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Branching Brownian Motion with Selection Pascal Maillard

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Branching Brownian Motion with Selection Pascal Maillard Branching Brownian motion with selection Pascal Maillard To cite this version: Pascal Maillard. Branching Brownian motion with selection. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 2012. English. tel-00741368 HAL Id: tel-00741368 https://tel.archives-ouvertes.fr/tel-00741368 Submitted on 12 Oct 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire de Université Pierre Probabilités et et Marie Curie Modèles Aléatoires École Doctorale Paris Centre Thèse de doctorat Discipline : Mathématiques présentée par Pascal Maillard Mouvement brownien branchant avec sélection (Branching Brownian motion with selection) dirigée par Zhan Shi Rapporteurs : M. Andreas Kyprianou University of Bath M. Ofer Zeitouni University of Minnesota & Weizmann Institute of Science Soutenue le 11 octobre 2012 devant le jury composé de : Mme Brigitte Chauvin Université de Versailles examinatrice M. Francis Comets Université Paris 7 examinateur M. Bernard Derrida Université Paris 6 et ENS examinateur M. Yueyun Hu Université Paris 13 examinateur M. Andreas Kyprianou University of Bath rapporteur M. Zhan Shi Université Paris 6 directeur Laboratoire de Probabilités et Modèles Aléatoires École Doctorale Paris Centre Université Pierre et Marie Curie Case courrier 188 4, place Jussieu 4, place Jussieu 75 005 Paris 75 252 Paris cedex 05 À mes parents, qui ont fait pousser la plante avec amour À Pauline, le soleil qui la fait vivre et lui est source de joie et de bonheur Remerciements 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. Merci à Louigi Addario-Berry, Elie Aïdékon, Louis-Pierre Arguin, Jean Bérard, Nathanaël Berestycki, Jason Schweinsberg, Damien Simon, Olivier Zindy pour m’avoir soutenu, conseillé, éclairé. Merci encore à Yacine B, Pierre B, Reda C, Leif D, Xan D, Nic F, Clément F, Patrick H, Mathieu J, Cyril L, Thomas M, Bastien M et Matthew R pour ce “aahh” de compréhension survenu après une discussion spontanée ou après des heures de travail en commun. Merci aux anciens et actuels doctorants du LPMA pour un quotidien stimulant, agréable et drôle et pour avoir supporté mes mauvaises blagues au déjeuner et mes questions idiotes au GTT (ou l’inverse). Merci également à l’équipe administrative du LPMA pour leur travail efficace et leur dévouement. Merci, Danke, Hvala, Thank you à mes amis, mathématiciens ou non, de Paris ou d’ailleurs, que j’ai eu la chance de rencontrer sur mon chemin. Merci aux superbes femmes de Pierrepont et à JP. Danke an meine Familie aus dem Norden. Et finalement, les plus grands remerciements vont à Gérard, Elsbeth, Tobi, Karin et André pour leur soutien inconditionnel, et surtout à Pauline, la seule personne qui sache aussi bien me faire tourner la tête qu’y mettre de l’ordre quand elle tourne en rond. Mouvement brownien branchant avec sélection Résumé Dans cette thèse, le mouvement brownien branchant (MBB) est un système aléatoire de particules, où celles-ci diffusent sur la droite réelle selon des mouvements browniens et branchent à taux constant en un nombre aléatoire de particules d’espérance supérieure à 1. Nous étudions deux modèles de MBB avec sélection : le MBB avec absorption à une droite espace-temps et le N-MBB, où, dès que le nombre de particules dépasse un nombre donné N, seules les N particules les plus à droite sont gardées tandis que les autres sont enlevées du système. Pour le premier modèle, nous étudions la loi du nombre de particules absorbées dans le cas où le processus s’éteint presque sûrement, en utilisant un lien entre les équations de Fisher–Kolmogorov–Petrovskii–Piskounov (FKPP) et de Briot–Bouquet. Pour le deuxième modèle, dont l’étude représente la plus grande partie de cette thèse, nous donnons des asymp- totiques précises sur la position du nuage de particules quand N est grand. Plus précisément, nous montrons qu’elle converge à l’échelle de temps log3 N vers un processus de Lévy plus une dérive linéaire, tous les