An Inverse Balayage problem for Brownian Motion

Abstract

Let $B$ be a standard $n$-dimensional Brownian motion, let $A$ be compact and let $\nu$ be a probability measure on $\partial A$. We treat the following inverse exit problem: describe the set $M(\nu)$ of all probability measures $\mu$ on $A$ such that $P^\mu\{B(T)\in \cdot\} = \nu(\cdot)$, where $T$ is the time of first exit from $A$. Elements of $M(\nu)$ are characterized in terms of integrals of harmonic functions with respect to them. For $n = 1$, extreme points of $M(\nu)$ are computed in closed form; for $n \geqslant 2$, extreme points of $M(\nu)$ are characterized. Geophysical and potential-theoretic aspects of the problem are discussed.