Extremal Particles in Branching Brownian Motion - Part 1

Transcription

1 Extremal Particles in Branching Brownian Motion - Part 1 Sandra Kliem (Part 1) und Kumarjit Saha (Part 2) Singapore - Genealogies of Interacting Particle Systems

2 Overview: Part 1 Part 1: Extremal Particles in Branching Brownian Motion. Part 2: Branching Brownian Motion under Selection. Branching Brownian motion (BBM) - Definition and basic properties. The F-KPP equation and its connection to BBM. The distribution of the maximum of BBM. Extremal particles in BBM Genealogy of extremal particles of BBM (cf. Arguin, Bovier and Kistler, [ABK11]); Poissonian statistics in the extremal process of BBM (cf. [ABK12]). Remark: Following the bibliography, further slides are given relating to The extremal process of BBM (cf. [ABK13]); BBM seen from its tip (cf. Aïdékon, Berestycki, Brunet and Shi, [ABBS13]); Additional details, with pointers given by. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

3 Branching Brownian Motion Definition (Branching Brownian Motion (BBM)) t = 0: single particle x 1 (0) starts at origin; moves as a Brownian motion (BM) in R 1 until after (at) time τ Exp(1) it splits into two identical particles that start (both) at x 1 (τ) and move as two independent BMs each. Repeat. The resulting process is a collection of a (random) number n(t) of particles ( x1 (t), x 2 (t),..., x n(t) (t) ) t 0 = { x k (t) : k n(t) } t 0. Note 1. n(τ) = 2. Note 2. Here we use w.l.o.g. binary branching. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

4 (see homepage of Matt Roberts, Univ. of Bath) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

5 (see homepage of Matt Roberts, Univ. of Bath) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

6 We have seen M (t) = maxu2n Xu (t) the position of the rightmost particle can be was alive at time s. studied through the analysis of the KPP equation. In this section we start our ext ploration of the extremal point process of the branching Brownian motion by looking ne Mtat=themax Xu (t). asymptotic behavior of M (t). u N(t) Figure 3.1:Asymptotics Position of M. (see J. Berestycki (Lecture Notes, Topics on BBM, t (Image tiroberts@gmail.com (McGill) for M the frontier ofbybbm Roberts) 9th September, 2011 Image by Matt Roberts) 3.1 Kolmogorov s and Bramson s result As we have seen in the previous chapter, u(t, x) = P0 (M (t) x) solves the F-KPP Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

7 Remark (see Bovier, [B15] for more respectively for references to literature) 1 There are connections to spin glass theory; in particular, Generalised Random Energy models (GREM). 2 Many results can be extended to branching random walk. 3 Connection to extremes of the free Gaussian random field in d = 2. 4 Can be extended to variable speed BBM. Remark (Genealogies of BBM) Let d(x k (t), x l (t)) inf{0 s t : x k (s) x l (s)} = time (from 0) to MRCA unique time where the most recent common ancestor split = time of death of longest surviving ancestor of both particles. A BBM can then also be constructed as follows. Construct first a continuous time Galton-Watson tree with binary branching. Let I(t) be the set of its leaves at time t. Then, conditional on the GW-process, BBM is a Gaussian process x k (t), k I(t) with E[x k (t)] = 0 and Cov(x k (t), x l (t)) = d(x k (t), x l (t)). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

8 First Properties 1 E[n(t)] = e t. 2 e t n(t) is a martingale that converges, a.s. and in L 1, to an exponential r.v. of parameter 1. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

9 The F-KPP equation Consider the partial differential equation u t = 1 2 u xx + u 2 u, u = u(t, x) [0, 1], t 0, x R, u(0, x) = f (x). (1) Set v = 1 u. Then this is a special case of the Kolmogorov-Petrovskii-Piskunov-(KPP)- equation (also known as the Kolmogorov- or Fisher-equation). Let { x k (t) : k n(t) } be a BBM starting at 0 and f : R [0, 1]. Then t 0 n(t) u(t, x) = E f (x x k (t)) k=1 is the solution to the F-KPP equation (1) with u(0, x) = f (x). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

10 The F-KPP equation Consider the partial differential equation u t = 1 2 u xx + u 2 u, u = u(t, x) [0, 1], t 0, x R, u(0, x) = f (x). (1) Set v = 1 u. Then this is a special case of the Kolmogorov-Petrovskii-Piskunov-(KPP)- equation (also known as the Kolmogorov- or Fisher-equation). Let { x k (t) : k n(t) } be a BBM starting at 0 and f : R [0, 1]. Then t 0 n(t) u(t, x) = E f (x x k (t)) k=1 is the solution to the F-KPP equation (1) with u(0, x) = f (x). Idea of proof: Branching property: Let p t(x) = 1 2πt e x2 /2t denote the Heat kernel. Then (use that τ Exp(1), i.e. f τ (s) = 1 {s 0} e s and P(τ > t) = e t ) t u(t, x) = e t p t(z)f (x z)dz + e s p s(z)u 2 (t s, x z)dzds. 0 Now differentiate w.r.t. t, use integration by parts and t pt(x) = 2 p t (x). x x 2 Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

