PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

Abstract

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.