Harmonic measure distributions of planar domains: a survey

Given a domain \(\Omega \) in the complex plane and a basepoint \(z_0\in \Omega \), the harmonic measure distribution function \(h:(0,\infty )\rightarrow [0,1]\) of the pair \((\Omega ,z_0)\) maps each radius \(r > 0\) to the harmonic measure of the part of the boundary \(\partial \Omega \) within distance r of the basepoint. This function was first introduced by Walden and Ward, inspired by a question posed by Stephenson, as a signature that encodes information about the geometry of \(\Omega \). It has subsequently been studied in various works. Two main goals of harmonic measure distribution studies are (1) to understand precisely what can be determined about a domain from its h-function, and (2) given a function \(f:(0,\infty )\rightarrow [0,1]\), to determine whether there exists a pair \((\Omega ,z_0)\) that has f as its h-function. In this survey paper, we present key examples of h-functions and summarize results related to these two goals. In particular, we discuss what is known about uniqueness of domains that generate h-functions, necessary conditions and sufficient conditions for a function to be an h-function, asymptotic behavior of h-functions, and convergence results involving h-functions. We also highlight current open problems.