11 u t = 1 n(t) 2 uxx + u2 u, u(0, x) = f (x) has as solution u(t, x) = E f (x x k (t)). k=1 Example 1. Let M(t) max k n(t) x k (t). With f (x) = 1 [0, ) (x) we obtain n(t) u(t, x) = E 1 [0, ) (x x k (t)) k=1 n(t) = E k=1 1 {xk (t) x} = P ( max k n(t) x k(t) x ) = P(M(t) x) = F M(t) (x). Example 2. Let f (x) = e φ(x), φ C c +. Set P t n(t) k=1 δ x k (t). Then n(t) u(t, x) = E e φ(x x k (t)) = E [ e φ(x z)p ] t(dz). k=1 Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

12 The F-KPP equation revisited Consider v(t, x) 1 u(t, x) instead of u(t, x). Then v(t, x) solves v t = 1 2 v xx v 2 + v = 1 2 v xx + v(1 v), v(0, x) = 1 u(0, x). (2) The Feynman-Kac formula yields the following representation for the linear equation Namely, v t = 1 2 v xx + k(t, x)v, [ ( t v(t, x) = E exp 0 ( t = E x [exp 0 v 0 (x) = v(0, x). ) ] k(t s, B x (s))ds v(0, B x (t)) ) ] k(t s, B(s))ds v(0, B(t)). Here, (B x (t)) t 0 is a BM, starting in x R. Bramson [B83] sets k(t, x) 1 v(t, x) with v solving (2). Then ( t ) ] ( ) v(t, x) = E x [exp 1 v(t s, B(s)) ds v(0, B(t)) 0 ( implicit description of v ). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

13 ( t v(t, x) = E x [exp 0 ( t = E x [exp 0 ) ] k(t s, B(s))ds v(0, B(t)) ) ] ( ) 1 v(t s, B(s)) ds v(0, B(t)). Observations: (recall: 1 v(t, x) = u(t, x), u(0, x) = 1 [0, ) (x)) v(t, x) is the weighted average of the different sample paths of BM. 0 < k(t s, B(s)) < 1, can show: lim y k(r, y) ɛ for r r(ɛ), lim y k(r, y) = 1; i.e., for y large, weighting of path nearly maximal, for y small, insignificant. Transition near k(r, y) = 1/2 u(r, y) = 1/2 (preview: m(t) = sup{x : u(t, x) 1/2} + O(1)). Bramson [B83] distinguishes paths x(s), 0 s t according to ( t ) ( t ) exp k(t s, x(s)) ds e t or exp k(t s, x(s)) ds e t. 0 0 Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

14 How to calculate v(t, x)? [ ( t v(t, x) = E exp 0 [ [ ( t = E E exp = ) ] k(t s, B x (s))ds v(0, B x (t)) 0 ) ]] k(t s, B x (s))ds v(0, B x (t)) B x (t) [ /2t ( t v(0, y) e (x y)2 E exp k(t s, z t x,y (s))ds) ] dy, 2πt where z t x,y denotes a Brownian bridge starting at x (at time 0) and ending at y (at time t). A Brownian bridge has the distribution of a BM starting at x, conditional on being in y at time t. z t 0,0 (s) B0 (s) s t B0 (t), 0 s t is a Gaussian process (all finite-dim. distrib.s are normally distributed), a.s. continuous on [0, t], a strong Markov process and indep. of B 0 (t). z t x,y (s) D = z t 0,0 (s) + s t y + t s t x, 0 s t. Var(z t s(t s) 0,0 (s)) = t 0 with a maximum in the middle, Var(z t 0,0 (t/2)) = t/4.... Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

15 (Brownian Bridge, cf. Figure-Sample-path-examples-of-a-Brownian-bridge-for-different-initial-and-final-states) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

16 The distribution of the maximum of BBM Let us return to Example 1. Then u(t, x) = P(M(t) x) = P ( max k n(t) x k (t) x ) solves (1) with u(0, x) = 1 [0, ) (x). As a result: 0 < u(t, x) < 1 for all x R, t > 0. Take as centering term, m(t) 2t log(t), (in i.i.d. case 2t log(t) 1 ) then m(t) = sup{x : u(t, x) 1/2} + O(1) (cf. Bramson [B83], Roberts [R13]) and P(M(t) m(t) x) = u(t, m(t) + x) w(x) unif. in x as t. (3) Here, w(x) is the unique (up to translation) solution of the equation 1 2 w xx + 2w x + w 2 w = 0 satisfying 0 < w(x) < 1 for all x R and w(x) 0 as x, w(x) 1 as x. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

17 The derivative martingale Lalley and Sellke [LS87] show: Let n(t) ( Z(t) 2t xk (t) ) 2t xk ) e 2( (t) k=1 be the so-called derivative martingale, then Z = lim Z(t) (4) t exists and is strictly positive a.s. Moreover, for some C > 0, P(M(t) m(t) x) t w(x) = E[ exp ( CZe 2x )] [ ( ) 2(x = E exp e log(cz) )]. 2 (5) Note 1. The so-called Gumbel distribution has cumulative distribution function F G (x) = P(G x) = exp ( e (x µ)/β) = exp ( e µ/β e x/β) with parameters µ R, β > 0. Hence, w(x) represents a random shift of the Gumbel distribution with µ = 0 and β = 1/ 2. Note 2. 1 w(x) Cxe 2x for x. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

