On particle trajectories and genealogy in branching Brownian motion and branching random walk
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Amphithéâtre Daurat bâtiment U3
Institut de Mathématiques de Toulouse
Composition of the committee:
- Nina Gantert, Reviewer, Technical University of Munich
- Yueyun Hu, Reviewer, Sorbonne Paris North University
- Xinxin Chen, Examiner, Beijing Normal University
- Julien Barral, Examiner, Sorbonne Paris North University
- Agnès Lagnoux, Examiner, University of Toulouse - Jean Jaurès
- Bastien Mallein, Examiner, University of Toulouse
- Pascal Maillard, PhD Supervisor, University of Toulouse
- Michel Pain, Co-supervisor, University of Toulouse
Abstract:
Branching Brownian motion and branching random walk are stochastic processes, the former in continuous time, the latter in discrete time, that describe the evolution of a population of particles. These particles are assumed to reproduce and move independently on the real line. In this thesis, we study the asymptotic behavior of the two processes as time t tends to infinity, through three main themes, described below.
First, we consider the following question, motivated by statistical physics. If two particles are drawn independently from the population at time t, according to a high-temperature Gibbs law, what is, for 0 < a < 1, the probability that their most recent common ancestor has lived after time at? In a joint work with Michel Pain, we provide precise estimates for this probability in the setting of branching Brownian motion, conditionally on the process and non-conditionally. Two temperature sub-phases arise, but surprisingly, the threshold is not the same in both cases.
Next, we study upper deviations of the maximal particle, that is, the event where its position is at distance x above its expectation, where x grow to infinity with time t. Since the regime where x is proportional to t has already been treated in the literature, we focus on the regime where x is much smaller than t, called moderate. In the case of branching Brownian motion, we provide an asymptotic equivalent for the upper moderate deviation probability. As a byproduct of our analysis, we obtain information on the trajectorial and genealogical behavior contributing to such deviations.
In a joint work with Lianghui Luo, we also study the upper moderate deviation probabilities in the framework of branching random walk. Our assumptions on the reproduction law are very general, which leads us to state an analogous result for the sub-regime where x is at most of order √t. We further apply this result to prove the convergence in law of the centered maximum of a two-speed branching random walk and to describe its limit.
Finally, in a joint work with Gabriel Flath and Julien Berestycki, we investigate the trajectories drawn by the ancestors of front particles, that is, particles at distance of order √t from the maximum. We provide a precise description of such trajectories in the form of an almost sure convergence result.