Mouvement brownien branchant avec s´ election Soutenance de th` ese de Pascal M AILLARD effectu´ ee sous la direction de Zhan S HI Jury Brigitte C HAUVIN , Francis C OMETS , Bernard D ERRIDA , Yueyun H U , Andreas K YPRIANOU , Zhan S HI Rapporteurs Andreas K YPRIANOU , Ofer Z EITOUNI Universit´ e Pierre et Marie Curie 11 octobre 2012
Thesis structure Introduction + 3 chapters: The number of absorbed individuals in branching Brownian motion 1 with a barrier Branching Brownian motion with selection of the N right-most 2 particles A note on stable point processes occurring in branching Brownian 3 motion Pascal M AILLARD Mouvement brownien branchant avec s´ election 2 / 33
Thesis structure Introduction + 3 chapters: The number of absorbed individuals in branching Brownian motion 1 with a barrier Branching Brownian motion with selection of the N right-most 2 particles A note on stable point processes occurring in branching Brownian 3 motion In this presentation: Chapters 1 and 2. Pascal M AILLARD Mouvement brownien branchant avec s´ election 2 / 33
Introduction Outline 1 Introduction 2 Branching Brownian motion with absorption 3 BBM with constant population size 4 Perspectives Pascal M AILLARD Mouvement brownien branchant avec s´ election 3 / 33
Introduction Branching Brownian motion (BBM) position x Definition A particle performs standard Brownian motion started at a time point x ∈ R . Pascal M AILLARD Mouvement brownien branchant avec s´ election 4 / 33
Introduction Branching Brownian motion (BBM) x position Definition A particle performs standard Brownian motion started at a time ~exp( β ) point x ∈ R . With rate β , it branches, i.e. it dies and spawns L offspring ( L being a random variable). Pascal M AILLARD Mouvement brownien branchant avec s´ election 4 / 33
Introduction Branching Brownian motion (BBM) x position Definition A particle performs standard Brownian motion started at a time ~exp( β ) point x ∈ R . With rate β , it branches, i.e. it dies and spawns L offspring ( L being a random variable). Each offspring repeats this process independently of the others. . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 4 / 33
Introduction Branching Brownian motion (BBM) x position Definition A particle performs standard Brownian motion started at a time ~exp( β ) point x ∈ R . With rate β , it branches, i.e. it dies and spawns L offspring ( L being a random variable). Each offspring repeats this process independently of the others. − → A Brownian motion indexed by a tree . . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 4 / 33
Introduction Branching Brownian motion (BBM) (2) x position Context An example of a multitype branching process (type time ~exp( β ) space: R ) . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 5 / 33
Introduction Branching Brownian motion (BBM) (2) x position Context An example of a multitype branching process (type time ~exp( β ) space: R ) Discrete counterpart: branching random walk . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 5 / 33
Introduction Branching Brownian motion (BBM) (2) x position Context An example of a multitype branching process (type time ~exp( β ) space: R ) Discrete counterpart: branching random walk Interpretations: Model for an asexual population undergoing mutation (position = fitness) Spin glass (with infinitely deep hierarchy) Directed polymer on a tree Prototype of a travelling . . . wave Pascal M AILLARD Mouvement brownien branchant avec s´ election 5 / 33
Introduction Branching Brownian motion (BBM) (3) We always suppose m := E [ L ] − 1 > 0. Right-most particle Let R t be the position of the right-most particle. Then, as t → ∞ , almost surely on the event of survival, R t � 2 β m . → t Picture by ´ Eric Brunet Pascal M AILLARD Mouvement brownien branchant avec s´ election 6 / 33
Introduction Branching Brownian motion (BBM) (3) We always suppose m := E [ L ] − 1 > 0. Right-most particle Let R t be the position of the right-most particle. Then, as t → ∞ , almost surely on the event of survival, R t � 2 β m . → t Convention Picture by ´ Eric Brunet We will henceforth set β = 1 / ( 2 m ) . Pascal M AILLARD Mouvement brownien branchant avec s´ election 6 / 33