18 (Gumbel distribution, cf. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

19 Additionally, [LS87] showed Theorem Suppose two independent BBMs ( X A 1 (t),..., X A n A (t) (t)) and ( X B 1 (t),..., X B n B (t) (t)) are started at 0 respectively x < 0. Then, with probability 1, there exist finite random times t n, n N, t n such that for all n N. M A (t n ) < M B (t n ) Idea of proof. Use (3), i.e. lim t P(M(t) m(t) x) = w(x). Corollary Every particle born in a BBM has a descendant particle in the lead at some future time. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

20 Extremal particles in BBM (cf. [ABK11] and [ABK12]) Arguin, L.-P. and Bovier, A. and Kistler, N. Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 (2011), Arguin, L.-P. and Bovier, A. and Kistler, N. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012), We are interested in the limit (t ) of the extremal process E t δ xk (t) m(t) δ xk (t). k n(t) Recall: The centering term m(t) satisfies k n(t) m(t) = 2t 3 2 log(t) = sup{x : u(t, x) 1/2} + O(1). 2 By the previous Remark on Genealogies of BBM, for a given realization of the branching, the genealogical distances d(x k (t), x l (t)) = inf{0 s t : x k (s) x l (s)} = time (from 0) to MRCA. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

21 Genealogy of extremal particles of BBM Theorem (Genealogy of Extremal Particles, Theorem 2.1, [ABK11]) For any compact set D R, lim sup P ( 1 k, l n(t) : r t>3r x k (t), x l (t) D and d(x k (t), x l (t)) (r, t r) ) = 0. Conclusion: The MRCA of extremal particles at time t splits/branches off with high probability at a time 1 in the interval (0, r) ( very early branching ) or 2 in the interval (t r, t) ( very late branching ) L. P. ARGUIN, A. BOVIER, AND N. KISTLER space s 7! s t m.t/ m.t/ extremal particles free evolution r t r t time no branching FIGURE Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles 2.4. Evolution in BBM of the system. 3. August, / 36

22 1658 L. P. ARGUIN, A. BOVIER, AND N. KISTLER space s 7! s t m.t/ m.t/ extremal particles free evolution r no branching t r t time FIGURE 2.4. Evolution of the system. (Figure 2.4 of [ABK11]) Brunet and Derrida also conjecture that the extremal process of BBM retains Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

23 Poissonian statistics in the extremal process of BBM W.l.o.g., order the (centered) particles in decreasing order, i.e. x 1 (t) x 2 (t) x n(t) (t). (6) Let { } d(xk (t), x l (t)) D(t) = {D kl (t)} k,l n(t). t k,l n(t) Definition Let 0 < q < 1. The q-thinning E (q) t of the pair (E t, D(t)) is defined as follows: Consider the equivalence classes of particles alive at time t with MRCA at a time later than q t, i.e. k q l D kl (t) q. Select the maximal (according to (6)) particle within each class. Then E (q) t is the point process of these representatives. Note 1. This extends to q = q(t) (0, 1). Note 2. The thinning map (E t, D(t)) E (q) (t) is continuous (on the space of pairs (X, Q), X ordered positions, Q symm. matrix with entries in [0, 1] and transitive op. Q ij q). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

24 Theorem (Theorem 2, [ABK12]) For any 0 < q < 1, the processes E (q) t converge in law to the same limit, E 0. Also, lim lim E (1 r/t) t = E 0. r t Moreover, conditionally on Z, (the limit of the derivative martingale, cf. (4)), E 0 = PPP ( CZ 2e 2x dx ) where C > 0 is the constant appearing in (5). (Figure 2 in [ABK12]) 1698 L.-P. ARGUIN, A. BOVIER AND N. KISTLER (PPP stands for Poisson Point Process ) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

25 (Figure 1 of [ABK13]) Conclusion: The particles Fig. 1atOnset theofrontier extremal ofprocess BBM for large times can be constructed as follows: 1 set down so-called cluster extrema according to E 0, that is, according to a randomly carriedshifted over to models PPP with infinite exponential levels of branching (x R) such density; as BBM or the continuous 2 attach random to each energy cluster models extrema studied ain cluster. [15]. BBM is a case right at the borderline where correlations just start to effect the extremes and the structure of the extremal process. Note 1. Particles Results oninthe one extremes clusteroflie BBM to the allowleft one(in to peek space) intoof theits world corresponding beyond the simple cluster extrema Poisson(Poissonian structures andparticle). hopefully open the gate towards the rigorous understanding of Note 2. Heuristic complex extremal for PP-structure: structures. the ancestors of the extremal particles evolve Mathematically, BBM offers a spectacular interplay between probability and nonlinear p.d.e s, as was noted already by McKean [37]. On one hand, the proof of independently for the time-interval [r, t r]. Theorem Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

