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Capacitary moduli for Lévy processes and intersections

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  • Rosen, Jay

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.

Suggested Citation

  • Rosen, Jay, 2000. "Capacitary moduli for Lévy processes and intersections," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 269-285, October.
    • Handle: RePEc:eee:spapps:v:89:y:2000:i:2:p:269-285
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    References listed on IDEAS

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    1. Rosen, Jay, 1986. "Tanaka's formula for multiple intersections of planar Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 23(1), pages 131-141, October.
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