Let D be a planar domain and z be a point in D . The harmonic measure distribution function h D z ( 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 . This paper is concerned with the following inverse problem: given a suitable function h , does there exist a domain D such that h = h D 0 ? To answer this, we first extend the concept of the h -function of a domain to one of a stopping time τ . Then, using the conformal invariance of Brownian motion, we solve the inverse problem for the h -functions of stopping times. In many cases, the stopping time is the exit time of a domain D , and in these cases D solves the original inverse problem. It also can happen that the stopping time constructed is not the exit time of a plane domain, although it can still be interpreted as the exit time of a domain on a Riemann surface; in these cases, the original inverse problem is not completely solved. We will illustrate our methods with a large family of examples.
Markowsky et al. (2026) studied this question.