deux explicites, confirmant des prévisions de Brunet, Derrida, Mueller et Munier. Cette étude contribue à la compréhension de fronts du type FKPP sous l’influence de bruit. Enfin, une troisième partie montre le lien qui existe entre le MBB et des processus ponctuels stables. Mots-clefs Mouvement brownien branchant, sélection, équation de Fisher–Kolmogorov–Petrovskii– Piskounov (FKPP) bruitée, équation de Briot–Bouquet, mesure aléatoire stable. Branching Brownian motion with selection Abstract In this thesis, branching Brownian motion (BBM) is a random particle system where the particles diffuse on the real line according to Brownian motions and branch at constant rate into a random number of particles with expectation greater than 1. We study two models of BBM with selection: BBM with absorption at a space-time line and the N-BBM, where, as soon as the number of particles exceeds a given number N, only the N right-most particles are kept, the others being removed from the system. For the first model, we study the law of the number of absorbed particles in the case where the process gets extinct almost surely, using a relation between the Fisher–Kolmogorov–Petrovskii–Piskounov (FKPP) and the Briot–Bouquet equations. For the second model, the study of which represents the biggest part of the thesis, we give a precise asymptotic on the position of the cloud of particles when N is large. More precisely, we show that it converges at the timescale log3 N to a Lévy process plus a linear drift, both of them explicit, which confirms a prediction by Brunet, Derrida, Mueller and Munier. This study contributes to the understanding of travelling waves of FKPP type under the influence of noise. Finally, in a third part we point at the relation between the BBM and stable point processes. Keywords Branching Brownian motion, selection, Fisher–Kolmogorov–Petrovskii–Piskounov (FKPP) equation with noise, Briot–Bouquet equation, stable random measure. Contents Introduction 11 1 Number of absorbed individuals in branching Brownian motion with a barrier 23 1 Introduction . 23 2 First results by probabilistic methods . 25 2.1 Notation and preliminary remarks . 25 2.2 Branching Brownian motion with two barriers . 26 2.3 Proof of Proposition 2.1 . 29 3 The FKPP equation . 29 4 Proof of Theorem 1.1 . 32 5 Preliminaries for the proof of Theorem 1.2 . 33 5.1 Notation . 33 5.2 Complex differential equations . 34 5.3 Singularity analysis . 35 5.4 An equation for continuous-time Galton–Watson processes . 36 6 Proof of Theorem 1.2 . 37 7 Appendix . 44 7.1 A renewal argument for branching diffusions . 44 7.2 Addendum to the proof of Theorem 1.1 . 45 7.3 Reduction to Briot–Bouquet equations . 46 7.4 Inversion of some analytic functions . 47 Acknowledgements . 49 2 Branching Brownian motion with selection of the N right-most particles 51 1 Introduction . 51 1.1 Heuristic ideas and overview of the results . 52 1.2 Notation guide . 55 2 Brownian motion in an interval . 56 2.1 A function of Jacobi theta-type . 57 2.2 Brownian motion killed upon exiting an interval . 57 2.3 The Brownian taboo process . 59 3 Preliminaries on branching Markov processes . 63 3.1 Definition and notation . 63 3.2 Stopping lines . 65 3.3 Many-to-few lemmas and spines . 66 3.4 Doob transforms . 68 4 BBM with absorption at a critical line . 69 4.1 Proof of Proposition 4.1 . 71 5 BBM in an interval . 73 5.1 Notation . 73 5.2 The processes Zt and Yt .......................... 74 5.3 The number of particles . 76 5.4 The particles hitting the right border . 79 5.5 Penalizing the particles hitting the right border . 81 6 BBM with absorption before a breakout . 84 6.1 Definitions . 84 6.2 The time of the first breakout . 88 6.3 The particles that do not participate in the breakout . 91 6.4 The fugitive and its family . 94 7 The B-BBM . 98 7.1 Definition of the model . 98 7.2 Proof of Proposition 7.3 . 101 7.3 Proof of Theorems 7.1 and 7.2 . 105 8 The B5-BBM . 108 8.1 Definition of the model . 108 8.2 Preparatory lemmas . 110 5 8.3 The probability of G1 . 114 8.4 Proofs of the main results .
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