Path Decompositions 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 (Probab Theory Relat Fields 4:197–217, 2004) and Le Jan (Ann Probab 38:1280–1319, 2010; Markov Paths, Loops and Fields. École d’été Saint-Flour XXXVIII 2008. Lecture Notes in Mathematics vol 2026. Springer, Berlin, 2011), in particular on a result discovered by Lupu (Mém Soc Math Fr (N.S.) 158, 2018) identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.