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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.05792 |
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| _version_ | 1866917714914705408 |
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| author | Berestycki, Julien Tough, Oliver |
| author_facet | Berestycki, Julien Tough, Oliver |
| contents | The $N$-branching Brownian motion with selection ($N$-BBM) is a particle system consisting of $N$ independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We establish the following selection principle: as $N \rightarrow \infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of the associated free boundary PDE. This resolves an open question going back at least to \cite[p.19]{Maillard2012} and \cite{GroismanJonckheer}, and follows a recent related result by the second author establishing a similar selection principle for the so-called Fleming-Viot particle system \cite{Tough23}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Selection principle for the $N$-BBM Berestycki, Julien Tough, Oliver Probability Analysis of PDEs 60J80, 35R35, 35Q70 The $N$-branching Brownian motion with selection ($N$-BBM) is a particle system consisting of $N$ independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We establish the following selection principle: as $N \rightarrow \infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of the associated free boundary PDE. This resolves an open question going back at least to \cite[p.19]{Maillard2012} and \cite{GroismanJonckheer}, and follows a recent related result by the second author establishing a similar selection principle for the so-called Fleming-Viot particle system \cite{Tough23}. |
| title | Selection principle for the $N$-BBM |
| topic | Probability Analysis of PDEs 60J80, 35R35, 35Q70 |
| url | https://arxiv.org/abs/2407.05792 |