Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

Abstract

We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet problem in which the Laplacian is replaced by the fractional Laplacian. 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 on which the Dirichlet problem is defined. As this algorithm would otherwise never end, it is truncated when the `walk-on-spheres' comes within ϵ>0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α∈(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 which does not depend on the point of issue of the stable process.