Polynomial slowdown in an angle-dependent 2d branching Brownian motion
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917478501711872 |
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| author | Berestycki, Julien Geldbach, David Pain, Michel |
| author_facet | Berestycki, Julien Geldbach, David Pain, Michel |
| contents | We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(θ)$ which depends only on the angle $θ$ of the particle. We assume that $b$ is maximal when $θ=0$, which is the preferred direction for breeding. Furthermore we assume that $b(θ) = 1 - β\abs{θ}^α+ O(θ^2)$, as $θ\to 0$, for $α\in (2/3,2)$ and $β>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-α)/(2+α)} - \left(\frac{3}{2\sqrt{2}} - \fracα{2\sqrt{2}(2+α)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial slowdown in an angle-dependent 2d branching Brownian motion Berestycki, Julien Geldbach, David Pain, Michel Probability Analysis of PDEs 60J80, 60J65, 35R99 We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(θ)$ which depends only on the angle $θ$ of the particle. We assume that $b$ is maximal when $θ=0$, which is the preferred direction for breeding. Furthermore we assume that $b(θ) = 1 - β\abs{θ}^α+ O(θ^2)$, as $θ\to 0$, for $α\in (2/3,2)$ and $β>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-α)/(2+α)} - \left(\frac{3}{2\sqrt{2}} - \fracα{2\sqrt{2}(2+α)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator. |
| title | Polynomial slowdown in an angle-dependent 2d branching Brownian motion |
| topic | Probability Analysis of PDEs 60J80, 60J65, 35R99 |
| url | https://arxiv.org/abs/2506.10623 |