We provide a probabilistic interpretation of a class of natural capacities on Euclidean space in terms of the level sets of a suitably chosen multiparameter additive Lévy process X. We also present several probabilistic applications of the aforementioned potential-theoretic connections. They include areas such as intersections of Lévy processes and level sets, as well as Hausdorff dimension computations.
"Level Sets of Additive Lévy Processes." Ann. Probab. 30 (1) 62 - 100, January 2002. https://doi.org/10.1214/aop/1020107761