Additive Functionals of Several Lévy Processes and Intersection Local Times

Abstract

Different extensions of an isomorphism theorem of Dynkin are developed and are used to study two distinct but related families of functionals of Lévy processes; $n$-fold “near-intersections” of a single Lévy process and continuous additive functionals of several independent Lévy processes. Intersection local times for $n$ independent Lévy processes are also studied. They are related to both of the above families. In all three cases sufficient conditions are obtained for the almost sure continuity of these functionals in terms of the almost sure continuity of associated Gaussian chaos processes. Concrete suffcient conditions are given for the almost sure continuity of these functionals of Lévy processes.