Next Article in Journal
A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings
Next Article in Special Issue
Pre-Metric Spaces Along with Different Types of Triangle Inequalities
Previous Article in Journal / Special Issue
Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
Article Menu

Export Article

Open AccessArticle
Axioms 2018, 7(2), 28; https://doi.org/10.3390/axioms7020028

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.
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)
View Full-Text   |   Download PDF [281 KB, uploaded 1 May 2018]   |  

Abstract

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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Rasila, A.; Sottinen, T. Yukawa Potential, Panharmonic Measure and Brownian Motion. Axioms 2018, 7, 28.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top