Revista de la
Unión Matemática Argentina
New harmonic-measure distribution functions of some simply connected planar regions in the complex plane
Arunmaran Mahenthiram

Volume 68, no. 2 (2025), pp. 459–483    

Published online (final version): September 3, 2025

https://doi.org/10.33044/revuma.4038

Download PDF

Abstract

Consider a Brownian particle released from a fixed point $z_0$ in a region $\Omega$. The harmonic-measure distribution function, or $h$-function, $h(r)$, expresses the probability that the Brownian particle first hits the boundary $\partial\Omega$ of the region $\Omega$ within distance $r$ of $z_0$. In this paper, we compute the $h$-function of several new planar simply connected two-dimensional regions by using two different methods, both involving conformal maps. We also explain the asymptotic behaviour at certain values of $r$ where two different regimes meet. Moreover, for some regions, we examine how the behaviour of $h(r)$ changes when part of the boundary changes.

References

  1. L. Greco, Brownian motion in the complex plane, 2014, Kenyon Summer Science Scholars Program. Paper 256. Available at https://digital.kenyon.edu/summerscienceprogram/256.
  2. C. C. Green, M. A. Snipes, L. A. Ward, and D. G. Crowdy, Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains, Proc. R. Soc. A 478 no. 2259 (2022), article no. 20210832.  DOI  MR  Zbl
  3. S. Kakutani, Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo 20 no. 10 (1944), 706–714.  DOI  MR  Zbl
  4. A. Mahenthiram, Harmonic-measure distribution functions, and related functions, for simply connected and multiply connected two-dimensional regions, Ph.D. thesis, University of South Australia, 2022. Available at https://hdl.handle.net/11541.2/33203.
  5. A. Mahenthiram, Computing $h$-functions of some planar simply connected two-dimensional regions, Taiwanese J. Math. 27 no. 5 (2023), 931–952.  DOI  MR  Zbl
  6. S. Matsumoto, What do random walks tell us about the shapes of regions in the complex plane?, Poster, Harvey Mudd College Celebration of Student Research, 2011.
  7. M. A. Snipes and L. A. Ward, Harmonic measure distributions of planar domains: A survey, J. Anal. 24 no. 2 (2016), 293–330.  DOI  MR  Zbl
  8. B. L. Walden and L. A. Ward, Distributions of harmonic measure for planar domains, in 16th Rolf Nevanlinna Colloquium (Joensuu, 1995), de Gruyter, Berlin, 1996, pp. 289–299.  MR  Zbl