Spinning Brownian motion II

Abstract

In a previous paper, we established strong existence and uniqueness for a reflected diffusion $(X,S)$ with values in $\bar D\times \mathbbm{R}^p$, solving the following pair of stochastic differential equations: $$ dX_t = \sigma(X_t)dB_t + \vec{\gamma}(X_t,S_t)dL_t, $$ $$ dS_t = [\vec{g}(X_t) - S_t ] dL_t. $$ Here $L_t$ is the boundary local time of $X_t$, and $\vec{\gamma}$ points uniformly into the domain $D$. The process $(X,S)$ is called spinning Brownian motion (sBm). In this article, we prove uniqueness of the stationary distribution of spinning Brownian motion by studying excursions away from the boundary and finding and exit system for these excursions in terms of the local time $L_t$ and an excursion measure. The exit system is used to obtain a conditioned version of sBm by patching "almost" independent excursions, from which we can deduce that the stationary distribution of sBm is nowhere singular to Lebesgue measure. We also show that the support of the stationary distribution is a convex set determined by the vector field $\vec{g}$ and provide some conjectures on the marginal of $S$ through examples.