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Central limit theorem for branching Brownian motions in random environment. (English) Zbl 1171.60024

Summary: We introduce a model of branching Brownian motions in time-space random environment associated with the Poisson random measure. We prove that, if the randomness of the environment is moderated by that of the Brownian motion, the population density satisfies a central limit theorem and the growth rate of the population size is the same as its expectation with strictly positive probability. We also characterize the diffusive behavior of our model in terms of the decay rate of the replica overlap. On the other hand, we show that, if the randomness of the environment is strong enough, the growth rate of the population size is strictly less than its expectation almost surely. To do this, we use a connection between our model and the model of Brownian directed polymers in random environment introduced by F. Comets and N. Yoshida [Commun. Math. Phys. 254, No. 2, 257–287 (2005 ; Zbl 1128.60089)].

MSC:

60K37 Processes in random environments
60J65 Brownian motion
60F05 Central limit and other weak theorems
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1128.60089
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References:

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