We provide a probabilistic interpretation of a class of natural capacities on Eu- clidean space in terms of the level sets of a suitably chosen multiparameter additive L evy process X. We also present several probabilistic applications of the aforementioned potential-theoretic connections. They include areas such as intersections of L evy processes and level sets, as well as Hausdor dimension computations. Keyword and Phrases Additive Stable Processes, Potential Theory