Lecture Notes in Mathematics 2151

Transcription

1 Lecture Notes in Mathematics 2151 Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg More information about this series at

2 Saint-Flour Probability Summer School The Saint-Flour volumes are reflections of the courses given at the Saint-Flour Probability Summer School. Founded in 1971, this school is organised every year by the Laboratoire de Mathématiques (CNRS and Université Blaise Pascal, Clermont-Ferrand, France). It is intended for PhD students, teachers and researchers who are interested in probability theory, statistics, and in their applications. The duration of each school is 13 days (it was 17 days up to 2005), and up to 70 participants can attend it. The aim is to provide, in three high-level courses, a comprehensive study of some fields in probability theory or Statistics. The lecturers are chosen by an international scientific board. The participants themselves also have the opportunity to give short lectures about their research work. Participants are lodged and work in the same building, a former seminary built in the 18th century in the city of Saint-Flour, at an altitude of 900 m. The pleasant surroundings facilitate scientific discussion and exchange. The Saint-Flour Probability Summer School is supported by: Université Blaise Pascal Centre National de la Recherche Scientifique (C.N.R.S.) Ministère délégué à l Enseignement supérieur et à la Recherche For more information, see Christophe Bahadoran [email protected] Arnaud Guillin [email protected] Laurent Serlet [email protected] Université Blaise Pascal Aubière cedex, France

3 Zhan Shi Branching Random Walks École d Été de Probabilités de Saint-Flour XLII 2012

4 Zhan Shi Laboratoire de Probabilités et ModJeles Aléatoires Université Pierre et Marie Curie Paris, France ISSN ISSN (electronic) Lecture Notes in Mathematics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification: 60J80, 60J85, 60G50, 60K37 Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (

5 To the memory of my teacher, Professor Marc Yor ( ) 但 去 莫 复 问, 白 云 无 尽 时 - Œ 唐 王 维 送 别

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7 Preface These notes attempt to provide an elementary introduction to the one-dimensional discrete-time branching random walk and to exploit its spinal structure. They begin with the case of the Galton Watson tree for which the spinal structure, formulated in the form of the size-biased tree, is simple and intuitive. Chapter 3 is devoted to a few fundamental martingales associated with the branching random walk. The spinal decomposition is introduced in Chap. 4, first in its more general form, followed by two important examples. This chapter gives the most important mathematical tool of the notes. Chapter 5 forms, together with Chap. 4, the main part of the text. Exploiting the spinal decomposition theorem, we study various asymptotic properties of the extremal positions in the branching random walk and of the fundamental martingales. The last part of the notes presents a brief account of results for a few related and more complicated models. The lecture notes by Berestycki [43] and Zeitouni [235] give a general and excellent account of, respectively, branching Brownian motion and the F-KPP equation and branching random walks with applications to Gaussian free fields. I would like to deeply thank Yueyun Hu; together we wrote about 20 papers in the last 20 years, some of them strongly related to the material presented here. I am grateful to Élie Aïdékon, Julien Berestycki, Éric Brunet, Xinxin Chen, Bernard Derrida, Gabriel Faraud, Nina Gantert, and Jean-Baptiste Gouéré for stimulating discussions, to Bastien Mallein and Michel Pain for great assistance in the preparation of the present notes, and to Christian Houdré for correcting my English with patience. I wish to thank Laurent Serlet and the Scientific Board of the École d été de probabilités de Saint-Flour for the invitation to deliver these lectures. Paris, France August 2015 Zhan Shi vii

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9 Contents 1 Introduction BranchingBrownianMotion Branching Random Walks TheMany-to-OneFormula Application: Velocity of the Leftmost Position Examples Notes Galton Watson Trees The Extinction Probability Size-Biased Galton Watson Trees Application:The Kesten StigumTheorem Notes Branching Random Walks and Martingales Branching Random Walks: Basic Notation TheAdditiveMartingale The Multiplicative Martingale TheDerivativeMartingale Notes The Spinal Decomposition Theorem Attaching a Spine to the Branching Random Walk Harmonic Functions and Doob s h-transform Change of Probabilities The Spinal Decomposition Theorem Proof of the Spinal Decomposition Theorem Example: Size-Biased Branching Random Walks Example:Abovea GivenValue Alongthe Spine Application:The Biggins MartingaleConvergenceTheorem Notes ix

10 x Contents 5 Applications of the Spinal Decomposition Theorem Assumption(H) Convergenceof the DerivativeMartingale LeftmostPosition: Weak Convergence Leftmost Position: Limiting Law Step 1: The DerivativeMartingaleis Useful Step 2: Proof of the KeyEstimate Step 3a: Proofof Lemma Step 3b: Proofof Lemma Step 4: The Role of the Non-lattice Assumption LeftmostPosition: Fluctuations Convergenceof the AdditiveMartingale The Genealogy of the Leftmost Position Proofof the Peeling Lemma Notes Branching Random Walks with Selection Branching Random Walks with Absorption The N-BRW The L-BRW Notes Biased Random Walks on Galton Watson Trees A Simple Example TheSlow Movement The Maximal Displacement FavouriteSites Notes A Sums of i.i.d. Random Variables A.1 TheRenewal Function A.2 Random Walks to Stay Above a Barrier References