Boundary behavior of a constrained Brownian motion between reflecting-repellent walls

TL;DR

This paper studies the boundary behavior of constrained Brownian motions within polyhedral domains, establishing conditions under which the process avoids non-smooth boundary parts and analyzing specific cases like Weyl chambers.

Contribution

It provides a unified framework using stochastic variational inequalities to analyze boundary hitting properties of reflected-repelled Brownian motions in polyhedral domains.

Findings

Reflected-repelled Brownian motion does not hit non-smooth boundary parts in polyhedra.

A sufficient condition for avoiding boundary faces is derived from the one-dimensional case.

Complete analysis of boundary attainability is given for radial Dunkl processes.

Abstract

Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and (or) singular repellent drift. When the domain is a polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for non-hitting a face of the polyhedron is derived from the one-dimensional case. A complete answer to the question of attainability of the walls of the Weyl chamber may be given for a radial Dunkl process.