Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 2, pp. 178--193

REFLECTING BROWNIAN MOTION IN A LIPSCHITZ DOMAIN AND A CONDITIONAL GAUGE THEOREM

By

S. RAMASUBRAMANIAN Indian Statistical Institute, Bangalore

SUMMARY. Let $\{P_{x}:x\in \overline{D}\}$ denote the reflecting Brownian motion in$\ D$ with normal reflection at the boundary where $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$. Let $% q(x)dx,c(x)d\sigma (x)$ belong to Kato class; consider the third boundary value problem for the operator $\left( \frac{1}{2}\Delta +q\right)$ in $D$ with boundary condition determined by $\left( \frac{\partial }{\partial n}% +c\right)$;(here $d\sigma$ denotes the surface area measure on $\partial D,$ and $n(\cdot )$ 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 $\zeta$ for $\{T_{t}\}$, we set \ $F(x,z)=\int_{0}^{\infty }\zeta (t,x,z)dt$, \ \ $x,z\in \overline{D}.$ It is proved that if \ $F(x,z)<\infty$ for some $x, z\in \overline{D}$ then the gauge $G$ is a bounded continuous function on $\overline{D}$, and that $% F(\cdot, \cdot )$ 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.

AMS (2000) subject classification. Primary 60J65; secondary 60J45.

Key words and phrases. Reflecting Brownian motion with normal reflection, Lipschitz domain, third boundary value problem, generalized Kato class, Feynman-Kac semigroup, gauge function, integral kernel, conditioned Brownian motion.

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