Abstract

We can encode information about the shape of the boundary of a planar region Ω by keeping track of the exit locations of Brownian particles released from a basepoint z0 in Ω. Tracking for each r > 0 the probability that the Brownian particle first exits the region Ω via a boundary point of Ω within the closed disc B(z0, r) of radius r about z0 yields the (“global”) h-function or harmonic-measure distribution function h® = hΩ,z0®. By contrast, tracking the probability that the Brownian particle first exits the closed disc B(z0, r) via a point on the boundary ∂Ω of Ω, as opposed to via a point on ∂B(z0, r)\∂Ω, yields the (“local”) g-function g® = gΩ,z0®. I will report the recently calculated h-functions and g-functions of several simply connected regions Ω, as well as the h-functions of multiply connected domains whose boundary components may be circles, intervals or polygons. Tools used include conformal mapping, properties of harmonic measure, and the prime function. This is joint work with Arunmaran Mahenthiram, Byron Walden and Christopher Green.