A. Bovier () Branching Brownian motion: extremal process and ergodic theorems

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1 Branching Brownian motion: extremal process and ergodic theorems Anton Bovier with Louis-Pierre Arguin and Nicola Kistler RCS&SM, Venezia,

2 Plan 1 BBM 2 Maximum of BBM 3 The Lalley-Sellke conjecture 4 The extremal process of BBM 5 Ergodic theorems 6 Universality

3 Branching Brownian motion Branching Brownian Motion Branching Brownian motion is one of the fundamental models in probability. It combines two classical objects:

4 Branching Brownian motion Branching Brownian Motion Branching Brownian motion is one of the fundamental models in probability. It combines two classical objects: Brownian motion Pure random motion

5 Branching Brownian motion Branching Brownian Motion Branching Brownian motion is one of the fundamental models in probability. It combines two classical objects: Brownian motion Galton-Watson process Pure random motion Pure random genealogy

6 Branching Brownian motion Branching Brownian motion Branching Brownian motion (BBM) combines the two processes: Each particle of the Galton-Watson process performs Brownian motion independently of any other. This produces an immersion of the Galton-Watson process in space.

7 Branching Brownian motion Branching Brownian motion Branching Brownian motion (BBM) combines the two processes: Each particle of the Galton-Watson process performs Brownian motion independently of any other. This produces an immersion of the Galton-Watson process in space. Picture by Matt Roberts, Bath Maury Bramson H. McKean A.V. Skorokhod J.E. Moyal BBM is the canonical model of a spatial branching process.

8 Branching Brownian motion Galton-Watson tree and corresponding BBM

9 Branching Brownian motion Galton-Watson tree and corresponding BBM

10 Branching Brownian motion BBM as Gaussian process

11 Branching Brownian motion BBM as Gaussian process Fix GW-tree. Label individuals at time t as i 1 (t),..., i n(t) (t)

12 Branching Brownian motion BBM as Gaussian process Fix GW-tree. Label individuals at time t as i 1 (t),..., i n(t) (t) d(i l (t), i k (t)) time of most recent common ancestor of i l (t) and i k (t)

13 Branching Brownian motion BBM as Gaussian process Fix GW-tree. Label individuals at time t as i 1 (t),..., i n(t) (t) d(i l (t), i k (t)) time of most recent common ancestor of i l (t) and i k (t) BBM is Gaussian process with covariance Ex k (t)x l (s) = d(i k (t), i l (s))

14 Branching Brownian motion BBM as Gaussian process Fix GW-tree. Label individuals at time t as i 1 (t),..., i n(t) (t) d(i l (t), i k (t)) time of most recent common ancestor of i l (t) and i k (t) BBM is Gaussian process with covariance BBM special case of models where Ex k (t)x l (s) = d(i k (t), i l (s)) Ex k (t)x l (t) = ta ( t 1 d(i k (t), i l (t)) ) for A : [0, 1] [0, 1].

15 Branching Brownian motion BBM as Gaussian process Fix GW-tree. Label individuals at time t as i 1 (t),..., i n(t) (t) d(i l (t), i k (t)) time of most recent common ancestor of i l (t) and i k (t) BBM is Gaussian process with covariance BBM special case of models where Ex k (t)x l (s) = d(i k (t), i l (s)) Ex k (t)x l (t) = ta ( t 1 d(i k (t), i l (t)) ) for A : [0, 1] [0, 1]. GREM models of spin-glasses.

16 Maximum of BBM First question: how big is the biggest?

17 Maximum of BBM First question: how big is the biggest? To compare:

18 Maximum of BBM First question: how big is the biggest? To compare: Single Brownian motion: [ P X (t) x ] t = 1 x ) exp ( z2 dz 2π 2

19 Maximum of BBM First question: how big is the biggest? To compare: Single Brownian motion: [ P X (t) x ] t = 1 x ) exp ( z2 dz 2π 2 e t = En(t) independent Brownian motions: [ P max x k(t) t 2 1 ] k=1,...,e t 2 2 ln t + x e 4πe 2x BBM Many BMs

20 Maximum of BBM The KPP-F equation

21 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation:

