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

In this article, we consider the problem of estimating the heatkernel on measure-metric spaces equipped with a resistance form.Such spaces admit a corresponding resistance metric that reflectsthe conductivity properties of the set. In this situation, ithas been proved that when there is uniform polynomial volumegrowth with respect to the resistance metric the behaviour ofthe on-diagonal part of the heat kernel is completely determinedby this rate of volume growth. However, recent results haveshown that for certain random fractal sets, there are globaland local (point-wise) fluctuations in the volume as r 0 andso these uniform results do not apply. Motivated by these examples,we present global and local on-diagonal heat kernel estimateswhen the volume growth is not uniform, and demonstrate thatwhen the volume fluctuations are non-trivial, there will benon-trivial fluctuations of the same order (up to exponents)in the short-time heat kernel asymptotics. We also provide boundsfor the off-diagonal part of the heat kernel. These resultsapply to deterministic and random self-similar fractals, andmetric space dendrites (the topological analogues of graph trees).