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

R. V. Ambartzumian

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 539-543

Résumé

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.