Yukawa potential, panharmonic measure and Brownian motion

Antti Rasila; Tommi Sottinen

doi:10.3390/axioms7020028

Yukawa potential, panharmonic measure and Brownian motion

Antti Rasila^*, Tommi Sottinen

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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.

Original language	English
Article number	28
Journal	Axioms
Volume	7
Issue number	2
DOIs	https://doi.org/10.3390/axioms7020028
State	Published - 1 May 2018
Externally published	Yes

Keywords

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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title = "Yukawa potential, panharmonic measure and Brownian motion",

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.",

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AU - Sottinen, Tommi

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AB - 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.

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KW - Brownian motion

KW - Duffin correspondence

KW - Harmonic measure

KW - Monte Carlo simulation

KW - Panharmonic measure

KW - Potential theory

KW - Walk-on-spheres algorithm

KW - Yukawa equation

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DO - 10.3390/axioms7020028

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SN - 2075-1680

VL - 7

JO - Axioms

JF - Axioms

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ER -

Yukawa potential, panharmonic measure and Brownian motion

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