An Inverse Balayage problem for Brownian Motion

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