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
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
http://recherche.math.univ-bpclermont.fr/stflour/stflour-en.php Christophe Bahadoran [email protected] Arnaud Guillin [email protected] Laurent Serlet [email protected]
Zhan Shi
Branching Random Walks
École d’Été de Probabilités
de Saint-Flour XLII – 2012
Zhan Shi
Laboratoire de Probabilités et ModelesJ Aléatoires
Université Pierre et Marie Curie Paris, France
ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics
ISBN 978-3-319-25371-8 ISBN 978-3-319-25372-5 (eBook) DOI 10.1007/978-3-319-25372-5
Library of Congress Control Number: 2015958655
Mathematics Subject Classification: 60J80, 60J85, 60G50, 60K37 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015
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To the memory of my teacher,
Professor Marc Yor (1949–2014)
但
去
莫复问，
白
云无尽时。
－
Œ唐
王维《送别》
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 Zhan Shi
August 2015
Contents
1 Introduction . . . 1
1.1 Branching Brownian Motion. . . 1
1.2 Branching Random Walks . . . 3
1.3 The Many-to-One Formula . . . 5
1.4 Application: Velocity of the Leftmost Position . . . 6
1.5 Examples.. . . 8
1.6 Notes . . . 10
2 Galton–Watson Trees . . . 11
2.1 The Extinction Probability .. . . 11
2.2 Size-Biased Galton–Watson Trees . . . 13
2.3 Application: The Kesten–Stigum Theorem . . . 16
2.4 Notes . . . 17
3 Branching Random Walks and Martingales . . . 19
3.1 Branching Random Walks: Basic Notation . . . 19
3.2 The Additive Martingale . . . 21
3.3 The Multiplicative Martingale . . . 22
3.4 The Derivative Martingale . . . 26
3.5 Notes . . . 27
4 The Spinal Decomposition Theorem . . . 29
4.1 Attaching a Spine to the Branching Random Walk . . . 29
4.2 Harmonic Functions and Doob’sh-Transform .. . . 30
4.3 Change of Probabilities . . . 31
4.4 The Spinal Decomposition Theorem . . . 33
4.5 Proof of the Spinal Decomposition Theorem . . . 35
4.6 Example: Size-Biased Branching Random Walks . . . 39
4.7 Example: Above a Given Value Along the Spine . . . 40
4.8 Application: The Biggins Martingale Convergence Theorem .. . . 42
4.9 Notes . . . 44
x Contents
5 Applications of the Spinal Decomposition Theorem. . . 45
5.1 Assumption (H). . . 45
5.2 Convergence of the Derivative Martingale . . . 47
5.3 Leftmost Position: Weak Convergence .. . . 54
5.4 Leftmost Position: Limiting Law . . . 62
5.4.1 Step 1: The Derivative Martingale is Useful . . . 63
5.4.2 Step 2: Proof of the Key Estimate . . . 65
5.4.3 Step 3a: Proof of Lemma 5.18 .. . . 71
5.4.4 Step 3b: Proof of Lemma 5.19. . . 76
5.4.5 Step 4: The Role of the Non-lattice Assumption . . . 79
5.5 Leftmost Position: Fluctuations. . . 82
5.6 Convergence of the Additive Martingale . . . 87
5.7 The Genealogy of the Leftmost Position . . . 88
5.8 Proof of the Peeling Lemma . . . 89
5.9 Notes . . . 97
6 Branching Random Walks with Selection . . . 99
6.1 Branching Random Walks with Absorption.. . . 99
6.2 TheN-BRW . . . 102
6.3 TheL-BRW . . . 104
6.4 Notes . . . 105
7 Biased Random Walks on Galton–Watson Trees . . . 107
7.1 A Simple Example . . . 107
7.2 The Slow Movement . . . 108
7.3 The Maximal Displacement. . . 110
7.4 Favourite Sites . . . 112
7.5 Notes . . . 113
A Sums of i.i.d. Random Variables . . . 115
A.1 The Renewal Function . . . 115
A.2 Random Walks to Stay Above a Barrier . . . 116
