Uniqueness of locally symmetric Brownian motion on Laakso spaces

TL;DR

This paper proves the existence and uniqueness of a locally symmetric Brownian motion on Laakso spaces, extending previous work on heat kernel estimates and diffusion processes on fractal-like spaces.

Contribution

It establishes the first rigorous proof of a unique local symmetry invariant diffusion on Laakso spaces, building on and extending prior analytical frameworks.

Findings

Existence of a local symmetry invariant diffusion on Laakso spaces

Uniqueness of the diffusion process under the given symmetry

Heat kernel estimates supporting the diffusion's properties

Abstract

We take the spaces introduced by Laakso in 2000 and building on the work of Barlow, Bass, Kumagai, and Teplyaev prove the existence and uniqueness of a local symmetry invariant diffusion via heat kernel estimates. This work also builds upon works of Cheeger, Barlow and Bass, as well as the author.