Computing harmonic-measure distribution functions of some multiply connected unbounded planar domains

Abstract

A Brownian particle released from a point z0in a two-dimensional region Ωwill move around randomly until it eventually hits the boundary of Ω. We are interested in the probability that it hits the boundary somewhere within distance rof the starting point z0, for each value of r. Putting together these probabilities for all values of rgives a function h(r)called the harmonic-measure distribution function or h-function of Ωwith respect to z0. This h-function encodes information about the shape of the boundary of Ω. In this paper, we compute the h-functions for some multiply connected planar regions whose boundary consists of collinear unequal slits or dissimilar discs. The key tool we use for computing these h-functions is a special function, called the Schottky-Kleinprime function. Furthermore, we have validated our results by simulating the random motion of Brownian particles in the regions mentioned above.