Harmonic-measure distribution functions of multiply connected domains with various geometries

The harmonic-measure distribution functions, or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -functions, associated with several classes of planar multiply connected domains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:math> and basepoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>Ω</mml:mi> </mml:math> locations are the principal objects of consideration in this paper. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -function with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> encodes the probability that a particle undergoing Brownian motion in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> first collides with the boundary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:math> within a certain distance from the basepoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -functions of multiply connected symmetrical rectilinear slit domains. In this paper, we generalize and extend the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -functions. Our examples yield solutions to instances of a variant of the conformal Skorokhod embedding problem.