Branching Brownian motion with selection

Abstract

Je tiens à remercier chaleureusement mes deux rapporteurs Andreas Kyprianou et Ofer Zeitouni ainsi que les examinateurs Brigitte Chauvin, Francis Comets, Bernard Derrida et Yueyun Hu. J'admire les travaux de chacun et c'est un grand honneur qu'ils aient accepté d'évaluer mon travail.Merci à mon directeur Zhan Shi qui m'a fait découvrir un sujet aussi riche et passionnant que le mouvement brownien branchant, qui m'a toujours soutenu et prodigué ses conseils, tout en m'accordant une grande liberté dans mes recherches. Je remercie également les autres professeurs de l'année 2008/2009 du Master 2 Processus Stochastiques à l'UPMC pour la formation exceptionnelle en probabilités sans laquelle cette thèse n'aurait pas été possible.Certaines autres personnes ont eu une influence directe sur mes recherches mathématiques. Je remercie vivement Julien Berestycki qui, par son enthousiasme infini, a su donner une impulsion nouvelle à mes recherches. Merci à Olivier Hénard, mon ami, pour avoir partagé à de nombreuses occasions sa fine compréhension des processus de branchement.

Front Propagation and Quasi-Stationary Distributions: Two Faces of the Same Coin

“…Bérard and Gouéré proved this shift in the velocity for a similar process. We refer to the work of Maillard [31] for the most detailed study of this process.…”

“…With a similar point of view in mind, Brunet, Derrida and coauthors [13,14,11,12] started in the nineties a study of the effect of microscopic noise in front propagation for equation (1) and related models, which resulted in a huge number of works that study the change in the behavior of the front when microscopic effects are taken into account. These works include both numerical and heuristic arguments [13,14,11,12,26] as well as rigorous proofs [4,5,16,31,32]. Before that, Bramson et.al [10] gave the first rigorous proof of a microscopic model for (1) that has a unique velocity for every initial condition.…”

Yaglom-type limit theorems for branching Brownian motion with absorption

“…In fact, there exists a 2.16 P R such that as λ Ñ 0, Remark 2.14. The result holds under weaker assumptions on the offspring distribution; see [41]. Also, an analogous result for branching random walk has been proven recently in [16].…”

“…Let f psq " Ers L s be the probability generating function of the offspring distribution, and let q be the smallest root of f psq " s, which is the extinction probability for a Galton-Watson process with offspring distribution pp k q 8 k"1 . Lemma 2.13 below is a direct consequence of Lemma 4.3 in Chapter 2 of [41]. As indicated there, it also follows from the following two results by de Haan's Tauberian Theorem (see Theorem 2 of [21]):…”

“…The results (2.14) and (2.15) have first been proved in [6] for binary branching and have been extended to general offspring distributions in Proposition 4.1 in Chapter 2 of [41]. The first part of Lemma 2.13 below is due to Neveu [46].…”