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

Abstract

A Brownian particle released from a point $z_0$ in a two-dimensional region Ω will 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 r of the starting point $z_0$, for each value of r. Putting together these probabilities for all values of r gives a function $h\left(r\right)$ called the harmonic-measure distribution function or h-function of Ω with respect to $z_0$. 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-Klein prime function. Furthermore, we have validated our results by simulating the random motion of Brownian particles in the regions mentioned above.