22 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2

23 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2 Fisher Kolmogorov Petrovsky

24 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2 Fisher Kolmogorov Petrovsky Fischer used this equation to model the evolution of biological populations. It accounts for:

25 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2 Fisher Kolmogorov Petrovsky Fischer used this equation to model the evolution of biological populations. It accounts for: birth: v,

26 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2 Fisher Kolmogorov Petrovsky Fischer used this equation to model the evolution of biological populations. It accounts for: birth: v, death: v 2,

27 Maximum of BBM The KPP-F equation One of the simplest reaction-diffusion equations is the Fisher-Kolmogorov-Petrovsky- Piscounov (F-KPP) equation: t v(x, t) = x v(x, t)+v v 2 Fisher Kolmogorov Petrovsky Fischer used this equation to model the evolution of biological populations. It accounts for: birth: v, death: v 2, diffusive migration: 2 x v.

28 Maximum of BBM KPP-F equation and the maximum of of BBM

29 Maximum of BBM KPP-F equation and the maximum of of BBM [ ] u(t, x) P max x k(t) x k=1...n(t)

30 Maximum of BBM KPP-F equation and the maximum of of BBM [ ] u(t, x) P max x k(t) x k=1...n(t) McKean, 1975: 1 u solves the F-KPP equation, i.e. { t u = x u + u 2 1 if x 0 u, u(0, x) = 0 if x < 0

31 Maximum of BBM KPP-F equation and the maximum of of BBM [ ] u(t, x) P max x k(t) x k=1...n(t) McKean, 1975: 1 u solves the F-KPP equation, i.e. { t u = x u + u 2 1 if x 0 u, u(0, x) = 0 if x < 0 Bramson, 1978: u(t, x+m(t)) ω(x), m(t) = 2t ln t

32 Maximum of BBM KPP-F equation and the maximum of of BBM [ ] u(t, x) P max x k(t) x k=1...n(t) McKean, 1975: 1 u solves the F-KPP equation, i.e. { t u = x u + u 2 1 if x 0 u, u(0, x) = 0 if x < 0 Bramson, 1978: u(t, x+m(t)) ω(x), m(t) = 2t ln t where ω(x) solves x ω + 2 x ω + ω 2 ω = 0.

33 The Lalley-Sellke conjecture The derivative martingale

34 The Lalley-Sellke conjecture The derivative martingale Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution [ ω(x) = E e CZe 2x ],

35 The Lalley-Sellke conjecture The derivative martingale Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution [ ω(x) = E e CZe 2x ], Z (d) = lim t Z(t), where Z(t) is the derivative martingale, Z(t) = k n(t) { 2t x k (t)}e 2{ 2t x k (t)}

36 The Lalley-Sellke conjecture The derivative martingale Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution [ ω(x) = E e CZe 2x ], Z (d) = lim t Z(t), where Z(t) is the derivative martingale, Z(t) = k n(t) { 2t x k (t)}e 2{ 2t x k (t)} Lalley-Sellke conjecture: P-a.s., for any x R, lim T 1 T 1I T { } = exp ( CZe ) 2x max n(t) 0 k=1 x k(t) m(t) x

37 The extremal process of BBM Looking at BBM from the top Closer look at the extremes: Zooming into the top Can we describe the asymptotic structure of the largest points, and their genealogical structure?

38 The extremal process of BBM Classical Poisson convergence for many BMs From classical extreme values statistics one knows: Let X i (t), i N, iid Brownian motions. Then, the point process e t ( ) P t δ Xi (t) 2t+ 1 2 ln t PPP 4πe x dx, 2 i=1 where PPP(µ) is Poisson point process with intensity measure µ.

39 The extremal process of BBM Extensions to correlated processes

40 The extremal process of BBM Extensions to correlated processes GREM [Derrida 82]: Recall Ex il (t)x ik (t) = ta ( t 1 d(i l (t), i k (t)) ). A increasing step function.

41 The extremal process of BBM Extensions to correlated processes GREM [Derrida 82]: Recall Ex il (t)x ik (t) = ta ( t 1 d(i l (t), i k (t)) ). A increasing step function. Extreme behaviour relatively insensitive to correlations: If A(x) < x, x (0, 1), then no change in the extremal process.

