Dirichlet processes and an intrinsic characterization of renormalized intersection local times

Abstract

We show that the nth order renormalized self-intersection local time γ n ( μ; t) for the symmetric stable process in R 2, where the n-fold multiple points are weighted by an arbitrary measure μ, can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random measure associated with γ n−1 ( μ; t).