26 m(t) = 2t log(t) E t i n(t) δ x i (t) m(t). Remark (Invariance property) The law of the limiting extremal process E = i N δ e i invariance property: For any s 0, satisfies the following E D = i,k δ ei +x (i) k (s) 2s, where {x (i) k (s) : k n(i) (s), s 0} i N are i.i.d. BBMs. Indeed, use that for t, m(t) = m(t s) + 2s + o(1). Then rewrite E t and take t. i n(t) = δ xi (t) m(t) = δ xi (t s)+x (i) i n(t s) k n (i) k (s) m(t) (s) δ xi (t s) m(t s)+x (i) i n(t s) k n (i) k (s) 2s+o(1) (s) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

27 Idea of proof of Theorem 2, [ABK12] Theorem 2, [ABK12] For any 0 < q < 1, the processes E (q) t converge in law to the same limit, E 0. Also, lim r lim t E (1 r/t) t = E 0. Moreover, conditionally on Z, E 0 = PPP ( CZ 2e 2x dx ). ( ) Genealogy of Extremal Particles, Theorem 2.1, [ABK11] For any compact set D R, lim r sup t>3r P ( 1 k, l n(t) : x k (t), x l (t) D and d(x k (t), x l (t)) (r, t r) ) = 0. ( ) Same limit E 0 for r t < q < 1 r t with r big enough follows from ( ). It remains to show ( ). This is done via convergence of Laplace functionals, that is, for φ C c + we claim that [ lim lim E e ] φ(x)e (1 r/t) t (dx) = E [e CZ ( ) 2e ] 1 e φ(x) 2x dx. r t Note. If X PPP(λ), then E[e φ(x)x (dx) ] = e (e φ(x) 1)λ(dx) (cf. [B15], Appendix). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

28 E t = k n(t) δ x k (t) m(t) with m(t) = 2t log(t) Conditional on the evolution of the BBM up to time r, we obtain { } D = x j (r) + M (j) (t r) m(t), where {x (j) k E (1 r/t) t j=1,...,n(r) (t)} k n (t), j N are i.i.d. BBM with M (j) (t) max (j) k n (j) (t) x (j) (t) L. P. ARGUIN, A. BOVIER, AND N. KISTLER k space s 7! s t m.t/ m.t/ extremal particles free evolution (Figure 2.4 of [ABK11]) Now, r t r t time no branching FIGURE 2.4. Evolution of the system. m(t) = 2r + m(t r) + o(1) for t. Brunet and Derrida also conjecture that the extremal process of BBM retains properties of Poisson point processes with exponential density, namely the invari- Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

29 P(M(t) m(t) x) = u(t, m(t) + x) w(x) unif. in x as t. m(r) = 2r log(r). As a result, [ lim E t e φ(x)e (1 r/t) t where M has law w. 2 (dx) ] = lim t E n(r) E [e φ(x j (r) 2r+M(t r) m(t r)+o(1)) ] j=1 n(r) = E E [e φ(x j (r) 2r+ M) ], j=1 Note 1. max j n(r) (x j (r) 2r) a.s. as r. Note 2. Now use asymptotics for M, i.e. 1 w(x) Cxe 2x for x. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

30 Idea of proof of Theorem 2.1, [ABK11] Localization of Paths of Extremal Particles Theorem (Genealogy of Extremal Particles, Theorem 2.1, [ABK11]) For any compact set D R, lim sup P ( 1 k, l n(t) : x k (t), x l (t) D and d(x k (t), x l (t)) (r, t r) ) = 0. r t>3r 1652 L. P. ARGUIN, A. BOVIER, AND N. KISTLER space U t;.s/ D s t m.t/ C O.s ;.t s/ / m.t/ D p 2t 3 2 p 2 log t s t m.t/ (Figure 2.1 of [ABK11]) r u t r u t 2 t FIGURE 2.1. The upper envelope. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36 time

31 For γ > 0, set f t,γ (s) { s γ, 0 s t 2, (t s) γ, t 2 s t. Then the upper envelope at time t is defined as U t,γ (s) s t m(t) + f t,γ(s). Theorem (Upper Envelope, Theorem 2.2 of [ABK11]) Let 0 < γ < 1/2. Let also y R and ɛ > 0 be given. There exists r u = r u (γ, y, ɛ) such that for r r u and for any t > 3r, P( k n(t) : x k (s) > y + U t,γ (s) for some s [r, t r]) < ɛ. Idea of proof. Discretize path and use P(M(t) > m(t) + x) t 1 w(x) Cxe 2x for x. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

32 Remark (Why is this useful?) For E[n(t)] = e t i.i.d. BMs, m(t) = 2t 1 log(t) 2 2 E[#{k n(t) : B k (t) > m(t)}] et e m(t)2 2t 2πt and = et 2πt e 2t 2 2 2t log(t) log(t)2 8 2t = 1 e log(t) 2πt 2 log(t)2 16t = 1 e log(t)2 16t 2π = O(1). For BBM, m(t) = 2t 3 log(t) 2 2. For et i.i.d. BMs we now get E[#{k n(t) : B k (t) > m(t)}] = 1 e 3 log(t) 2 9 log(t)2 16t 2πt Now, for a BM B t starting in 0, P(B s U t,γ (s), r s t r B t = m(t)) = P(z t 0,m(t) (s) U t,γ(s), r s t r) 1 t. = O(t). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

