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

 Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 539–543 (Mi tvp551)

Short Communications

On an application of the connection between the Brownian motion and the Dirichlet problem

R. V. Ambartzumian

Armenian Academy of Sciences, Calculating Center

Abstract: It is known that in the domain $G$ with a piecewise smooth boundary $\Gamma$ the solution $f(P)$ of the Dirichlet problem with continuous boundary values $f(S)$, $S\in\Gamma$ , can be represented in the form
$$f(P)=\int_\Gamma u(P,S)f(S) dS$$
where $u(P,S)$ is the probability density for a brownian particle to be absorbed at a point $S\in\Gamma$ starting from a point $P$ of the domain $G$ with the absorbing boundary $\Gamma$.
It is shown that the construction of the function $u_0(P,S)$ for a domain $G_0$ which splits into two non-intersecting domains $G_1$ and $G_2$ with common boundary points and with known functions $u_1(P,S)$ and $u_2(P,S)$ is reduced to solving some Fredholm integral equation of the second kind.
The uniqueness of the solution of this integral equation is proved.

Full text: PDF file (1104 kB)

English version:
Theory of Probability and its Applications, 1965, 10:3, 490–493

Bibliographic databases:

Citation: R. V. Ambartzumian, “On an application of the connection between the Brownian motion and the Dirichlet problem”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 539–543; Theory Probab. Appl., 10:3 (1965), 490–493

Citation in format AMSBIB
\Bibitem{Amb65} \by R.~V.~Ambartzumian \paper On an application of the connection between the Brownian motion and the Dirichlet problem \jour Teor. Veroyatnost. i Primenen. \yr 1965 \vol 10 \issue 3 \pages 539--543 \mathnet{http://mi.mathnet.ru/tvp551} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=189133} \zmath{https://zbmath.org/?q=an:0158.35604} \transl \jour Theory Probab. Appl. \yr 1965 \vol 10 \issue 3 \pages 490--493 \crossref{https://doi.org/10.1137/1110059}