Particle picture approach to the self-intersection local time of density processes in S′(Rd)

For a Poisson high-density system of independent motions in Rd we consider the corresponding density process as the limit of fluctuations or, equivalently, the limit of the “total charge” if each particle is equipped with a random charge. We prove that under fairly general assumptions on the motions and on the intensity measure of the system, the self-intersection local time (SILT) of the density process can be expressed by means of intersections of pairs of evolving particles. This result helps to understand the interpretation and meaning of SILT. As an example, we discuss the cases of symmetric α-stable motions and fractional Brownian motions in detail.