Additive Lévy processes arise naturally in the studies of the Brownian sheet, intersections of Lévy processes and so on. In this talk, we present some recent results on the novel connections between an additive Lévy process X in \Rd, and a class of energy forms and their corresponding capacities. As applications of these results, we study the Hausdorff dimension and exact capacity of random sets associated to an ordinary Lévy process and solve several long-standing problems in the folklore of the theory of Lévy processes.
This talk is based on joint papers with D. Khoshnevisan and Y. Zhong.