Capacitary moduli for Levy processes and intersections

[摘要] We introduce the concept of capacitary modulus for a set Lambda subset of or equal to R-d, 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 L-2 inner product (h,f). We show that for a large class of Levy 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 u(1)(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 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. (C) 2000 Elsevier Science B.V. All rights reserved.

[发布日期] 2000-10-01

[关键词] capacitary modulus;Levy process;intersection