The number of absorbed individuals in branching brownian motion with a barrier

www.imstat.org/aihp 2013, Vol. 49, No. 2, 428–455

DOI:10.1214/11-AIHP451

© Association des Publications de l’Institut Henri Poincaré, 2013

The number of absorbed individuals in branching Brownian motion with a barrier

Pascal Maillard

aLaboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris VI, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail:pascal.maillard@upmc.fr

Received 8 April 2010; revised 22 July 2011; accepted 29 July 2011

Abstract. We study supercritical branching Brownian motion on the real line starting at the origin and with constant driftc. At the pointx >0, we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring.

It is known that there is a critical driftc₀, such that this process becomes extinct almost surely if and only ifc≥c₀. In this case, if Z_xdenotes the number of individuals absorbed at the barrier, we give an asymptotic forP (Z_x=n)asngoes to infinity. Ifc=c₀ and the reproduction is deterministic, this improves upon results of L. Addario-Berry and N. Broutin [1] and E. Aïdékon [2] on a conjecture by David Aldous about the total progeny of a branching random walk. The main technique used in the proofs is analysis of the generating function ofZ_xnear its singular point 1, based on classical results on some complex differential equations.

Résumé. Nous étudions le mouvement brownien branchant sur-critique sur la droite réelle, issu de l’origine et avec une dérive constantec. Au pointx >0, nous ajoutons une barrière absorbante, c’est-à-dire les individus qui touchent la barrière sont tués instantanément et sans se reproduire. Il est connu qu’il existe une dérive critiquec₀tel que ce processus s’éteint presque surement si et seulement sic≥c₀. Dans ce cas, si on note parZ_xle nombre d’individus absorbés en la barrière, nous donnons un équivalent deP (Z_x=n)quandntend vers l’infini. Sic=c₀et la reproduction est déterministe, ceci améliore des résultats de L. Addario- Berry et N. Broutin [1] et E. Aïdékon [2] sur une conjecture de David Aldous concernant la progéniture totale d’une marche aléatoire branchante. La technique principale utilisée dans les preuves est l’analyse de la fonction génératrice deZ_xau voisinage de son point singulier 1, basée sur des résultats classiques concernant certaines équations differéntielles dans le champ complexe.

MSC:Primary 60J80; secondary 34M35

Keywords:Branching Brownian motion; Galton–Watson process; Briot–Bouquet equation; FKPP equation; Travelling wave; Singularity analysis of generating functions

1. Introduction

We define branching Brownian motion as follows. Starting with an initial individual sitting at the origin of the real line, this individual moves according to a 1-dimensional Brownian motion with driftcuntil an independent exponentially distributed time with rate 1. At that moment it dies and producesL(identical) offspring,Lbeing a random variable taking values in the non-negative integers withP (L=1)=0. Starting from the position at which its parent has died, each child repeats this process, all independently of one another and of their parent. For a rigorous definition, see for example [10].

We assume thatm=E[L] −1∈(0,∞), which means that the process is supercritical. At position x >0, we add anabsorbing barrier, i.e. individuals hitting the barrier are instantly killed without producing offspring. Kesten proved [19] that this process becomes extinct almost surely if and only if the driftc≥c₀=√

2m(he actually needed E[L²]<∞for the “only if” part, but we are going to prove that the statement holds in general). A conjecture of David

Aldous [3], originally stated for the branching random walk, says that the numberN_x of individuals that have lived during the lifetime of the process satisfiesE[N_x]<∞andE[N_xlog⁺N_x] = ∞in the critical speed area (c=c₀), andP (N_x> n)∼Kn^−γ in the subcritical speed area (c > c0), with someK >0, γ >1. For the branching random walk, the conjecture of the critical case was proven by Addario-Berry and Broutin [1] for general reproduction laws satisfying a mild integrability assumption. Aïdékon [2] refined the results for constantLby showing that there are positive constantsρ, C₁, C₂, such that for everyx >0, we have

C1xe^ρx

n(logn)²≤P (N_x> n)≤ C2xe^ρx

n(logn)² for largen.

