Brownian Motion and Characterization of Domains (Open)

Summary: Given a plane domain $\Omega$ bounded by a regular Jordan curve $\Gamma$ and a Lebesgue measurable subset $E$ of $\Gamma$, let $P(E;z)$ denote the probability that a Brownian particle starting at $z \in \Omega$ hits by boundary $\Gamma$ for the first time at a point in $E$. The problem calls for a characterization of the domains such that for arbitrary $z \in \Omega$ and $0 < C < |\Gamma|$, the optimization problem $\sup\{P(E;z): |E| = C\}$ is solved by taking $E$ to be an appropriate single arc of $\Gamma$ with measure $C$.

Classification: Primary, differential equations; Secondary, PDE

Note from the Editor: Professor Berrone has kindly pointed out that a response to question (i) can be found in his paper Characterization of domains through families of measures, which was published in Demonstratio Mathematica, XXXVI, No. 2 (2003), pp. 313--328

Lucio R. Berrone
Universidad Nacional de Rosario

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