Reflected Brownian motion on simple nested fractals

TL;DR

This paper establishes the existence and properties of reflected Brownian motion on simple nested fractals, providing geometric conditions and constructing transition densities with regularity properties.

Contribution

It introduces a method to define reflected Brownian motion on nested fractals and derives conditions for its well-posedness and regularity.

Findings

Existence of reflected Brownian motion on nested fractals.

Construction of transition probability densities with regularity.

Identification of geometric conditions for well-defined reflection.

Abstract

We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions on the fractal under which this projection can be well defined. They are illustrated by various specific examples. We first construct a proper version of the transition probability densities for reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman-Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property