Continuous-state branching processes, extremal processes and super-individuals

Transcription

1 Continuous-state branching processes, extremal processes and super-individuals Clément Foucart Université Paris 13 with Chunhua Ma Nankai University Workshop Berlin-Paris Berlin 02/11/2016

2 Introduction Consider a continuous-state branching process (X t (x), t 0) starting from x. Questions Can we characterize the growth rate of a supercritical CSBP with infinite mean (no Malthusian parameter)? Can we characterise the decay rate of a subcritical CSBP with infinite variation? Consider a continuous branching population encoded by a flow of CSBPs (as Bertoin Le Gall 2000). Questions How are organized the growth and the decay locally in the population? Do all families evolve at the same scale or some initial individuals (super-individuals) have progenies growing faster than all the others?

3 Definition (CSBP) A positive Markov process (X t (x), t 0) with X 0 (x) = x 0 is a CSBP if for any y R + (X t (x + y), t 0) d = (X t (x), t 0) + ( X t (y), t 0) where ( X t (y), t 0) is an independent copy of (X t (y), t 0). This ensures the existence of a map t v t (λ) s.t. E[e λxt(x) ] = exp( xv t (λ)) and v s+t (λ) = v s v t (λ), Theorem (Jirina (58), Lamperti (67)) There exists Ψ of the form Ψ(q) = σ2 2 q2 + γq + such that dvt(λ) dt + 0 = Ψ(v t (λ)). ( e qx 1 + qx1 {x 1} ) π(dx)

4 Asymptotic behaviors Proposition (Grey 74) Supercritical case: Ψ (0) [, 0[, The largest root of Ψ, called ρ, is in (0, + ] X t (x) 0 with probability e xρ X t (x) + with probability 1 e xρ Non-explosion: P( t, X t (x) < ) = 1 dq 0 Ψ(q) = +. Subcritical case: Ψ (0) 0, The largest root of Ψ is 0 X t (x) 0 a.s. Persistence: P( t; X t (x) > 0) = 1 + dq Ψ(q) = +

5 Continuous population model Definition (Flow of CSBPs: Bertoin Le Gall 2000, Dawson Li 2012, Duquesne Labbé 2014) Let N Ψ a measure on D(R +, R + ) s.t. N Ψ ( ) = lim x PΨ x ( ). ab Consider N = i I δ (x i,x i ) a PPP over R + D with intensity dx N Ψ (dx ). For all x 0, let X 0 (x) = x and for all t > 0, X t (x) = x i x X i t x 0 1 for all t 0 (X t (x), x 0) is a (càdlàg) subordinator with Laplace exponent λ v t (λ) for any y x, (X t (y) X t (x), t 0) is a CSBP(Ψ) started from y x, independent of (X t (x), t 0). a in the infinite variation case. b In the finite variation case, there is an other Poisson representation

6 The flow (X t (x), t 0, x 0) provides a continuous population: The individual y is a descendant at time t of the individual x living at time 0 if X t (x ) < y < X t (x) X t (x) = X t (x) X t (x ) is the progeny of x at time t. population size X t (x) X t (u) At time t x X j t = X t(x j) X t(x j) x j X t(x i) x i X i t = X t(x i) 0 t time 0 x i x j x

7 Super-individuals In a non-explosive CSBP with infinite mean (Ψ(u)/u ) u 0+ and in a persistent CSBP with infinite variation (Ψ(u)/u + ), some individuals have a progeny that u + overwhelms the total progeny of all individuals below them. Definition The individual x is a super-individual if X lim t(x) t + X t(x ) Denote by S the set of super-individuals { } X t (x) S := x > 0; lim t + X t (x ) = +. = + a.s. There is an order between super-individuals: if x 1, x 2 S and x 1 x 2, then Xt(x 1) X t(x 2 ) Xt(x 2 ) X t(x 2 ) 0.

