Computation of Riesz $\alpha$-capacity $C_\alpha$ of general sets in $\mathbb{R}^d$ using stable random walks

Abstract

A method for computing the Riesz $\alpha$-capacity, $0 < \alpha \le 2$, of a general set $K \subset \mathbb{R}^d$ is given. The method is based on simulations of isotropic $\alpha$-stable motion paths in $d$-dimensions. The familiar Walk-On-Spheres method, often utilized for simulating Brownian motion, is modified to a novel Walk-In-Out-Balls method adapted for modeling the stable path process on the exterior of regions ``probed'' by this type of generalized random walk. It accounts for the propensity of this class of random walk to jump through boundaries because of the path discontinuity. This method allows for the computationally efficient simulation of hitting locations of stable paths launched from the exterior of probed sets. Reliable methods of computing capacity from these locations are given, along with non-standard confidence intervals. Illustrative calculations are performed for representative types of sets K, where both $\alpha$ and $d$ are varied.