Collision Local Times and Measure-Valued Processes

Abstract

AbstractWe consider two independent Dawson-Watanabe super-Brownian motions,Y1andY2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that ifd ≤ 5then with positive probability there exist timestsuch that the closed supports ofintersect; whereas ifd> 5 then no such intersections occur. For the cased≤ 5, we construct a continuous, non-decreasing measure–valued processL(Y1,Y2), thecollision local time,such that the measure defined by, is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition ofL(Y1,Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions withpoint interactions.In the course of our proofs we obtain smoothness results for the random measuresthat are uniform int.These theorems use a nonstandard description ofYiand are of independent interest.