| This dissertation studies two different types of interaction of diffusion processes with the boundary of a domain D ⊆ Rn , which is assumed to be bounded, and of class C 2( Rn ). 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: Eu,v= 12D1u1 updx+12 D1u˙t&ar; vrx sdx, 1 where t&ar;:6 D→Rn is tangential to ∂D, and u, v belong to the Sobolev space W1,2( D). The reference measure rho(x)dx is assumed to be given by a harmonic function rho whose gradient ∇rho is uniformly bounded. It is shown that such process X admits a Skorohod decomposition dXt=dBt+n&ar; +t&ar; XtdLt. 2 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 Xt in a bounded domain D of class C2( Rn ) that behaves very much like oblique reflected Brownian motion, except that the directions of reflection depend on an external parameter St called spin. The spin is allowed to change only when the process Xt 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: dXt=s&parl0;Xt &parr0;dBt+n&ar;&parl0;X t&parr0;dLt+t&ar; &parl0;Xt,St&parr0;dLt dSt=&sqbl0;g&ar; &parl0;Xt&parr0;-St&sqbr0;dLt 3 where Lt is the local time process of Xt, n&ar; is the interior unit normal to ∂D, and t&ar; is a vector field perpendicular to nˆ. The function sigma(·) is a non-degenerate (n x n)-matrix valued function, and t&ar; (·) and g&ar; (·) are Lipschitz bounded vector fields. We prove that a unique strong solution to (3) exists as the limit of a family of processes ( Xepsilon, Sepsilon) that satisfy an equation like (3), but in which the spin component dS has a noise epsilondW. With this added noise, the process (Xepsilon, Sepsilon ) 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 Xt as a component of an exist system for Xt. |