AssumingLconstant has the advantage thatN_x is directly related to the numberZ_x of individualsabsorbed at the barrierbyN_x−1=(Z_x−1)(L/(L−1)), hence it is possible to studyN_xthroughZ_x.

In this sense, Neveu [23] had already proven the critical case conjecture for branching Brownian motion since he showed that the processZ=(Z_x)_x≥0is actually a continuous-time Galton–Watson process of finite expectation, but withE[Z_xlog⁺Z_x] = ∞for everyx >0, ifc=c₀.

LetN= {1,2,3, . . .}andN0= {0} ∪N. Define the infinitesimal transition rates (see [4], p. 104, Eq. (6), or [14], p. 95)

q_n=lim

x↓0

1

xP (Z_x=n), n∈N0\ {1}. We propose a refinement of Neveu’s result:

Theorem 1.1. Assumec=c₀.Assume thatE[L(logL)^2+ε]<∞for someε >0.Then we have asn→ ∞, ∞

k=n

q_k∼ c0

n(logn)² and P (Z_x> n)∼ c0xe^c0x

n(logn)² for eachx >0.

The heavy tail ofZxsuggests that its generating function is amenable to singularity analysis in the sense of [12].

This is in fact the case in both the critical and subcritical cases if we impose a stronger condition upon the offspring distribution and leads to the next theorem.

Definef (s)=E[s^L]the generating function of the offspring distribution. Denote byδthe span ofL−1, i.e. the greatest positive integer, such thatL−1 is concentrated onδZ. Letλc≤λcbe the two roots of the quadratic equation λ²−2cλ+c²₀=0 and denote byd=^λ_λ^c_c the ratio of the two roots. Note thatc=c₀if and only ifλ_c=λ_cif and only ifd=1.

Theorem 1.2. Assume that the law ofLadmits exponential moments,i.e.that the radius of convergence of the power seriesE[s^L]is greater than1.

− In the critical speed area(c=c0),asn→ ∞,

qδn+1∼ c₀

δn²(logn)² and P (Zx=δn+1)∼ c₀xe^c0x

δn²(logn)² for eachx >0.

− In the subcritical speed area(c > c0)there exists a constantK=K(c, f ) >0,such that,asn→ ∞,

qδn+1∼ K

n^d+1 and P (Zx=δn+1)∼e^λcx−e^λcx λc−λc

K

n^d+1 for eachx >0.

Furthermore,q_δn+k=P (Z_x=δn+k)=0for alln∈Zandk∈ {2, . . . , δ}.

Remark 1.3. The idea of using singularity analysis for the study of Zx comes from Robin Pemantle’s(unfinished) manuscript[24]about branching random walks with Bernoulli reproduction.

Remark 1.4. Since the coefficients of the power seriesE[s^L]are real and non-negative,Pringsheim’s theorem(see e.g. [13],TheoremIV.6,p. 240)entails that the assumption in Theorem1.2is verified if and only iff (s)is analytic at1.

Remark 1.5. Letβ >0andσ >0.We consider a more general branching Brownian motion with branching rate given byβand the drift and variance of the Brownian motion given bycandσ²,respectively.Call this process the(β, c, σ )- BBM(the reproduction is still governed by the law ofL,which is fixed).In this terminology,the process described at the beginning of this section is the(1, c,1)-BBM.The(β, c, σ )-BBM can be obtained from(1, c/(σ√

β),1)-BBM by rescaling time by a factorβand space by a factorσ/√

β.Therefore,if we add an absorbing barrier at the pointx >0, the(β, c, σ )-BBM gets extinct a.s.if and only ifc≥c₀=σ√

2βm.Moreover,if we denote byZ^{(β,c,σ )}_x the number of particles absorbed atx,we obtain that

Z_x^{(β,c,σ )}

x≥0 and Z^(1,c/(σ

√β),1) x√

β/σ

x≥0 are equal in law.

