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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 3, Pages 557–563 (Mi tvp2668)

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:
$$\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}=0,\quad\frac{\partial h}{\partial v}|_{L\setminus\Delta}=0$$
and prove that each non-negative solution of this problem may be uniquely represented in the form ($*$).

Full text: PDF file (502 kB)

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

Bibliographic databases:

Citation: A. L. Rozental', “On conditional brownian motions with oblique reflection, which correspond to inaccessible singular points”, Teor. Veroyatnost. i Primenen., 17:3 (1972), 557–563; Theory Probab. Appl., 17:3 (1973), 528–535

Citation in format AMSBIB
\Bibitem{Roz72} \by A.~L.~Rozental' \paper On conditional brownian motions with oblique reflection, which correspond to inaccessible singular points \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 3 \pages 557--563 \mathnet{http://mi.mathnet.ru/tvp2668} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=307366} \zmath{https://zbmath.org/?q=an:0285.60066} \transl \jour Theory Probab. Appl. \yr 1973 \vol 17 \issue 3 \pages 528--535 \crossref{https://doi.org/10.1137/1117063}