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Article

Yukawa Potential, Panharmonic Measure and Brownian Motion

1
Department of Mathematics and Systems Analysis, School of Science, Aalto University, P.O. Box 1100, FIN-00076 Aalto, Finland
2
Department of Mathematics and Statistics, Faculty of Technology, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland
*
Author to whom correspondence should be addressed.
Axioms 2018, 7(2), 28; https://doi.org/10.3390/axioms7020028
Received: 9 April 2018 / Revised: 24 April 2018 / Accepted: 25 April 2018 / Published: 1 May 2018
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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. View Full-Text
Keywords: potential theory; Brownian motion; Duffin correspondence; harmonic measure; Bessel functions; Monte Carlo simulation; panharmonic measure; walk-on-spheres algorithm; Yukawa equation potential theory; Brownian motion; Duffin correspondence; harmonic measure; Bessel functions; Monte Carlo simulation; panharmonic measure; walk-on-spheres algorithm; Yukawa equation
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MDPI and ACS Style

Rasila, A.; Sottinen, T. Yukawa Potential, Panharmonic Measure and Brownian Motion. Axioms 2018, 7, 28. https://doi.org/10.3390/axioms7020028

AMA Style

Rasila A, Sottinen T. Yukawa Potential, Panharmonic Measure and Brownian Motion. Axioms. 2018; 7(2):28. https://doi.org/10.3390/axioms7020028

Chicago/Turabian Style

Rasila, Antti, and Tommi Sottinen. 2018. "Yukawa Potential, Panharmonic Measure and Brownian Motion" Axioms 7, no. 2: 28. https://doi.org/10.3390/axioms7020028

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