On conditional brownian motions with oblique reflection, which correspond to inaccessible singular points

A. L. Rozental'

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 557-563

Résumé

Let $D$ be a two-dimensional domain bounded by a smooth contour $L$, $v(z)$ be a vector field at points of $L$ directed inward $D$, $\Delta$ be a finite set of discontinuity points of $v(z)$ and $X$ be a Brownian motion in $D$ with reflection away from $L\setminus\Delta$ in the direction of $v(z)$. We construct subprocesses of $X$ corresponding to inaccessible points of $\Delta$ and investigate the behaviour of their trajectories. This construction enables us to investigate the boundary value problem: $$ \frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}=0,\quad\frac{\partial h}{\partial v}\bigg|_{L\setminus\Delta}=0 $$ and prove that each non-negative solution of this problem may be uniquely represented in the form ($*$).