Absolute continuity results for superprocesses with some applications

Abstract

Let ${X^1}$ and ${X^2}$ be instances of a measure-valued Dawson-Watanabe $\xi$-super process where the underlying spatial motions are given by a Borel right process, $\xi$, and where the branching mechanism has finite variance. A necessary and sufficient condition on $X_0^1$ and $X_0^2$ is found for the law of $X_s^1$ to be absolutely continuous with respect to the law of $X_t^2$. The conditions are the natural absolute continuity conditions on $\xi$, but some care must be taken with the set of times $s$, $t$ being considered. The result is used to study the closed support of super-Brownian motion and give sufficient conditions for the existence of a nontrivial "collision measure" for a pair of independent super-Lévy processes or, more generally, for a super-Lévy process and a fixed measure. The collision measure gauges the extent of overlap of the two measures. As a final application, we give an elementary proof of the instantaneous propagation of a super-Lévy process to all points to which the underlying Lévy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.