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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.00532 |
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| _version_ | 1866915230583357440 |
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| author | He, Hui |
| author_facet | He, Hui |
| contents | We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Belloum and Mallein in \cite{BeMa21} showed that the maximum position $M_t$ of all particles alive at time $t$, suitably centered by a deterministic function $m_t$, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as $t\rightarrow\infty$, \[ {\mathbb P}(M_t\geq θm_t),\quad θ>1. \] We shall show that the decay rate function exhibits phase transitions depending on certain relations between $θ$, the variance of the underlying Brownian motion and the branching rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviations for the maximum of a reducible two-type branching Brownian motion He, Hui Probability We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Belloum and Mallein in \cite{BeMa21} showed that the maximum position $M_t$ of all particles alive at time $t$, suitably centered by a deterministic function $m_t$, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as $t\rightarrow\infty$, \[ {\mathbb P}(M_t\geq θm_t),\quad θ>1. \] We shall show that the decay rate function exhibits phase transitions depending on certain relations between $θ$, the variance of the underlying Brownian motion and the branching rate. |
| title | Large deviations for the maximum of a reducible two-type branching Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2408.00532 |