A method of solution for the inverse problem for h-functions of planar Brownian motion

2024/12/07 by Markowsky, Greg, McDonald, Clayton
#Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.05764

Abstract

Given a planar domain D, the harmonic measure distribution function h_D(r), with base point z, is the harmonic measure with pole at z of the parts of the boundary which are within a distance r of z. Equivalently it is the probability Brownian motion started from z first strikes the boundary within a distance r from z. We call h_D the h-function of D, this function captures geometrical aspects of the domain, such as connectivity, or curvature of the boundary. This paper is concerned with the inverse problem: given a suitable function h, does there exist a domain D such that h = h_D? To answer this, we first extend the concept of a h-function of a domain to one of a stopping time τ . By using the conformal invariance of Brownian motion we solve the inverse problem for that of a stopping time. The associated stopping time will be the projection of a hitting time of the real line. If this projection corresponds to the hitting time of a domain D, then this technique solves the original inverse problem. We have found a large family of examples such that the associated stopping time is that of a hitting time.

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