33 The upper envelope can be replaced by a lower entropic envelope E: For α > 0 let E t,α (s) s t m(t) f t,α(s). Theorem (Entropic Repulsion, Theorem 2.3 of [ABK11]) Let D R be a compact set and 0 < α < 1/2. Set D sup{x D}. For any ɛ > 0 there exists r e = r e (α, D, ɛ) such that for r r e and t > 3r, P( k n(t) : x k (t) m(t) + D, but s [r, t r] : x k (s) D + E t,α (s)) < ɛ L. P. ARGUIN, A. BOVIER, AND N. KISTLER space U t;.s/ D s t m.t/ C O.s ;.t s/ / m.t/ D p 2t 3 2 p 2 log t E t;.s/ D s t m.t/ O.s ;.t s/ / re t r e time (Figure 2.2 of [ABK11]) t FIGURE 2.2. Entropic repulsion. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

34 Together with a lower envelope we get with 0 < α < 1 2 < β < 1, 1656 L. P. ARGUIN, A. BOVIER, AND N. KISTLER space m.t/ E t;.s/ admissible region E t;ˇ.s/ (Figure 2.3 of [ABK11]) r FIGURE 2.3. The tube. t r t time The proof of the assertion is straightforward from Theorems 2.3 and 2.5 by Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

35 Remark (How to use this 3 ) The expected number of pairs of particles of BBM whose (respective) path (x(s)) 0 s t satisfies some conditions for s [r, t r], say Σ [r,t r] t, is E[#{(k, l) : k l, x k ( ), x l ( ) Σ [r,t r] t }] t =C(e t ) 2 ds e s dy p s (y)p(x( ) Σ [r,t r] x(s) = y)p(x( ) Σ [r s,t] x(s) = y). 0 how many pairs at time t on average; condition on splitting at time s and at position y. If first particle satisfies the condition on [0, t], then the second one automatically satisfies it on [0, s]. If we include a condition on genetic distance, we get E[#{(k, l) : k l, x k ( ), x l ( ) Σ [r,t r] t, d(x k (t), x l (t)) [r, t r]}] t r = Ce t ds e t s dy p s (y)p(x Σ [r,t r] t x(s) = y)p(x Σ [s,t r] t t r Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36 t x(s) = y).

36 Bibliography Arguin, L.-P. and Bovier, A. and Kistler, N. Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 (2011), Arguin, L.-P. and Bovier, A. and Kistler, N. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012), Arguin, L.-P. and Bovier, A. and Kistler, N. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013), Aïdékon, E. and Berestycki, J. and Brunet, É. and Shi, Z. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013), Bovier, A. From spin glasses to branching Brownian motion and back? Lecture Notes in Math (2015), Bramson, M. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44, Number 285 (1983). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

37 Chauvin, B. and Rouault, A. KPP Equation and Supercritical Branching Brownian Motion in the Subcritical Speed Area. Application to Spatial Trees. Probab. Theory Related Fields 80 (1988), Gouéré, J.-B. Branching Brownian motion seen from its leftmost particle (following Arguin-Bovier-Kistler and Aïdékon-Berestycki-Brunet-Shi). Astérisque 361 (2014), Lalley, S. P. and Sellke, T. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 (1987), Neveu, J. Multiplicative martingales for spatial branching processes. Progr. Probab. Statist. 15 (1988), Roberts, M.I. A simple path to asymptotics for the frontier of a branching Brownian motion. Ann. Probab. 41 (2013), Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

38 The extremal process of BBM (cf. [ABBS13] and [ABK13]) Arguin, L.-P. and Bovier, A. and Kistler, N. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013), Aïdékon, E. and Berestycki, J. and Brunet, É. and Shi, Z. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013), Both articles give a description of the (weak w.r.t. φ C c + ) limit (t ) of the extremal process E t δ xk (t) m(t) δ xk (t). k n(t) k n(t) See Gouéré [G14] for a (french) review that presents and compares both approaches. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

39 [ABK13] The extremal process of BBM Remark Bovier [B15] discusses the following results in detail. There are also other representations. 4 The proofs rely on the consideration of the respective Laplace functionals. Theorem (Theorem 3.1 (Existence of the limit), [ABK13]) The point process E t converges in law to a point process E. Idea of proof: Example 2 for the F-KPP equation. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

40 Theorem 2, [ABK12] For any 0 < q < 1, the processes E (q) t converge in law to the same limit, E 0. Also, lim r lim t E (1 r/t) t = E 0. Moreover, conditionally on Z, E 0 = PPP ( CZ 2e 2x dx ), where C > 0 is the constant appearing in (5). E t δ xk (t) m(t) with m(t) = 2t log(t). k n(t) Definition (Cluster-extrema) Conditionally on the limiting derivative martingale Z, consider the PPP with C as in (5). P Z i N δ pi D = PPP ( CZ 2e 2x dx ) (7) Definition (Clusters) Let Ē t k n(t) δ x k (t) 2t. Conditionally on {max k n(t) x k (t) 2t 0}, the process Ē t converges to a point process Ē = j δ ξ j. Now define the point process of the gaps by D δ j, j ξ j max ξ j. (8) j j Note. D is a point process on (, 0] with an atom at 0. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

