Estimates of the transition densities for the reflected Brownian motion on simple nested fractals

TL;DR

This paper derives precise two-sided estimates for transition densities of reflected Brownian motion on simple nested fractals, comparing them with free Brownian motion densities, enhancing understanding of stochastic processes on fractal structures.

Contribution

It provides the first sharp two-sided estimates for transition densities of reflected Brownian motion on simple nested fractals, extending existing results to a broader class of fractals.

Findings

Sharp two-sided estimates for $g_M(t,x,y)$ and $g_M(t,x,y)-g(t,x,y)$

Comparison between reflected and free Brownian motion densities

Applicable to a large class of planar simple nested fractals

Abstract

We give sharp two-sided estimates for the functions $g_M(t,x,y)$ and $g_M(t,x,y)-g(t,x,y)$, where $g_M(t,x,y)$ are the transition probability densities of the reflected Brownian motion on a $M$-complex of size $M \in \mathbb{Z}$ of an unbounded planar simple nested fractal and $g(t,x,y)$ are the transition probability densities of the `free' Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.