An evolution equation for the intersection local times of superprocesses

Summary

Introduction

The primary aim of this paper is to establish evolution equations for the intersection local time (ILT) of the super Brownian motion and certain super stable processes. We shall proceed by carefully defining the requisite concepts and giving all of our main results in the Introduction, while leaving the proofs for later sections. The Introduction itself is divided into four sections, which treat, in turn, the definition of the superprocesses that will interest us, the definition of ILT and some previous results, our main result – a Tanaka-like evolution equation for ILT – and an Itô formula for measure-valued processes along with a description of how to use it to derive the evolution equation. Some technical lemmas make up Section 2 of the paper, while Section 3 is devoted to proofs.

In order to conserve space, we shall motivate neither the study of superprocesses per se – other than to note that they arise as infinite density limits of infinitely rapidly branching stochastic processes – nor the study of ILT – other than to note that this seems to be important for the introduction of an intrinsic dependence structure for the spatial part of a superprocess. Good motivational and background material on superprocesses can be found in Dawson (1978, 1986), Dawson, Iscoe and Perkins (1989), Ethier and Kurtz (1986), Roelly-Coppoletta (1986), Walsh (1986) and Watanabe (1968), as well as other papers in this volume. Material on ILT can be found in Adler, Feldman and Lewin (1991), Adler and Lewin (1991), Adler and Rosen (1991), Dynkin (1988) and Perkins (1988).