Gaussian Processes and Local Times of Symmetric Lévy Processes

Abstract

We give a relatively simple proof of the necessary and sufficient condition for the joint continuity of the local times of symmetric Levy processes. This result was obtained in 1988 by M. Barlow and J. Hawkes without requiring that the Levy processes be symmetric. In 1992 the authors used a very different approach to obtain necessary and sufficient condition for the joint continuity of the local times of strongly symmetric Markov processes, which includes symmetric Levy processes. Both the 1988 proof and the 1992 proof are long and difficult. In this paper the 1992 proof is significantly simplified. This is accomplished by using two recent isomorphism theorems, which relate the local times of strongly symmetric Markov processes to certain Gaussian processes, one due to N. Eisenbaum alone and the other to N. Eisenbaum, H. Kaspi, M. B. Marcus, J. Rosen, and Z. Shi. Simple proofs of these isomorphism theorems are given in this paper.