From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Alban, Alexander Bovier, Anton Gros, Annabell Hartung, Lisa |
| author_facet | Alban, Alexander Bovier, Anton Gros, Annabell Hartung, Lisa |
| contents | We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function.
We consider general speed functions lying strictly below their concave hull and piecewise linear, concave speed functions.
In the first case, the log-correction for the order of the maximum depends only on the rate of convergence of the speed function near 0 and 1 and exhibits a smooth interpolation between the correction in the i.i.d. case, $\frac{1}{2\sqrt{2}} \ln t$, and that of standard BBM, $\frac{3}{2\sqrt{2}} \ln t$.
In the second case, we describe the order of the maximum in dependence of the form of speed function and show that any log-correction larger than $\frac{3}{2\sqrt{2}} \ln t$ can be obtained.
In both cases, we prove that the limiting law of the maximum and the extremal process essentially coincide with those of standard BBM, using a first and second moment method which relies on the localisation of extremal particles.
This extends the results of Bovier and Hartung for two-speed BBM. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03580 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion Alban, Alexander Bovier, Anton Gros, Annabell Hartung, Lisa Probability 60J80, 60G70, 82B44 We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function. We consider general speed functions lying strictly below their concave hull and piecewise linear, concave speed functions. In the first case, the log-correction for the order of the maximum depends only on the rate of convergence of the speed function near 0 and 1 and exhibits a smooth interpolation between the correction in the i.i.d. case, $\frac{1}{2\sqrt{2}} \ln t$, and that of standard BBM, $\frac{3}{2\sqrt{2}} \ln t$. In the second case, we describe the order of the maximum in dependence of the form of speed function and show that any log-correction larger than $\frac{3}{2\sqrt{2}} \ln t$ can be obtained. In both cases, we prove that the limiting law of the maximum and the extremal process essentially coincide with those of standard BBM, using a first and second moment method which relies on the localisation of extremal particles. This extends the results of Bovier and Hartung for two-speed BBM. |
| title | From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion |
| topic | Probability 60J80, 60G70, 82B44 |
| url | https://arxiv.org/abs/2405.03580 |