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SUMMARY:On particle trajectories and genealogy in branching Brownian motio
 n and branching random walk
DTSTART:20260605T073000Z
DTEND:20260605T103000Z
DTSTAMP:20260614T015800Z
UID:indico-event-16619@indico.math.cnrs.fr
DESCRIPTION:Speakers: Louis Chataignier\n\nComposition of the committee:- 
 Nina Gantert\, Reviewer\, Technical University of Munich- Yueyun Hu\, Revi
 ewer\, Sorbonne Paris North University- Xinxin Chen\, Examiner\, Beijing N
 ormal University- Julien Barral\, Examiner\, Sorbonne Paris North Universi
 ty- Agnès Lagnoux\, Examiner\, University of Toulouse - Jean Jaurès- Bas
 tien Mallein\, Examiner\, University of Toulouse- Pascal Maillard\, PhD Su
 pervisor\, University of Toulouse- Michel Pain\, Co-supervisor\, Universit
 y of Toulouse\nAbstract:\nBranching Brownian motion and branching random w
 alk are stochastic processes\, the former in continuous time\, the latter 
 in discrete time\, that describe the evolution of a population of particle
 s. These particles are assumed to reproduce and move independently on the 
 real line. In this thesis\, we study the asymptotic behavior of the two pr
 ocesses as time t tends to infinity\, through three main themes\, describe
 d below.\nFirst\, we consider the following question\, motivated by statis
 tical physics. If two particles are drawn independently from the populatio
 n 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 esti
 mates for this probability in the setting of branching Brownian motion\, c
 onditionally on the process and non-conditionally. Two temperature sub-pha
 ses arise\, but surprisingly\, the threshold is not the same in both cases
 .\nNext\, we study upper deviations of the maximal particle\, that is\, th
 e 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 t
 o t has already been treated in the literature\, we focus on the regime wh
 ere x is much smaller than t\, called moderate. In the case of branching B
 rownian motion\, we provide an asymptotic equivalent for the upper moderat
 e deviation probability. As a byproduct of our analysis\, we obtain inform
 ation on the trajectorial and genealogical behavior contributing to such d
 eviations.\nIn a joint work with Lianghui Luo\, we also study the upper mo
 derate 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 orde
 r √t. We further apply this result to prove the convergence in law of th
 e centered maximum of a two-speed branching random walk and to describe it
 s limit.\nFinally\, in a joint work with Gabriel Flath and Julien Berestyc
 ki\, we investigate the trajectories drawn by the ancestors of front parti
 cles\, that is\, particles at distance of order √t from the maximum. We 
 provide a precise description of such trajectories in the form of an almos
 t sure convergence result.\n\nhttps://indico.math.cnrs.fr/event/16619/
LOCATION:Amphithéâtre Daurat bâtiment U3 (Institut de Mathématiques de
  Toulouse)
URL:https://indico.math.cnrs.fr/event/16619/
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