Random Fourier series and continuous additive functionals of Levy processes on the torus

• 成果类型: Article
• 署名作者: Marcus, MB; Rosen, J
• 署名单位: City University of New York (CUNY) System; College of Staten Island (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 1178-1218
• 关键词: local-times

摘要

Let X be an exponentially killed Levy process on T-n, the n-dimensional torus, that satisfies a sector condition. (This includes symmetric Levy processes.) Let F-e denote the extended Dirichlet space of X. Let h is an element of F-e and let {h(y), y is an element of T-n} denote the set of translates of h. That is, h(y)(.) = h(. - y). We consider the family of zero-energy continuous additive functions {N-t([hy]), (y, t) is an element of T-n x R(+)} as defined by Fukushima. For a very large class of random functions h we show that J(rho)(T-n) = integral(log N-rho(T-n, epsilon))(1/2) d epsilon < .infinity is a necessary and sufficient condition for the family {N-t([hy)], (y, t) is an element of T-n x R(+)} to have a continuous version almost surely. Here N-rho(T-n, epsilon) is the minimum number of balls of radius epsilon in the metric rho that covers T-n, where the metric rho is the energy metric. We argue that this condition is the natural extension of the necessary and sufficient condition for continuity of local times of Levy processes of Barlow and Hawkes. Results on the bounded variation and p-variation (in t) of N-t([hy]), for y fixed, are also obtained for a large class of random functions h.