Harmonic-measure distribution functions of multiply connected domains with various geometries
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2025
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Green, C.C.
Mahenthiram, A.
Ward, L.A.
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Journal article
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2025; 481(2321, article no. 20240392):1-29
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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.
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Copyright 2025 The Authors