluctuations of the same order (up to exponents) in the short-time heat kernel asymptotics. We also provide bounds for the off-diagonal part of the heat kernel. These results apply to deterministic and random self-similar fractals, and metric space dendrites (the topological analogues of graph trees).