Yukawa potential, panharmonic measure and Brownian motion

Antti Rasila*, Tommi Sottinen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure, which is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon-Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm.

Original languageEnglish
Article number28
Issue number2
StatePublished - 1 May 2018
Externally publishedYes


  • Bessel functions
  • Brownian motion
  • Duffin correspondence
  • Harmonic measure
  • Monte Carlo simulation
  • Panharmonic measure
  • Potential theory
  • Walk-on-spheres algorithm
  • Yukawa equation


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