Capacitary moduli for Lévy processes and intersections

We introduce the concept of capacitary modulus for a set Λ⊆Rd, which is a function h that provides simple estimates for the capacity of Λ with respect to an arbitrary kernel f, estimates which depend only on the L2 inner product (h,f). We show that for a large class of Lévy processes, which include the symmetric stable processes and stable subordinators, a capacitary modulus for the range of the process is given by its 1-potential density u1(x), and a capacitary modulus for the intersection of the ranges of m independent such processes is given by the product of their 1-potential densities. The uniformity of estimates provided by the capacitary modulus allows us to obtain almost-sure asymptotics for the probability that one such process approaches within ε of the intersection of m other independent processes, conditional on these latter processes. Our work generalizes that of Pemantle et al. (1996) on the range of Brownian motion.