On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities

Abstract

\S 1. Introduction. Let $D$ be an arbitrary bounded domain of the N-dimensional Euclidean space $R^{N}(N\geqq 1)$ . A function $G_{\alpha}(x, y)$ defined for $\alpha>0,$ $x,$ $y\in D,$ $x\neq y$ will be called a resolvent density on $D$ , if it satisfies that, $G_{\alpha}(x, y)\geqq 0,$ $\alpha\int_{D}G_{\alpha}(x, z)dz\leqq 1$ and $G_{\alpha}(x, y)-G_{\beta}(x, y)+(\alpha-\beta)\int_{D}G_{\alpha}(x, z)G_{\beta}(z, y)dz=0$ for all $\alpha>0,$ $\beta>0$ and $x,$ $y$ $\in D,$ $\chi\neq y$ . Denote by $G_{a}^{0}(x, y)$ the resolvent density corresponding to the absorbing barrier Brownian motion on $D^{1)}$ .