In particular,if we denote the infinitesimal transition rates of(Z_x^{(β,c,σ )})_x≥0byq_n^{(β,c,σ )},forn∈N0\ {1},then we have q_n^{(β,c,σ )}=lim

x↓0

1 xP

Z^{(β,c,σ )}_x =n

=

√β σ lim

x↓0

σ x√

βP Z^(1,c/(σ

√β),1) x√

β/σ =n

=

√β σ q^(1,c/(σ

√β),1)

n .

One therefore easily checks that the statements of Theorems1.1and1.2are still valid for arbitraryβ >0andσ >0, provided that one replaces the constantsc0, λc, λc, Kbyc0/σ²,λc/σ²,λc/σ²,^√_σ^βK(c/(σ√

β), f ), respectively.

Remark 1.6. After submission of this article,Yang and Ren published an article[25]which permits to weaken the hypothesis in Theorem1.1:It is enough to assume thatE[L(logL)²]<∞.In our proof,one needs to replace the reference[20]by[25]and use TheoremBof[7]instead of our Lemma4.1,in order to obtain(4.6).

The content of the paper is organised as follows: In Section2we derive some preliminary results by probabilis- tic means. In Section3, we recall a known relation between Z_x and the so-called Fisher–Kolmogorov–Petrovskii–

Piskounov (FKPP) equation. Section4is devoted to the proof of Theorem1.1, which draws on a Tauberian theorem and known asymptotics of travelling wave solutions to the FKPP equation. In Section5 we review results about complex differential equations, singularity analysis of generating functions and continuous-time Galton–Watson pro- cesses. Those are needed for the proof of Theorem1.2, which is done in Section6.

2. First results by probabilistic methods

The goal of this section is to prove

Proposition 2.1. Assumec > c₀andE[L²]<∞.There exists a constantC=C(x, c, L) >0,such that P (Z_x> n)≥ C

n^d for largen.

This result is needed to assure that the constantK in Theorem1.2is non-zero. It is independent from Sections3 and4and in particular from Theorem1.1. Its proof is entirely probabilistic and follows closely [2].

2.1. Notation and preliminary remarks

Our notation borrows from [20]. Anindividualis an element in the space of Ulam–Harris labels

U=

n∈N0

Nⁿ,

which is endowed with the ordering relationsand≺defined by

uv ⇐⇒ ∃w∈U: v=uw and u≺v ⇐⇒ uvandu=v.

The space of Galton–Watson trees is the space of subsetst⊂U, such that∅∈t,v∈tifv≺uandu∈tand for every uthere is a numberL_u∈N0, such that for allj∈N,uj∈tif and only ifj≤L_u. Thus,L_uis the number of children of the individualu.

Branching Brownian motion is defined on the filtered probability space (T,F, (Ft), P ). Here, T is the space of Galton–Watson trees with each individual u∈thaving a mark(ζ_u, X_u)∈R⁺×D(R⁺,R∪ {Δ}), whereΔis a cemetery symbol andD(R⁺,R∪{Δ})denotes the Skorokhod space of cadlag functions fromR⁺toR∪{Δ}. Here,ζu

denotes the life length andXu(t)the position ofuat timet, or of its ancestor that was alive at timet. More precisely, for v∈t, letd_v=

wvζ_w denote the time of death andb_v=d_v−ζ_v the time of birth ofv. ThenX_u(t)=Δfor t≥d_uand ifvuis such thatt∈ [b_v, d_v), thenX_u(t)=X_v(t).

The sigma-fieldFt contains all the information up to timet, andF=σ (

t≥0Ft).

Let y, c∈R andL be some random variable taking values in N0\ {1}.P =P^y,c,L is the unique probability measure, such that, starting with a single individual at the pointy,

− Each individual moves according to a Brownian motion with driftcuntil an independent time ζ_u following an exponential distribution with parameter 1.

− At the time ζ_u the individual dies and leaves L_u offspring at the position where it has died, withL_u being an independent copy ofL.

