Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains

The harmonic-measure distribution function, or h-function, of a planar domain Omega subset of C with respect to a basepoint z(0) is an element of Omega is a signature that profiles the behaviour in Omega of a Brownian particle starting from z(0). Explicit calculation of h-functions for a wide array of simply connected domains using conformal mapping techniques has allowed many rich connections to be made between the geometry of the domain and the behaviour of its h-function. Until now, almost all h-function computations have been confined to simply connected domains. In this work, we apply the theory of the Schottky-Klein prime function to explicitly compute the h-function of the doubly connected slit domain C\([-1/2, -1/6] boolean OR [1/6, 1 /2]). In view of the connection between the middle-thirds Cantor set and highly multiply connected symmetric slit domains, we then extend our methodology to explicitly construct the h-functions associated with symmetric slit domains of arbitrary even connectivity. To highlight both the versatility and generality of our results, we graph the h-functions associated with quadruply and octuply connected slit domains.

Automated summary

The harmonic-measure distribution function, or h-function, is crucial for understanding the behavior of Brownian particles in planar domains. Previous research has focused on simply connected domains, but this study introduces a new calculation method and extends it to doubly and multiply connected domains, resulting in satisfactory results.