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.

翻译：本文提出了一种计算一般集合 $K \subset \mathbb{R}^d$ 的 Riesz $\alpha$-容量（$0 < \alpha \le 2$）的方法。该方法基于各向同性 $\alpha$-稳定运动路径在 $d$ 维空间中的模拟。通常用于模拟布朗运动的著名“游走-球-上”（Walk-On-Spheres）方法被改进为一种新颖的“游走-球-内-外”（Walk-In-Out-Balls）方法，适用于建模此类广义随机游走“探测”区域外部的稳定路径过程。该方法考虑了这类随机游走因路径不连续性而穿越边界的倾向。这一方法使得从探测集合外部发射的稳定路径的击中位置的计算模拟高效可行。本文还给出了基于这些位置计算容量的可靠方法及非标准置信区间。针对代表性类型的集合 $K$（其中 $\alpha$ 和 $d$ 均变化）进行了示例性计算。