8 Supercritical CSBP with finite mean Definition (Bertoin et al. 2008) An individual x is prolific if X t (x) + a.s. P := {x > 0; X t (x) + } Proposition ( Grey 74+Bertoin et al Duquesne Labbé 2014) Assume Ψ (0+) (, 0). Almost-surely, for all x > 0, v t (λ)x t (x ) W λ x and v t (λ)x t (x) W λ x where (W λ x, x 0) is a càdlàg subordinator λ v t (λ), the inverse of λ v t (λ) P = {x > 0; Wx λ > 0} and S P is degenerate.

9 Supercritical CSBP with infinite mean Theorem (preliminary version, Grey 77, F. Ma 16) Suppose Ψ (0+) = and 0 Ψ(u) =. Fix λ 0 (0, ρ), and ( define G(y) := exp ) λ 0 du y Ψ(u) for y (0, ρ). Then, for all x 0, almost-surely ( ) e t G 1 X ρ t(x) Z x. t + du G is decreasing and slowly varying at 0 {Z x = 0} = {X t (x) 0} P(Z x z) = exp( xg 1 (z)) with G 1 (z) = v log(1/z) (λ 0 ). Example (Neveu s mechanism) Ψ(u) = u log u for which ρ = 1. Fix λ 0 = 1 e, G(z) = log(1/z)

10 Question What is the nature of the process (Z x, x 0)? Definition (extremal process= subordinator for the max operator ) A process (Z x, x 0) is an extremal-f process if { P(Zx z) = F (z) x Lemma Z x+y = Z x Z y a.s. where Z y (Z u ) 0 u x and Z y d = Z y (Z x, x 0) is an extremal-f process with F (z) = e v log(1/z)(λ 0 ). Proof. X t (x + y) = X t(y) + X t (x) with X t(y) = X t (x + y) X t (x). = e t 1 G( X ) t(x+y) e t 1 G( X ) t(x) e t 1 G( X t (y)) thus Z x+y Z x Z y a.s. but Z x+y d = Zx Z y

11 Fact ( Lévy-Itô decomposition of extremal processes) Consider a PPP with intensity dx µ over R + R. The process of its records is a càdlàg extremal-f process with F (z) = e µ(z). Theorem (F. Ma 2016, supercritical part) There exists M := i I δ (x i,z i ) a PPP(dx µ(dz)) with µ(z) = v log(1/z) (λ 0 ) such that almost-surely, for all x 0 e t G ( ) 1 X t (x) ρ Z x = sup Z i t + x i x µ has total mass µ(0) = ρ (0, + ] and has no atom. Moreover, Proposition Almost-surely, if Z i > Z j then X t (x j )/ X t (x i ) 0, P = {x i ; Z i > 0, i I } and S P = {x > 0; Z x > 0} a.s.

12 Poisson representation population size X t (x) X t (u) At time t x (x j, Z j ) (x k, Z k ) (x i, Z i ) X t(x j) X t(x i) X t(x i) X t(x j) X t(x k) 0 t time 0 x i x k denotes (x l, Z l ) such that Z l = 0 (non prolific) denotes (x i, Z i ) not a partial record (prolific non superprolific) denotes (x i, Z i ) partial record: (superprolific) x l x

13 Subcritical case Assume Ψ (0+) 0. Since for all x 0, X t (x) 0 a.s. there is no prolific individual in the population. Recall { } X t (x) S := x > 0; lim t + X t (x ) = +. A super-individual is an individual whose decay is much slower than the decay of all individuals below it.

14 Subcritical CSBP with finite variation Definition (variation) For any Ψ, Ψ(u) lim =: d = + 1 u + u {σ>0} + γ + 1 Proposition (Grey 74 + Duquesne Labbé 2014) 0 xπ(dx) R {+ }. Assume d R. For all x, (X t (x), t 0) is persistent and almost-surely, for all x 0 v t (λ)x t (x) t + V λ x and v t (λ)x t (x ) t + V λ x. where (V λ x, x 0) is a càdlàg subordinator. Thus S is degenerate.

