Walk-on-Cubes Monte Carlo Simulation for Nonisotropic Fractional Laplace, Helmholtz, and Yukawa Equations

Jul 16, 2026

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The work studies nonisotropic fractional counterparts of the Laplace, Helmholtz, and Yukawa equations by connecting the nonisotropic fractional Laplace operator to a symmetric α-stable Lévy process with independent identically distributed components. A Duffin correspondence is provided for the Yukawa equation, and a Feynman–Kac reconstruction is developed for the Helmholtz equation. Using these links, the paper derives a Walk-on-Cubes algorithm to simulate solutions of the Helmholtz and Yukawa problems.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages29

Reading Time1 h 13 min

Introduction

Duffin Correspondence

Theoretical Background and Simulation

arXiv :2607 . 10528v1 [math .NA] 12 Jul 2026 WALK-ON-CUBES MONTE CARLO SIMULATION FOR NONISOTROPIC FRACTIONAL LAPLACE, HELMHOLTZ, AND YUKAWA EQUATIONS ANTTI RASILA Department of Mathematics with Computer Science, Guangdong Technion Israel Institute of Technology, Shantou, Guangdong 515063, P. R. China Department of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003, Israel TOMMI SOTTINEN School of Technology and Innovations, University of Vaasa, P. O. Box 700, FIN-65101 Vaasa, Finland YAOTONG YUAN Department of Mathematics with Computer Science, Guangdong Technion Israel Institute of Technology, Shantou, Guangdong 515063, P. R. China Department of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003, Israel Abstract. We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman–Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric α-stable L´evy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations. E-mail addresses: [antti.rasila@iki.fi](antti.rasila@iki.fi) ; [antti.rasila@gtiit.edu.cn](antti.rasila@gtiit.edu.cn) , [tommi.sottinen@uwasa.fi](tommi.sottinen@uwasa.fi) , [yuan09517@gtiit.edu.cn](yuan09517@gtiit.edu.cn). Date: July 14, 2026 . 2020 Mathematics Subject Classification. 65C05; 35J05; 35Q40; 68U20 . Key words and phrases. α-stable L´evy process, Monte Carlo method, nonisotropic fractional Laplace operator, Helmholtz equation, Yukawa equation. 2 RASILA, SOTTINEN, AND YUAN 1. Introduction Let x = (x1 , ... , xd) ∈ Rd . Let α ∈ (0 , 2) and let Aαx be the nonisotropic fractional Laplacian: d Aαx =X −(−∆xi)α/2 , i=1 where −(−∆xi)α/2 is the one-dimensional fractional Laplacian defined by the singular integral (1.1) (−∆xi)α/2f (xi ) = Cα Z− f|~~ ~~x(xii~~ ~~−)~~ ~~−y|~~ ~~f1+(yα) dy. Here the positive one-dimensional normalization is (1.2) Cα := 2απΓ1/2(|(1Γ(αα/)/22)~~ ~~|) = α2απ112ΓΓ((1(~~ ~~1−+αα/)2/)2) . Let D ⊂ Rd be a bounded domain, and let λ ∈ R . We consider the Dirichlet problem of finding for a given g : Dc → R a solution u: D → R such that (1.3) (1.4) Aαxu (x) = λu (x) if x ∈ D, u (x) = g (x) if x ∈ Dc. If λ < 0 (1.3)–(1.4) is called the fractional Helmholtz equation, if λ > 0 it is called the fractional Yukawa equation, and for λ = 0 it is called the fractional Laplace equation. Recently Kyprianou et al. [15] studied the isotropic fractional Laplace equation and provided the Walk-on-Spheres algorithm for its simulation. Our work is related to theirs, with two major differences: first, we consider thenonisotropic case corresponding to independent identically distributed component L´evy process as opposed to the isotropic L´evy process. Second, we also consider fractional Helmholtz and Yukawa equations. To our knowledge, the nonisotropic operator Aαx is relatively unstudied in the literature. One related study is Dybiec and Szczepaniec [11] . The rest of the paper is organized as follows: In Section 2 we provide the Duffin correspondence relating the fractional Laplace equation and the fractional Yukawa equation. In Section 3 we provide the theoretical background for our Monte Carlo simulation, including the obstruction to the same spatial Duffin lifting in the Helmholtz regime. Finally, in Section 4 we introduce our Walk-on-Cubes algorithm and provide a simulation. 2. Duffin Correspondence Duffin [10] provided a correspondence that transforms the classical Dirichlet problem of the Yukawa equation into a Dirichlet problem of the Laplace equation. This was later studied and extended to the Helmholtz equation in [17 , 22 , 23] . WALK-ON-CUBES SIMULATION 3 Since the operato

FAQs

What do the paper’s nonisotropic fractional Laplace, Helmholtz, and Yukawa equations refer to?

They are fractional analogs of the classical Laplace, Helmholtz, and Yukawa Dirichlet problems, distinguished by the sign of the parameter λ (λ<0 Helmholtz, λ>0 Yukawa, λ=0 Laplace).

How is the Yukawa equation related to the fractional Laplace equation?

Through a Duffin correspondence that transforms the Dirichlet problem for the Yukawa equation into an equivalent fractional Laplace-type problem using a one-dimensional eigenfunction lifting.

How does the Walk-on-Cubes algorithm enable simulation in this setting?

The paper uses the relation to a symmetric α-stable Lévy process and the established reconstruction/correspondence results to construct a Walk-on-Cubes procedure that simulates solutions for the Helmholtz and Yukawa equations.

Jul 16, 2026

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