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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.04527 |
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| _version_ | 1866917858829664256 |
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| author | Mercer, Jacob |
| author_facet | Mercer, Jacob |
| contents | $N$-Brownian bees is a branching-selection particle system in $\mathbb{R}^d$ in which $N$ particles behave as independent binary branching Brownian motions, and where at each branching event, we remove the particle furthest from the origin. We study a variant in which $d=1$ and particles have an additional drift $μ\in\mathbb{R}$. We show that there is a critical value, $μ_c^N$, and three distinct regimes (sub-critical, critical, and super-critical) and we describe the behaviour of the system in each case. In the sub-critical regime, the system is positive Harris recurrent and has an invariant distribution; in the super-critical regime, the system is transient; and in the critical case, after rescaling, the system behaves like a single reflected Brownian motion.
We also show that the critical drift $μ_c^N$ is in fact the speed of the well-studied $N$-BBM process, and give a rigorous proof for the speed of $N$-BBM, which was missing in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle Mercer, Jacob Probability $N$-Brownian bees is a branching-selection particle system in $\mathbb{R}^d$ in which $N$ particles behave as independent binary branching Brownian motions, and where at each branching event, we remove the particle furthest from the origin. We study a variant in which $d=1$ and particles have an additional drift $μ\in\mathbb{R}$. We show that there is a critical value, $μ_c^N$, and three distinct regimes (sub-critical, critical, and super-critical) and we describe the behaviour of the system in each case. In the sub-critical regime, the system is positive Harris recurrent and has an invariant distribution; in the super-critical regime, the system is transient; and in the critical case, after rescaling, the system behaves like a single reflected Brownian motion. We also show that the critical drift $μ_c^N$ is in fact the speed of the well-studied $N$-BBM process, and give a rigorous proof for the speed of $N$-BBM, which was missing in the literature. |
| title | Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle |
| topic | Probability |
| url | https://arxiv.org/abs/2412.04527 |