Walk-on-Cubes Monte Carlo Simulation for nonisotropic fractional Laplace, Helmholtz, and Yukawa equations

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 $\alpha$-stable Lévy 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.