Spinning Brownian motion

TL;DR

This paper establishes the existence, uniqueness, and stationary distribution properties of a novel spinning Brownian motion process in a bounded domain, where reflection direction depends on a spin parameter that updates at the boundary.

Contribution

It introduces a new class of reflection processes with spin-dependent reflection directions and proves their well-posedness and stationary distribution uniqueness.

Findings

Proved strong existence and uniqueness of the process.

Established the stationary distribution's uniqueness.

Provided examples of stationary distributions.

Abstract

We prove strong existence and uniqueness for a reflection process $X$ in a smooth, bounded domain $D$ that behaves like obliquely-reflected-Brownian-motion, except that the direction of reflection depends on a (spin) parameter $S$, which only changes when $X$ is on the boundary of $D$ according to a physical rule. The process $(X,S)$ is a degenerate diffusion. We show uniqueness of the stationary distribution by using techniques based on excursions of $X$ from $\partial D$, and an associated exit system. We also show that the process admits a submartingale formulation and use related results to show examples of the stationary distribution.