Unbiased `walk-on-spheres' Monte Carlo methods for the fractional Laplacian

TL;DR

This paper develops an unbiased Monte Carlo `walk-on-spheres` algorithm for solving boundary value problems involving the fractional Laplacian, accommodating jumps in stable processes and enabling simulations in disconnected domains.

Contribution

It introduces an unbiased `walk-on-spheres` Monte Carlo method for the fractional Laplacian that handles jumps and disconnected domains, extending classical methods.

Findings

Algorithm terminates almost surely in finite steps.

Handles disconnected domains via jump processes.

Provides a practical simulation approach for fractional PDEs.

Abstract

We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called `walk-on-spheres` algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the `walk-on-spheres` comes within epsilon > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic alpha-stable process with alpha in (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.