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The support of measure-valued branching processes in a random environment. (English) Zbl 0853.60065

A one-dimensional super-Brownian motion in a catalytic medium (random but constant in time) is considered. More precisely, the catalysts are given by an infinitely divisible random measure on \(\mathbb{R}\) with independent increments. Sufficient conditions are given for the global support of the process to be compact, or non-compact, respectively. For instance, it is compact if the catalysts are stable and the initial measure has compact support. On the other hand, if the catalysts are “very rarified”, the global support is non-compact. Here the catalysts might even be dense in \(\mathbb{R}\).

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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