The support of measure-valued branching processes in a random environment

We consider the one-dimensional catalytic branching process introduced by Dawson and Fleischmann, which is a modification of the super-Brownian motion. The catalysts are given by a nonnegative infinitely divisible random measure with independent increments. We give sufficient conditions for the global support of the process to be compact, and sufficient conditions for noncompact global support. Since the catalytic process is related to the heat equation, compact support may be surprising. On the other hand, the super-Brownian motion has compact global support. We find that all nonnegative stable random measures lead to compact global support, and we give an example of a very rarified Levy process which leads to noncompact global support.