Reflections at infinity of time changed RBMs on a domain with Liouville branches

TL;DR

This paper studies the extensions of time-changed reflecting Brownian motions in unbounded domains with Liouville branches, characterizing possible boundary behaviors and associated Dirichlet spaces.

Contribution

It provides a classification of symmetric conservative diffusion extensions of time-changed RBMs on domains with Liouville branches, linking extensions to partitions of domain ends.

Findings

Finite number of diffusion extensions characterized by domain ends.

Extended Dirichlet spaces are subspaces of BL(D) involving Sobolev spaces and approaching probabilities.

Extensions are independent of the specific time change used.

Abstract

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset {\mathbb R}^d$ with $N$ number of Liouville branches. We consider a diffusion $X$ on $\overline D$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space ${\rm BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.