Short Communications

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

A. L. Rozental'

Moscow

Abstract: 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 ($*$).

Received: 29.10.1970

 English version: Theory of Probability and its Applications, 1973, 17:3, 528–535

Bibliographic databases:

• 
•