− Each child ofurepeats this process, all independently of one another and of the past of the process.

Note that oftencandLare regarded as fixed andyas variable. In this case, the notationP^yis used. In the same way, expectation with respect toP is denoted byEorE^y.

A common technique in branching processes since [21] is to enhance the spaceT by selecting an infinite genealog- ical line of descent from the ancestor∅, called thespine. More precisely, ifT ∈T andtits underlying Galton–Watson tree, thenξ =(ξ₀, ξ₁, ξ₂, . . .)∈U^N0 is aspineofT ifξ₀=∅and for everyn∈N0,ξ_n+1is a child ofξ_nint. This gives the space

T= (T , ξ )∈T ×U^N0: ξ is a spine ofT

of marked trees with spine and the sigma-fieldsFandFt. Note that if(T , ξ )∈T, thenT is necessarily infinite.

Assume from now on thatm=E[L] −1∈(0,∞). LetN_t be the set of individuals alive at timet. Note that every Ft-measurable functionf:T →Radmits a representation

f (T , ξ )=

u∈N_t

f_u(T )1u∈ξ,

wherefuis anFt-measurable function for everyu∈U. We can therefore define a measurePon(T,F, (Ft))by

TfdP=e^−mt

T

u∈N_t

f_u(T )P (dT ). (2.1)

It is known [20] that this definition is sound and thatPis actually a probability measure with the following properties:

− UnderP, the individuals on the spine move according to Brownian motion with driftcand die at an accelerated ratem+1, independent of the motion.

− When an individual on the spine dies, it leaves a random number of offspring at the point where it has died, this number following the size-biased distribution ofL. In other words, letLbe a random variable withE[f (L)] = E[f (L)L/(m+1)]for every positive measurable function f. Then the number of offspring is an independent copy ofL.

− Amongst those offspring, the next individual on the spine is chosen uniformly. This individual repeats the behaviour of its parent.

− The other offspring initiate branching Brownian motions according to the lawP.

Seen as an equation rather than a definition, (2.1) also goes by the name of “many-to-one lemma.”

2.2. Branching Brownian motion with two barriers

We recall the notationP^y from the previous subsection for the law of branching Brownian motion started aty∈R andE^ythe expectation with respect toP^y. Recall the definition ofPand defineP^yandE^yanalogously.

Leta, b∈Rsuch thaty∈(a, b). Letτ=τ_a,bbe the (random) set of those individuals whose paths enter(−∞, a]∪

[b,∞)and all of whose ancestors’ paths have stayed inside(a, b). Foru∈τ we denote byτ (u)the first exit time from(a, b)byu’s path, i.e.

τ (u)=inf t≥0:X_u(t) /∈(a, b)

=min t≥0: X_u(t)∈ {a, b} ,

and setτ (u)= ∞foru /∈τ. The random setτ is an (optional) stopping line in the sense of [10].

Foru∈τ, defineX_u(τ )=X_u(τ (u)). Denote byZ_a,bthe number of individuals leaving the interval(a, b)at the pointa, i.e.

Z_a,b=

u∈τ

1Xu(τ )=a.

Lemma 2.2. Assume|c|> c₀and defineρ=

c²−c²₀.Then E^y[Z_a,b] =e^c(a−y)sinh((b−y)ρ)

sinh((b−a)ρ). If,furthermore,V =E[L(L−1)]<∞,then

E^y Z²_a,b

= 2Ve^c(a−y) ρsinh³((b−a)ρ)

sinh

(b−y)ρ

y a

e^c(a−r)sinh²

(b−r)ρ sinh

(r−a)ρ dr +sinh

(y−a)ρ

b y

e^c(a−r)sinh³

(b−r)ρ dr

+E^y[Z_a,b].

