Stationary distribution for spinning reflecting diffusions

Abstract

Thesis (Ph.D.)--University of Washington, 2012This dissertation studies two different types of interaction of diffusion processes with the boundary of a domain $DsubRR^n$, which is assumed to be bounded, and of class $C^2(RR^n)$. The first process that is studied is obliquely reflected Brownian motion, and it is constructed as the unique Hunt process $X$ properly associated with the following Dirichlet form: \begin{align} \label{eq:abs_df} \tag{1} \E(u,v) = \frac12\int_D \nabla u\nabla(u\rho) dx + \frac12\int_D \nabla u \cdot\vec{\tau}\ v\ \rho(x)\sigma(dx), \end{align} where \vec\tau:\partial D\to\RR^n is tangential to $\partial D$, and $u,v$ belong to the Sobolev space $W^{1,2}(D)$. The reference measure $\rho(x)dx$ is assumed to be given by a harmonic function $\rho$ whose gradient $\nabla\rho$ is uniformly bounded. It is shown that such process $X$ admits a Skorohod decomposition \begin{align} \label{eq:abs_skorohod} \tag{2} dX_t = dB_t + [\vec{n}+\vec\tau](X_t)dL_t. \end{align} Moreover, we show that the unique stationary distribution of $X$ is the measure given by $\rho(x)dx$. In the second part of the dissertation, we present a new reflection process $X_t$ in a bounded domain $D$ of class C^2(\RR^n) that behaves very much like oblique reflected Brownian motion, except that the directions of reflection depend on an external parameter $S_t$ called spin. The spin is allowed to change only when the process $X_t$ is on the boundary of $D$. The pair $(X,S)$ is called spinning Brownian motion and is found as the unique strong solution to the following stochastic differential equation: % %Let $D\subseteq\mathbb{R}^n$ be an open $C^2$ domain, and let $B_t$ be a $n$-dimensional Brownian motion. A pair $(X_t,S_t)$ is called spinning Brownian motion (sBM) if it solves the following stochastic differential equation \begin{align} \label{eq:abs_sbm} \tag{3} \left\{ \begin{array}{rl} dX_t &= \sigma(X_t)dB_t + \vec{n}(X_t)dL_t + \vec \tau (X_t,S_t)dL_t \\ dS_t &= \spar{\vec{g}(X_t) - S_t } dL_t \end{array} \right. \end{align} where $L_t$ is the local time process of $X_t$, $\vec{n}$ is the interior unit normal to $\partial D$, and $\vec\tau$ is a vector field perpendicular to $\hat n$. The function $\sigma(\cdot)$ is a non-degenerate $(n\times n)$-matrix valued function, and $\vec{\tau}(\cdot)$ and $\vec g(\cdot)$ are Lipschitz bounded vector fields. % We prove that a unique strong solution to \eqref{eq:abs_sbm} exists as the limit of a family of processes (X^\e,S^\e) that satisfy an equation like \eqref{eq:abs_sbm}, but in which the spin component $dS$ has a noise \e dW. With this added noise, the process (X^\e,S^\e) is an obliquely reflected Brownian motion in an unbounded domain. % It is also shown that spinning Brownian motion has a unique stationary distribution. The main tool of the proof is excursion theory, and an identification of the Local time of $X_t$ as a component of an exist system for $X_t$