Capacitary moduli for Lévy processes and intersections

Jay Rosen

Stochastic Processes and their Applications, 2000, vol. 89, issue 2, 269-285

Abstract: We introduce the concept of capacitary modulus for a set , which is a function h that provides simple estimates for the capacity of [Lambda] 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 [var epsilon] 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.

Keywords: Capacitary; modulus; Lévy; process; Intersection (search for similar items in EconPapers)
Date: 2000
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