### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 145-163 |

Number of pages | 19 |

Journal | Journal of Statistical Physics |

Volume | 136 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2009 |

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### Keywords

- Branching Brownian motion
- Brownian directed polymer
- Central limit theorem
- Phase transition
- Poisson random measure
- Random environment

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*136*(1), 145-163. https://doi.org/10.1007/s10955-009-9774-5

**Central limit theorem for branching Brownian motions in random environment.** / Shiozawa, Yuichi.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 136, no. 1, pp. 145-163. https://doi.org/10.1007/s10955-009-9774-5

}

TY - JOUR

T1 - Central limit theorem for branching Brownian motions in random environment

AU - Shiozawa, Yuichi

PY - 2009/7

Y1 - 2009/7

N2 - 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.

AB - 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.

KW - Branching Brownian motion

KW - Brownian directed polymer

KW - Central limit theorem

KW - Phase transition

KW - Poisson random measure

KW - Random environment

UR - http://www.scopus.com/inward/record.url?scp=70349337345&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349337345&partnerID=8YFLogxK

U2 - 10.1007/s10955-009-9774-5

DO - 10.1007/s10955-009-9774-5

M3 - Article

AN - SCOPUS:70349337345

VL - 136

SP - 145

EP - 163

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -