Path decompositions of perturbed reflecting Brownian motions

TL;DR

This paper investigates the path structure of perturbed reflecting Brownian motions at key hitting times and minima, utilizing loop soup techniques and excursion laws to deepen understanding of their stochastic properties.

Contribution

It introduces a novel analysis of PRBM path decompositions at hitting times and minima using loop soups and excursion measures, building on recent theoretical developments.

Findings

Characterization of PRBM excursions above past minima

Connection between PRBM excursions and Brownian loop measures

New insights into the stochastic structure of perturbed reflecting Brownian motions

Abstract

We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by Lawler and Werner [10] and Le Jan [13]-[14], in particular on a result discovered by Lupu [15] identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.