ABSOLUTE CONTINUITY RESULTS FOR SUPERPROCESSES WITH SOME APPLICATIONS

Let X1 and X2 be instances of a measure-valued Dawson-Watanabe zeta-super process where the underlying spatial motions are given by a Borel right process, zeta, and where the branching mechanism has finite variance. A necessary and sufficient condition on X(O)1 and X(O)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 zeta, 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-Levy processes or, more generally, for a super-Levy 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-Levy process to all points to which the underlying Levy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.