42 The extremal process of BBM Extensions to correlated processes GREM [Derrida 82]: Recall Ex il (t)x ik (t) = ta ( t 1 d(i l (t), i k (t)) ). A increasing step function. Extreme behaviour relatively insensitive to correlations: If A(x) < x, x (0, 1), then no change in the extremal process. Poisson cascades: If A takes only finitely many values, and A(x) > x; for some x (0, 1), the extremal process is known (Derrida, B-Kurkova) and given by Poisson cascade process.

43 The extremal process of BBM Extensions to correlated processes GREM [Derrida 82]: Recall Ex il (t)x ik (t) = ta ( t 1 d(i l (t), i k (t)) ). A increasing step function. Extreme behaviour relatively insensitive to correlations: If A(x) < x, x (0, 1), then no change in the extremal process. Poisson cascades: If A takes only finitely many values, and A(x) > x; for some x (0, 1), the extremal process is known (Derrida, B-Kurkova) and given by Poisson cascade process. Borderline: If A takes only finitely many values, and A(x) x, for all x [0, 1], but A(x) = x, for some x (0, 1), the extremal process is again Poisson, but with reduced intensity (B-Kurkova).

44 The extremal process of BBM Extensions to correlated processes GREM [Derrida 82]: Recall Ex il (t)x ik (t) = ta ( t 1 d(i l (t), i k (t)) ). A increasing step function. Extreme behaviour relatively insensitive to correlations: If A(x) < x, x (0, 1), then no change in the extremal process. Poisson cascades: If A takes only finitely many values, and A(x) > x; for some x (0, 1), the extremal process is known (Derrida, B-Kurkova) and given by Poisson cascade process. Borderline: If A takes only finitely many values, and A(x) x, for all x [0, 1], but A(x) = x, for some x (0, 1), the extremal process is again Poisson, but with reduced intensity (B-Kurkova). What happens at the natural border A(x) = x??

45 The extremal process of BBM The life of BBM

46 The extremal process of BBM The life of BBM General principle: follow history of the leading particles! There are three phases with distinct properties and effects:

47 The extremal process of BBM The life of BBM General principle: follow history of the leading particles! There are three phases with distinct properties and effects: the early years

48 The extremal process of BBM The life of BBM General principle: follow history of the leading particles! There are three phases with distinct properties and effects: the early years midlife

49 The extremal process of BBM The life of BBM General principle: follow history of the leading particles! There are three phases with distinct properties and effects: the early years midlife before the end

50 The extremal process of BBM The life of BBM General principle: follow history of the leading particles! There are three phases with distinct properties and effects: the early years midlife before the end Let us look at them...

51 The extremal process of BBM The early years... Randomness persists for all times from what happened in the early history:

52 The extremal process of BBM The early years... Randomness persists for all times from what happened in the early history: Both

53 The extremal process of BBM The early years... Randomness persists for all times from what happened in the early history: Both and occur with positive probability, independent of t! In the second case, all particles at time r have the same chance to have offspring that is close to the maximum.

54 The extremal process of BBM The early years... Randomness persists for all times from what happened in the early history: Both and occur with positive probability, independent of t! In the second case, all particles at time r have the same chance to have offspring that is close to the maximum. Two consequences: the random variable Z, the derivative martingale particles near the maximum at time t can have common ancestors at finite, t-independent times (when t ).

55 The extremal process of BBM...Midlife...

56 The extremal process of BBM...Midlife... Key fact: The function m(t) is convex: The function m(t)

57 The extremal process of BBM...Midlife... Key fact: The function m(t) is convex: Descendants of a particle maximal at time 0 s t cannot be maximal at time t! The function m(t)

58 The extremal process of BBM...Midlife... Key fact: The function m(t) is convex: Descendants of a particle maximal at time 0 s t cannot be maximal at time t! Particles realising the maximum at time t have ancestors at times s that are selected from the very many particles that are a lot below the maximum at time s. The function m(t)

59 The extremal process of BBM...Midlife... Key fact: The function m(t) is convex: Descendants of a particle maximal at time 0 s t cannot be maximal at time t! Particles realising the maximum at time t have ancestors at times s that are selected from the very many particles that are a lot below the maximum at time s. Offspring of the selected particles is atypical! The function m(t)

60 The extremal process of BBM...Midlife... Key fact: The function m(t) is convex: Descendants of a particle maximal at time 0 s t cannot be maximal at time t! Particles realising the maximum at time t have ancestors at times s that are selected from the very many particles that are a lot below the maximum at time s. Offspring of the selected particles is atypical! The function m(t) Only one descendant of a selected particle at times 0 s t can be at finite distance from the maximum at time t.