15 Subcritical process with infinite variation Theorem (F. Ma 2016, subcritical part) Suppose d = + and + du Ψ(u) = +. Fix λ 0 (0, + ) and ( define G(y) := exp ) y du λ 0 Ψ(u) on (0, + ). There exists M := i I δ (x i,z i ) a PPP(dx µ(dz)) with µ(z) = G 1 (z) = v log(z) (λ 0 ) such that almost-surely ( ) e t G 1 X t(x) Z x = sup xi x Z i for all x 0. t + µ(0, ) = and µ has no atom. Proposition S = {x > 0; Z x > 0} a.s.

16 Corollary (Supercritical case (Grey 79 for GW chains) ) Consider two independent CSBPs (X t (x), t 0), (Y t (y), t 0) non-explosive with infinite mean and same mechanism. Conditionally on {X t (x) + } {Y t(y) + }, t + t + X t (x) Y t (y) t + Corollary (Subcritical case) { + with probability x 0 with probability x+y y x+y. Consider two independent subcritical persistent CSBPs (X t (x), t 0), (Y t (y), t 0) with infinite variation and same mechanism X t (x) Y t (y) t + { + with probability x 0 with probability x+y y x+y.

17 Eve property Duquesne and Labbé (2014) have considered the following question: Question Does the population (encoded by a flow of CSBPs) concentrates on the progeny of a single individual? In other words, fix the initial size x, is there an individual e [0, x] ( the Eve), such that X t (e) X t (x) 1 a.s.? Corollary (Duquesne and Labbé 2014) In the case of infinite variation and infinite mean, the population has an Eve. In our framework, the Eve corresponds to the last super-individual in [0, x].

18 Neveu case Consider (X t (x), t 0) a CSBP of Neveu. It is non-explosive with infinite mean and persistent with infinite variation. A well-known result, attributed to Neveu (1992) (shown in Fleischmann, Sturm (2004)), states that for any fixed x e t log X t (x) t + Z x a.s. where Z x has a Gumbel law over R. By combining our results, we get: Proposition Almost-surely for all x 0, e t log X t (x) t + Z x where (Z x, x 0) is an extremal-λ process with Λ(z) = e e z z R. for

19 Conclusion and references In the non-persistent and explosive cases, extremal processes still arise, but through the times of explosion and absorption. When the reproduction has infinite mean, the infinite divisibility of the flow (X t (x), t 0, x 0) becomes the max-infinite divisibility of the process (Z x, x 0). Less clear in the subcritical setting... Bertoin, Le Gall, The Bolthausen-Sznitman coalescent and the genealogy of CSBPs, PTRF (2000) Duquesne, Labbé, On the Eve property for CSBP, Electron. J. Probab. (2014) Neveu, A CSBP in relation with the GREM model of spin glass theory, Rapport interne (unprinted:-( Ecole Polytechnique, 1992.

20 D. R. Grey, Asymptotic behaviour of CSBPs, JAP (1974), D. R. Grey, Almost-sure convergence in Markov branching process with infinite mean, J. Appl. Probability 14 (1977), D. R. Grey,On regular branching processes with infinite mean, SPA (1978/79), Thank you!!

21 Sketch of proof Fix x. 1 For all λ (0, ρ), v t (λ)x t (x) W λ x P(Wx λ = ) = e xλ. 2 If λ λ then W λ x W λ x {0, } a.s. and 3 Λ x := inf{λ (0, ρ) Q; W λ x = + } is a random variable! 4 Let λ (0, ρ). If λ < Λ x < λ, then for large t : v t (λ) 1/X t (x) and v t (λ ) 1/X t (x) = G(v t (λ)) G(1/X t (x)) G(v t (λ )) = G(λ) e t G(1/X t (x)) G(λ ) 5 If Λ x = ρ then X t (x) 0. This yields the a.s convergence: e t G(1/X t (x) ρ) Z x