Central Limit Theorem for Branching Brownian Motions in Random Environment
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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 Comets and Yoshida.
KeywordsBranching Brownian motion Random environment Poisson random measure Central limit theorem Phase transition Brownian directed polymer
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- 2.Bertin, P.: Positivity of the Lyapunov exponent for Brownian directed polymers in random environment in dimension one. Preprint Google Scholar
- 3.Bertin, P.: Free energy for Brownian directed polymers in random environment in dimension two. Preprint Google Scholar
- 8.Comets, F., Yoshida, N.: Some new results on Brownian directed polymers in random environment. Sūrikaisekikenkyūsho Kōkyūroku 1386, 50–66 (2004) Google Scholar
- 15.Nakashima, M.: Almost sure central limit theorem for branching random walks in random environment. Preprint Google Scholar
- 19.Shiozawa, Y.: Localization for branching Brownian motions in random environment (submitted) Google Scholar
- 22.Watanabe, S.: Limit theorems for a class of branching processes. In: Chover, J. (ed.) Markov Processes and Potential Theory, pp. 205–232. Wiley, New York (1967) Google Scholar
- 24.Yoshida, N.: Private communication (2008) Google Scholar