Introduction BBM ← → FKPP Let g : R → [ 0 , 1 ] be measurable. Define � � � u ( t , x ) = E x g ( X u ( t )) . u ∈N t Then u satisfies the following partial differential equation: Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP) equation � ∂ t u = 1 2 ∂ 2 x u + β ( E [ u L ] − u ) u ( 0 , x ) = g ( x ) (initial condition) The prototype of a parabolic PDE admitting travelling wave solutions. Pascal M AILLARD Mouvement brownien branchant avec s´ election 7 / 33
Introduction Selection 0 -x position Two models of BBM with selection : time y = - x + ct . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 8 / 33
Introduction Selection 0 -x position Two models of BBM with selection : BBM with absorption : Let f ( t ) be 1 time a continuous function (the y = - x + ct barrier ). Kill an individual as soon as its position is less than f ( t ) ( one-sided FKPP ). . . . Pascal M AILLARD Mouvement brownien branchant avec s´ election 8 / 33
Introduction Selection 0 -x position Two models of BBM with selection : BBM with absorption : Let f ( t ) be 1 time a continuous function (the y = - x + ct barrier ). Kill an individual as soon as its position is less than f ( t ) ( one-sided FKPP ). BBM with constant population 2 size ( N -BBM) : Fix N ∈ N . As . . . soon as the number of individuals exceeds N , kill the left-most individuals until the population size equals N ( noisy FKPP ). Pascal M AILLARD Mouvement brownien branchant avec s´ election 8 / 33
Branching Brownian motion with absorption Outline Introduction 1 2 Branching Brownian motion with absorption Results Proof idea 3 BBM with constant population size 4 Perspectives Pascal M AILLARD Mouvement brownien branchant avec s´ election 9 / 33
Branching Brownian motion with absorption Results Branching Brownian motion with absorption 0 -x position We take f ( t ) = − x + ct ( linear barrier ). Vast literature, known results (sample): time y = - x + ct almost sure extinction ⇔ c ≥ 1 ( c = 1: critical case c > 1: supercritical case) growth rates for c < 1. asymptotics for extinction . . . probability for c = 1 − ε , ε small We are interested in the number of absorbed individuals in the case c ≥ 1 (question raised by D. Aldous). Pascal M AILLARD Mouvement brownien branchant avec s´ election 10 / 33
Branching Brownian motion with absorption Results Our results (critical case) Let Z x denote the number of individuals absorbed at the line − x + ct . Theorem Assume that c = 1 and that E [ L ( log L ) 2 ] < ∞ . For each x > 0 , xe x P ( Z x > n ) ∼ n ( log n ) 2 , as n → ∞ . If, furthermore, E [ s L ] < ∞ for some s > 1 , then xe x P ( Z x = δ n + 1 ) ∼ as n → ∞ , δ n 2 ( log n ) 2 where δ is the span of L − 1 . Pascal M AILLARD Mouvement brownien branchant avec s´ election 11 / 33
Branching Brownian motion with absorption Results Our results (supercritical case) Theorem Assume that c > 1 and that E [ s L ] < ∞ for some s > 1 . Let λ c < λ c be the roots of the equation λ 2 − 2 c λ + 1 = 0 and define d = λ c /λ c . There ∃ K = K ( c , L ) > 0 , such that for all x > 0 , P ( Z x = δ n + 1 ) ∼ K ( e λ c x − e λ c x ) as n → ∞ . n d + 1 Pascal M AILLARD Mouvement brownien branchant avec s´ election 12 / 33
Branching Brownian motion with absorption Results Other studies Addario-Berry and Broutin (2011), A¨ ıd´ ekon (2010): Less precise tail estimates ( c = 1). A¨ ıd´ ekon, Hu and Zindy (2012+): Similar results for branching random walk ( c ≥ 1), with more explicit K . Pascal M AILLARD Mouvement brownien branchant avec s´ election 13 / 33
Branching Brownian motion with absorption Results Other studies Addario-Berry and Broutin (2011), A¨ ıd´ ekon (2010): Less precise tail estimates ( c = 1). A¨ ıd´ ekon, Hu and Zindy (2012+): Similar results for branching random walk ( c ≥ 1), with more explicit K . In contrast to the above papers, our proofs are entirely analytic. Strategy: derive asymptotics on the generating function of Z x near its singularity 1 (following an idea of R. Pemantle’s). Pascal M AILLARD Mouvement brownien branchant avec s´ election 13 / 33
Branching Brownian motion with absorption Proof idea The number of absorbed individuals 0 position -y -x Theorem (Neveu, 1988) ( Z x ) x ≥ 0 is a continuous-time Galton–Watson process. The time infinitesimal generating function a ( s ) = d E [ s Z x ] / d x admits the decomposition a = − ψ ′ ◦ ψ − 1 , . . . . . . where ψ is an FKPP travelling wave of speed c, i.e. . . . 2 ψ ′′ ( s ) − c ψ ′ ( s ) + β ( E [ s L ] − s ) = 0 , 1 . . . and ψ ( x ) ↑ 1 , as x → ∞ . Pascal M AILLARD Mouvement brownien branchant avec s´ election 14 / 33
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