Reflecting Brownian motion in a lipschitz domain and a conditional Gauge theorem

Abstract

Let $\{P_{x}\,\colon x\in \overline{D}\}$ denote the reflecting Brownian motion in D with normal reflection at the boundary where D is a bounded Lipschitz domain in ${\Bbb R}^{d}$. Let $q(x)dx,c(x)d\sigma (x)$ belong to Kato class; consider the third boundary value problem for the operator $(\frac{1}{2}\Delta +q)$ in D with boundary condition determined by $\left(\frac{\partial}{\partial n}+c\right)$; (here ds denotes the surface area measure on ∂D, and n(·) the inward normal). Let $\{T_{t}\}$ denote the corresponding Feynman-Kac semigroup and G the gauge function. After indicating a way of getting the integral kernel ζ for $\{T_{t}\}$, we set F(x, z) = $\int_{0}^{\infty}\zeta (t,x,z)dt$, x, z ε $\overline{D}$. It is proved that if F(x, z) < 8 for some x, z ε $\overline{D}$ then the gauge G is a bounded continuous function on $\overline{D}$, and that F(·, ·) is finite and continuous on $\{x\neq z\}$. A connection between F and conditioned Brownian motion is given; a consequence is that if the gauge for the third boundary value problem is finite then so is the gauge for the Dirichlet problem.