61 The extremal process of BBM...just before the end Any particle the arrives close to the maximum at time t can have produced offspring shortly before. These will be only a finite amount smaller then their brothers. Hence, particles near the maximum come in small families.

62 The extremal process of BBM The extremal process

63 The extremal process of BBM The extremal process Let E t n(t) δ xi (t) m(t) i=1

64 The extremal process of BBM The extremal process Let E t n(t) δ xi (t) m(t) i=1 Let Z be the limit of the derivative martingale, and set P Z = ( ) δ pi PPP CZe 2x dx i N

65 The extremal process of BBM The extremal process Let E t n(t) δ xi (t) m(t) i=1 Let Z be the limit of the derivative martingale, and set P Z = ( ) δ pi PPP CZe 2x dx i N Let L(t) { max j n(t) x j (t) > 2t } and

66 The extremal process of BBM The extremal process Let E t n(t) δ xi (t) m(t) i=1 Let Z be the limit of the derivative martingale, and set P Z = ( ) δ pi PPP CZe 2x dx i N Let L(t) { max j n(t) x j (t) > 2t } and (t) k δ xk (t) max j n(t) x j (t) conditioned on L(t).

67 The extremal process of BBM The extremal process Let E t n(t) δ xi (t) m(t) i=1 Let Z be the limit of the derivative martingale, and set P Z = ( ) δ pi PPP CZe 2x dx i N Let L(t) { max j n(t) x j (t) > 2t } and (t) k δ xk (t) max j n(t) x j (t) conditioned on L(t). Law of (t) under P ( L(t)) converges to law of point process,. Let (i) be iid copies of, with atoms (i) j.

68 The extremal process of BBM The extremal process

69 The extremal process of BBM The extremal process Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013)) With the notation above, the point process E t converges in law to a point process E, given by E i,j N δ pi + (i) j

70 The extremal process of BBM The extremal process Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013)) With the notation above, the point process E t converges in law to a point process E, given by E i,j N δ pi + (i) j

71 The extremal process of BBM The extremal process Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013)) With the notation above, the point process E t converges in law to a point process E, given by E i,j N δ pi + (i) j Similar result obtained independently by Aïdékon, Brunet, Berestycki, and Shi.

72 Ergodic theorems Ergodic theorem for the max Alternative look: what happens if we consider time averages? Naively one might expect a law of large numbers: lim T 1 T ( ) 1I T { } = E exp CZe 2x max n(t) 0 k=1 x k(t) m(t) x a.s. But this cannot be true!

73 Ergodic theorems Ergodic theorem for the max Alternative look: what happens if we consider time averages? Naively one might expect a law of large numbers: lim T 1 T ( ) 1I T { } = E exp CZe 2x max n(t) 0 k=1 x k(t) m(t) x a.s. But this cannot be true! Lalley and Sellke conjectured a random version: Theorem (Arguin, B, Kistler, 2012 (EJP 18)) P-a.s., for any x R, lim T 1 T 1I T { } = exp ( CZe ) 2x max n(t) 0 k=1 x k(t) m(t) x

74 Ergodic theorems Ergodic theorem for the extremal process We prove this conjecture and extend it to the entire extremal process:

75 Ergodic theorems Ergodic theorem for the extremal process We prove this conjecture and extend it to the entire extremal process: Theorem (Arguin, B, Kistler (2012)) E t converges P-almost surely weakly under time-average to the Poisson cluster process E Z. That is, P-a.s., f C c + (R), ( ) [ ( )] 1 exp f (y)e t,ω (dy) dt E exp f (y)e Z (dy) T Here E Z is the process E for given value Z of the derivative martingale, E is w.r.t. the law of that process, given Z.

