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

Abstract

The harmonic-measure distribution function, or h -function, of a planar domain Ω ⊂ C with respect to a basepoint z 0 ∈ Ω is a signature that profiles the behaviour in Ω 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 ] ∪ [ 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.