Explicit formulas for heat kernels on diamond fractals

TL;DR

This paper derives explicit formulas for heat kernels on a family of fractals lacking volume doubling, enabling new analysis of heat propagation and quantum equations on complex spaces.

Contribution

It provides the first explicit pointwise formulas for heat kernels on certain non-volume doubling fractals, expanding analytical tools for these spaces.

Findings

Explicit heat kernel formulas derived for fractals

Uniform continuity estimates of the heat kernel obtained

Fundamental solution for Schrödinger equation expressed

Abstract

This paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a $(\mathbb{N}\times\mathbb{N})$-parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property. The formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free Schr\"odinger equation. The results also open up the possibility to approach infinite dimensional spaces based on this model.