41 Theorem (Theorem 2.1 (Main Theorem), [ABK13]) Let P Z be as in (7) and let {D (i) : i N} be a family of independent copies of the gap-process (8). Then the point process E t converges in law as t to a Poisson cluster point process E given by E lim E D The extremal process of BBM t δ t pi. + (i) 539 j = i,j (Figure Fig. 11of Onset [ABK13]) of the extremal process Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

42 Remark [ABBS13] BBM seen from its tip The proofs use path localization and path decomposition techniques. J. Berestycki (Lecture Notes, Topics on BBM, http: // www. stats. ox. ac. uk/ ~ berestyc/ Articles/ EBP18_ v2. pdf) gives a good introduction in the underlying concepts. Notation (Change in Scaling) Note that a different scaling is used and instead of rightmost particles, leftmost are considered. Particles now follow a BM with drift 2 and variance 2, that is, replace B t by 2(B t 2t). (The exponential clocks for splitting-events still ring at rate 1 and a particle splits in two.) Instead of M(t) = max k n(t) x k (t) they consider min k n(t) x k (t). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

43 m(t) 2t log(t). The derivative mart. Z(t) n(t) k=1 ( 2t xk (t) ) 2t xk ) e 2( (t) satisfies Z = lim t Z(t). Particles now follow a BM with drift 2 and variance 2, that is, replace B t by 2(B t 2t). Consider min k n(t) x k (t). Remark (Consequences of Scaling) m(t) becomes m (t) = log(t). 1 n(t) Z(t) becomes 2 k=1 x k(t)e x k (t). [ABBS13] use Z (t) = n(t) k=1 x k(t)e x k (t) instead and thus CZ becomes (C/ 2)Z = C Z. E[Z t ] = 0 for all t 0. From now onwards, we use the notation of [ABBS13]. Definition (The additive martingale) The process M(t) n(t) k=1 e x k (t) is a martingale with E[M t ] = 1 for all t 0, the so-called additive martingale. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

44 E t k n(t) δ x k (t) m(t). Cluster-extrema P Z i N δp D i = PPP ( CZ 2e 2x dx ) = PPP ( e 2x+log(CZ) d( 2x) ). Theorem 2.1 (Main Theorem), [ABK13] Let P Z be as in (7) and let {D (i) : i N} be a family D of independent copies of the gap-process (8). Then E lim t E t = i,j δ p i + (i). j Let N (t) δ xk (t) m(t)+ log(cz). k n(t) Theorem (Theorem 2.1, [ABBS13] 5 ) As t the pair { N (t), Z(t)} converges jointly in distribution to {L, Z}, L and Z are independent and L is obtained as follows. (i) Define P a Poisson point measure on R, with intensity measure e x dx. (ii) For each atom x of P, we attach a point measure D (x) where D (x) are independent copies of a certain decoration point measure D. (iii) L is then the point measure corresponding to L δ x+y. x P y D (x) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

45 E. Aïdékon e (Y, Q) is the limit of the path s X 1,t (t s) X 1 (t) and of the points that have branched rece m X 1,t (Figure 1 of [ABBS13]) Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

46 Sandra Kliem (Univ. Duisburg-Essen) Extremal ParticlesFig. in1 BBM (Y, Q) is the limit of the path s X1,t (t s) 3. X1(t) August, and of the 2017 points that have45 branched / 36rece Notation Order the particles in increasing order, i.e. x 1 (t) x 2 (t) x n(t) (t). For s t, let x 1,t (s) denote the position at time s of the ancestor of x 1 (t), i.e. s x 1,t (s) is the path of the leftmost particle up until time t. Let Y t (s) x 1,t (t s) x 1 (t), s [0, t] the time reversed path back from the final position x 1 (t). Denote by < τ 2 (t) < τ 1 (t) t the successive splitting times along the path of the leftmost particle (enumerated backwards). The time at which x i (t) and x j (t) share their MRCA is denoted by τ i,j (t). Let N i (t) 1 j n(t):τ 1,j (t)=τ i (t) δ x j (t) x 1(t). Finally, let for 0 < η < t, D(t, η) δ 0 + i:τ i (t)>t η N i(t). 410 E. Aïdékon e

47 Theorem (Theorem 2.3, [ABBS13]) The following convergence holds jointly in distribution: lim lim ( (Yt (s), s [0, t]), D(t, η), x 1 (t) m(t) ) = ( (Y (s), s 0), D, W ), η t where the r.v. W is independent of the pair ((Y (s), s 0), D), and D is the point measure which appears in Theorem 2.1. Note. P(W (x) x) = 1 w( x/ 2) C x e x for x (cf. (5) and below). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

