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Superprocesses over a stochastic flow. (English) Zbl 1018.60052

Authors’ abstract: We study a specific particle system in which particles undergo random branching and spatial motion. Such systems are best described, mathematically, via measure valued stochastic processes. As is now quite standard, we study the so-called superprocess limit of such a system as both the number of particles in the system and the branching rate tend to infinity. What differentiates our system from the classical superprocess case, in which the particles move independently of each other, is that the motions of our particles are affected by the presence of a global stochastic flow. We establish weak convergence to the solution of a well-posed martingale problem. Using the particle picture formulation of the flow superprocess, we study some of its properties. We give formulas for its first two moments and consider two macroscopic quantities describing its average behavior, properties that have been studied in some detail previously in the pure flow situation, where branching was absent. Explicit formulas for these quantities are given and graphs are presented for a specific example of a linear flow of Ornstein-Uhlenbeck type.
Reviewer’s remark: The two open problems mentioned in Section 5 concerning a conditional log-Laplace approach conditioned on the random medium had been solved by J. Xiong [“ A stochastic log-Laplace equation” (WIAS Berlin, Preprint No. 859. 2003)].

MSC:

60G57 Random measures
60F05 Central limit and other weak theorems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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