Abstract:
The harmonic-measure distribution functions,
or h-functions, associated with several classes of
planar multiply connected domains Ω ⊂C and
basepoint z0 ∈Ω locations are the principal objects
of consideration in this paper. The h-function with
respect to Ω and z0 encodes the probability that
a particle undergoing Brownian motion in Ω first
collides with the boundary ∂Ω within a certain
distance from the basepoint z0 where it was initially
released. Recently, Green et al. (Green et al. 2022 Proc.
R. Soc. A 478, 20210832. (doi:10.1098/rspa.2021.0832))
derived the first explicit formulae for the h-functions
of multiply connected symmetrical rectilinear slit
domains. In this paper, we generalize and extend the
h-function calculations in Green et al. by considering
various types of planar domains—those whose
boundaries consist of either rectilinear slits or
circles—as well as different locations of the basepoint.
Throughout, we make judicious use of the Schottky-
Klein prime function and its associated theory to
derive analytical formulae for the h-functions. Our
examples yield solutions to instances of a variant of
the conformal Skorokhod embedding problem.