48 Construction of the decoration point measure D We will construct D conditional on Y. Notation Let b > 0, (B t, t 0) a BM and (R t, t 0) a three-dimensional Bessel process started from R 0 = 0 and independent of B. Let T b inf{t 0 : B t = b}. Set { B s, s [0, T b ], (Figure 1 and 2 of [ABBS13]) Γ (b) s b R s Tb, s T b. Branching Brownian motion seen from its tip E. Aïdékon et al.. (Y, Q) Sandra is the limit Kliem of the(univ. path s Duisburg-Essen) X 1,t (t s) X 1 (t) and of the points Extremal thatfig. have2 branched Particles The process recently inɣ BBM (b) 3. August, / 36

49 Construction of D (1) Construction of Y. For A C(R +, R) measurable, P(Y A, inf s 0 Y (s) db) = 1 c E with normalizing constant c. (2) Construction of D conditional on Y. [ e 2 0 P ( x 1(v) 2Γ (b) v ) dv 1 { 2Γ (b) A} Conditionally on the path Y, let π be a PPP on [0, ) with intensity 2 P(Y (τ) + x 1 (τ) > 0)dτ. For each point τ π start an independent BBM (NY (τ) (u), u 0) at position Y (τ) conditioned to have min N Y (τ) (τ) > 0. Then D δ 0 + NY (τ) (τ). τ π (Recall that Y (0) = 0 and that the path Y moves backwards in time, whereas the BBMs move forward in time.) ] Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

50 Spinal decomposition The process M(t) n(t) k=1 e x k (t) is a martingale with E[M t] = 1 for all t 0, the so-called additive martingale. Let Q be the probability measure s.t. Q Ft = M(t) P Ft, where P refers to the distribution of BBM and F t is the filtration of the BBM (under P) up to time t. Theorem (Theorem 5 of Chauvin and Rouault [CR88]) Q is the law of the following branching diffusion. (1) Let Ξ s {1,..., n(s)} denote the label of a distinguished particle at time s 0 with Q(Ξ t = i F t ) = e x i (t) M t. The process (Ξ s, s [0, t]) is called the spine. (2) The position of the spine (x Ξs (s), s [0, t]) is a driftless BM of variance 2. (3) The particle with label Ξ s at time s branches at (accelerated) rate 2 and gives birth to BBMs (with distribution P). Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

51 Lemma (Many-to-one-principle) Let Ψ i, i = 1,..., n(t) be F t -measurable r.v.s. Then E P Ψ i = E Q 1 Ψ i = E Q [e x Ξt (t) Ψ Ξt ]. M(t) i n(t) i n(t) Example We obtain P ( i n(t) : (x i,t (s), s [0, t]) A ) E [ i n(t) ] 1 ({xi,t (s),s [0,t]) A} = E [ e x ] Ξt (t) 1 {(xξs,s [0,t]) A} = E [ e 2B t 1 {( 2Bs,s [0,t]) A}]. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

52 1 A note on the i.i.d. case (cf. A. Bovier, Lecture Notes, Extreme values of random processes, Lemma 1.2.1) Let X 1 (t),..., X n (t) be n N i.i.d. normal r.v.s. Let b n log(log(n)) + log(4π) 2 log(n) 2 2 log(n) Then, for all x R, and a n 2 log(n). lim P(max X k(t) b n + x/a n ) = e e x. n k n For a BBM, E[n(t)] = e t. Set n = e t and consider B i, i N independent standard BMs. Then, for y x/ 2, ( lim P B k (t) max 2t log(t) ) n k e t t 2 log(4π) + 2x + 2t 2 2t ( = lim P max B k(t) 2t 1 log(t) ) t k e t 2 2y + O(1) + y = e e. 2 Recall, that for BBM, m(t) = 2t log(t) and P(M(t) m(t) x) t E [ e CZe 2x ]. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

53 2 Idea of proof of Theorem 2, [ABK12] - Heuristic 1 w(x) Cxe 2x for x. Suppose heuristically w (x) C 2xe 2x. Z(t) = n(t) ( k=1 2t xk (t) ) 2t xk ) e 2( (t) Z for t. [ lim E e ] n(r) φ(x)e (1 r/t) t (dx) = E E [e φ(x j (r) 2r+ M) ], t j=1 where M has law w and max j n(r) (x j (r) 2r) a.s. as r. Rewrite the above to (log(ab) = log(a) + log(b), log(x) (1 x) for 0 < x < 1) [ ])] [ n(r) j=1 (E [e E e log φ ( x j (r) 2r+ M ) E e [ ]] n(r) j=1 E 1 e φ ( x j (r) 2r+ M ) E [e n(r) j=1 [ E (1 e φ(x) )P( M= x j (r)+ 2r+dx) ] e (1 e φ(x) ) n(r) j=1 C 2 ( x j (r)+ 2r+x r E [ e C 2(1 e φ(x) )Ze 2x dx ]. ( ) ] )e 2 x j (r)+ 2r+x dx Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

