Heat kernel analysis on diamond fractals

TL;DR

This paper analyzes the heat kernel on a family of fractal-like spaces lacking volume doubling, providing bounds, regularity results, and applications to diffusion convergence.

Contribution

It offers the first detailed heat kernel analysis on these specific fractal spaces, including bounds and functional inequalities, despite the absence of volume doubling.

Findings

Uniform heat kernel bounds established

Lipschitz continuity of the heat kernel proved

Functional inequalities derived for diffusion analysis

Abstract

This paper presents a detailed analysis of the heat kernel on an $(\mathbb{N}\times\mathbb{N})$-parameter family of compact metric measure spaces, which do not satisfy the volume doubling property. In particular, uniform bounds of the heat kernel and its Lipschitz continuity, as well as the continuity of the corresponding heat semigroup are studied; a specific example is presented revealing a logarithmic correction. The estimates are further applied to derive several functional inequalities of interest in describing the convergence to equilibrium of the diffusion process.