{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:42:09Z","timestamp":1760197329112,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,5,1]],"date-time":"2018-05-01T00:00:00Z","timestamp":1525132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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\u2013Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm.<\/jats:p>","DOI":"10.3390\/axioms7020028","type":"journal-article","created":{"date-parts":[[2018,5,3]],"date-time":"2018-05-03T03:20:27Z","timestamp":1525317627000},"page":"28","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Yukawa Potential, Panharmonic Measure and Brownian Motion"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3797-942X","authenticated-orcid":false,"given":"Antti","family":"Rasila","sequence":"first","affiliation":[{"name":"Department of Mathematics and Systems Analysis, School of Science, Aalto University, P.O. Box 1100, FIN-00076 Aalto, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9983-9708","authenticated-orcid":false,"given":"Tommi","family":"Sottinen","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Technology, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"511","DOI":"10.2307\/1971428","article-title":"Harmonic measure and arclength","volume":"132","author":"Bishop","year":"1990","journal-title":"Ann. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gehring, F.W., and Hag, K. (2012). The Ubiquitous Quasidisk, American Mathematical Society. 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