76 Ergodic theorems Elements of the proof For ε > 0 and R T T, decompose 1 T ( exp T 0 ) f (y)e t,ω (dy) dt

77 Ergodic theorems Elements of the proof For ε > 0 and R T T, decompose 1 T ( exp T 0 ) f (y)e t,ω (dy) dt = 1 T εt ( ) exp f (y)e t,ω (dy) dt } 0 {{ } (I ):vanishes as ɛ 0

78 Ergodic theorems Elements of the proof For ε > 0 and R T T, decompose 1 T ( exp T 0 ) f (y)e t,ω (dy) dt = 1 εt ( ) exp f (y)e t,ω (dy) dt T 0 }{{} (I ):vanishes as ɛ T [ ( FRT ] E exp f (y)e t,ω (dy)) dt T εt }{{} (II ):what we want

79 Ergodic theorems Elements of the proof For ε > 0 and R T T, decompose 1 T ( exp T 0 ) f (y)e t,ω (dy) dt = 1 εt ( ) exp f (y)e t,ω (dy) dt T 0 }{{} (I ):vanishes as ɛ T [ ( FRT ] E exp f (y)e t,ω (dy)) dt T εt }{{} (II ):what we want T + 1 Y t (ω)dt T εt }{{} (III ): needs LLN

80 Ergodic theorems Elements of the proof: the LLN

81 Ergodic theorems Elements of the proof: the LLN ( T ) [ ( T FRT ] (III ) exp f (y)e t,ω (dy) E exp f (y)e t,ω (dy)) ɛt ɛt should vanish by a law of large numbers.

82 Ergodic theorems Elements of the proof: the LLN ( T ) [ ( T FRT ] (III ) exp f (y)e t,ω (dy) E exp f (y)e t,ω (dy)) ɛt ɛt should vanish by a law of large numbers. We use a criterion which is an adaptation of the theorem due to Lyons: Lemma Let {Y s } s R+ be a.s. uniformly bounded and E[Y s ] = 0 for all s. If T =1 [ 1 T E 1 T T 0 Y s ds 2] <, then 1 T Y s ds 0, T 0 a.s.

83 Ergodic theorems LLN Requires covariance estimate: Lemma Let Y s from (III). For R T = o( T ) with lim T R T = +, there exists κ > 0, s.t. E[Y s Y s ] Ce Rκ T for any s, s [εt, T ] with s s R T.

84 Universality Universality The new extremal process of BBM should not be limited to BBM: Branching random walk (Aïdekon, Madaule) Gaussian free field in d = 2 [Bolthausen, Deuschel, Giacomin, Bramson, Zeitouni,Biskup and Louisdor (!)...] Cover times of random walks [Ding, Zeitouni, Sznitman,...] Spin glasses with log-correlated potentials [Fyodorov, Bouchaud,..] and building block for further models: Extensions to stronger correlations: [Fang-Zeitouni 12]... beyond the borderline Extension back to spin glasses: some of the observations made give hope... see Louis-Pierre s talk

85 Universality References L.-P. Arguin, A. Bovier, and N. Kistler, The genealogy of extremal particles of branching Brownian motion, Commun. Pure Appl. Math. 64, (2011). L.-P. Arguin, A. Bovier, and N. Kistler, Poissonian statistics in the extremal process of braching Brownian motion, Ann. Appl. Probab. 22, (2012). L.-P. Arguin, A. Bovier, and N. Kistler, The extremal process of branching Brownian motion, to appear in Probab. Theor. Rel. Fields (2012). L.-P. Arguin, A. Bovier, and N. Kistler, An ergodic theorem for the frontier of branching Brownian motion, EJP 18 (2013). L.-P. Arguin, A, Bovier, and N. Kistler, An ergodic theorem for the extremal process of of branching Brownian motion, arxiv: , (2012). E. Aïdékon, J. Berestyzki, É. Brunet, Z. Shi, Branching Brownian motion seen from its tip, to appear in Probab. Theor. Rel. Fields (2012). Thank you for your attention!