54 3 Idea of proof of Theorem 2.1, [ABK11] - Heuristic Let k, l be such that d(x k (t), x l (t)) = s [r, t r]. Consider the case s t/2 and to simplify calculations s = O(t) in what follows. Due to entropic repulsion: w.l.o.g. and x k (t), x l (t) m(t) + D x k (s)(= x l (s)) < D + E t,α (s) = D + 2s s 3 t 2 log(t) sα 2 2s s α for some fixed 0 < α < 1/2 and for all s [r, t r]. For particles k and l to reach m(t) + D, the MRCA of k and l (at time s) must itself produce a BBM that after a time-interval of length t s has height at least ( m(t) min(d) ) ( 2s s α ) 2(t s) + s α. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

55 Number of possible choices for ancestor: At time s, there are on average e s particles. The chance of one particle (= BM) to reach (x k (s) =) 2s s α is of order 1 e 2s2 2 2ss α +s 2α 2s = 1 e s e 2s α e s2α 1 2, 2πs 2πs where 2α 1 < 0. In the product (more particles at this height as if we consider BBM) at time s we have on average at most of order e 2s α choices. Chance for ancestor (at time s) to have a child at height 2(t s) + s α (at time t): Starting with a single particle, the probability that BBM jumps this high in the time-interval t s is (use that t s r and s = O(t) and P(M(t) m(t) > x) = 1 w(x) Cxe 2x for x ) P(M(t s) 2(t s) + s α ) ( = P M(t s) m(t s) 3 ) 2 log(t s) + sα 2 ( 3 1 w 2 2 log(t s) + sα) C(s α + δ)e 2(s α +δ) e 2s α. Chance to have two (that split immediately): of order ( e 2s α ) 2. Overall chance: of order e 2s α, so negligible. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

56 4 Other Representations for the extremal process of BBM, (cf. [ABK13]) Theorem 2, [ABK12] For any 0 < q < 1, the processes E (q) t converge in law to the same limit, E 0. Also, lim r lim t E (1 r/t) t = E 0. Moreover, conditionally on Z, E 0 = PPP ( CZ 2e 2x dx ). Proposition (Proposition 3.2, [ABK13]) For φ C c + (R) and any x R, [ lim E e ] φ(y+x)e 2x ] t(dy) = E [e C(φ)Ze t where, for v(t, y) the solution of F-KPP with initial condition v(0, y) = e φ(y), 2 C(φ) = lim (1 v(t, y + 2t))ye 2y dy t π is a strictly positive constant depending on φ only. 0 Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

57 P Z D ( δ pi = PPP CZ 2e 2x dx ) i N Let Ē t k n(t) δ x k (t) 2t. Conditionally on {max k n(t) x k (t) 2t 0}, the process Ē t converges to a point process Ē = j δ ξ j. Now define the point process of the gaps by D j δ j, j ξ j max j ξ j. Note. D is a point process on (, 0] with an atom at 0. Theorem 2.1 (Main Theorem), [ABK13] Let P Z be as in (7) and let {D (i) : i N} be a family D of independent copies of the gap-process (8). Then E lim t E t = i,j δ p i + (i). j Let (η i : i N) be the atoms of a PPP on (, 0) with intensity measure 2 2x π ( x)e dx. For each i N consider independent BBMs with drift 2, i.e. (t) 2t : k n (i) (t)}. The auxiliary point process is defined as {x (i) k Π t δ 1 2 log(z)+η i +x (i) k (t) 2t. i,k Theorem (Theorem 3.6 (The auxiliary point process), [ABK13]) E = D lim t. t Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

58 5 Heuristic for appearance of the Poisson point measure Fix k 1. (Proposition 10.1 in [ABBS13]) Let H k be the set of particles (at position k) that hit the spatial position k first in their line of descent. Note: Conditionally on H k, the subtrees rooted at the points of H k are independent BBMs started at position k and at a random time (i.e. when the particle of H k hit k). Define H k #H k. Note: finite a.s. (use m(t) = + 3 log(t) and that we consider 2 the minimum of BBM) Now let Z = lim t Z(t) = lim t n(t) k=1 x k(t)e x k (t). Neveu ([N88], (5.4)) shows that Z k ke k H k. lim Z k = Z, a.s. k Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36

59 Let H k,t H k be the set of all particles that hit k before time t. For u H k,t, write x1 u (t) for the minimal position at time t of the particles which are descendants of u. If u H k \H k,t, let x1 u (t) = 0. Now define the point measures P k,t u H k δ x u 1 (t) m(t)+log(cz k ) and (Z k ke k H k and m(t) = 3 log(t), i.e. m(t + c) m(t) 0 for t ) 2 Pk, δ k+w (u) +log(cz k ), u H k where, conditionally on F Hk (sigma-algebra generated by the BBM when the particles are stopped upon hitting the position k), the W (u) are independent copies of the r.v. W. Proposition (Proposition 10.1 of [ABBS13]) The following convergences hold in distribution. lim t P k,t = Pk, and lim k (P k,, Z k ) = (P, Z) where P is as in Theorem 2.1 and P and Z are independent. Sandra Kliem (Univ. Duisburg-Essen) Extremal Particles in BBM 3. August, / 36