Towards computing the harmonic-measure distribution function for the middle-thirds Cantor set

Abstract

This paper is concerned with the numerical computation of the harmonic-measure distribution function, or $h$-function for short, associated with a particular planar domain. This function describes the hitting probability of a Brownian walker released from some point with the boundary of the domain. We use a fast and accurate boundary integral method for the numerical calculation of the $h$-functions for symmetric multiply connected slit domains with high connectivity. In view of the fact that the middle-thirds Cantor set $\mathcal{C}$ is a special limiting case of these slit domains, the proposed method is used to approximate the $h$-function for the set $\mathcal{C}$. We also numerically analyze some asymptotic features of the calculated $h$-functions.