Self-intersection local time of Additive levy process

Abstract

This article discusses the problem of existence of jointly continuous self-intersection local time for an additive levy process. Here, “local time” is understood in the sense of occupation density, and by an additive Levy process the authors mean a process X={X(t),t∈R+N)} which has the decomposition X=X1⊕X2⊕⋯⊕XN, each X· has the lower index αℓ,α=min {α1,⋯,αN}. Let Z=(Xt2−Xt1,⋯,Xtr−Xtr−1). They prove that if Nrα > d(r – 1), then a jointly continuous local time of Z, i.e. the self-intersection local time of X, can be obtained.