Proof. On the space T of marked trees with spine, define the random variable I byI =i if ξ_i ∈τ andI = ∞ otherwise. For an eventAand a random variableY writeE[Y, A]instead ofE[Y1A]. Then

E^y[Z_a,b] =E^y

u∈τ

1Xu(τ )=a

=E^y

e^{mτ (ξI)}, I <∞, X_ξI(τ )=a

by the many-to-one lemma extended to optional stopping lines (see [6], Lemma 14.1 for a discrete version). But since the spine follows Brownian motion with driftc, we haveI <∞,P-a.s. and the above quantity is therefore equal to

W^y,c

e^mT, BT =a ,

whereW^y,c is the law of standard Brownian motion with drift c started at y, (B_t)_t≥0 the canonical process and T =T_a,bthe first exit time from(a, b)ofB_t. By Girsanov’s theorem, and recalling thatm=c²₀/2, this is equal to

W^y

e^{c(BT−y)−(1/2)(c2−c20)T}, B_T =a ,

whereW^y=W^y,0. Evaluating this expression ([8], p. 212, Formula 1.3.0.5) gives the first equality.

Foru∈U, letΘ_ube the operator that maps a tree inT to its sub-tree rooted inu. Denote further byC_uthe set of u’s children, i.e.C_u= {uk: 1≤k≤L_u}. Then note that for eachu∈τ we have

Z_a,b=1+

v≺u

w∈Cv

wu

Z_a,b◦Θ_w,

hence E^y

Z_a,b²

=E^y

u∈τ

1X_u(τ )=aZa,b

=E^y[Za,b] +P^y

e^{mτ (ξI)}

v≺ξI

w∈Lv

wξI

Za,b◦Θw, Xξ_I(τ )=a

. (2.2)

Define theσ-algebras G=σ

X_ξI(t);t≥0 , H=G∨σ (ζ_v;v≺ξ_I), I=H∨σ

ξ, I, (L_v;v≺ξ_I) ,

such thatGcontains the information about the path of the spine up to the individual that quits(a, b)first,Hadds to G the information about the fission times on the spine andI adds toHthe information about the individuals of the spine and the number of their children. Now, conditioning onI and using the strong branching property, the second term in the last line of (2.2) is equal to

P^y

e^{mτ (ξI)}

v≺ξ_I

(L_v−1)E^Xv(dv−)[Z_a,b], X_ξI(τ )=a

(recall thatd_vis the time of death ofv). Conditioning onHand noting the fact thatL_vfollows the size-biased law of Lfor an individualvon the spine, yields

P^y

e^{mτ (ξI)}

v≺ξ_I

V

m+1E^XξI(dv)[Z_a,b], X_ξI(τ )=a

.

Finally, since underPthe fission times on the spine form a Poisson process of intensitym+1, conditioning onGand applying Girsanov’s theorem yields

W^y

e^{c(BT−y)−(1/2)ρ2T} T

0

V E^Bt[Z_a,b]dt, BT =a

=Ve^c(a−y) _b

a

E^r[Z_a,b]W^y

e^−(1/2)ρ2TL^r_T, B_T =a dr,

whereL^r_T is the local time of(Bt)at the timeT and the pointr. The last expression can be evaluated explicitly ([8],

p. 215, Formula 1.3.3.8) and gives the desired equality.

Corollary 2.3. Under the assumptions of Lemma2.2,for eachb >0there are positive constantsC_b⁽¹⁾,C_b⁽²⁾,such that asa→ −∞,

(a) E⁰[Z_a,b] ∼C_b⁽¹⁾e^(c+ρ)a,

(b) ifc > c₀,E⁰[Z_a,b² ] ∼C_b⁽²⁾e^(c+ρ)aand (c) ifc <−c₀,E⁰[Z_a,b² ] ∼C_b⁽²⁾e^2(c+ρ)a.

The following result is well known and is only included for completeness. We emphasize that the only moment assumption here ism=E[L] −1∈(0,∞). Recall thatZ_xdenotes the number of particles absorbed atx of a BBM started at the origin. For|c| ≥c0, defineλ_cto be the smaller root ofλ²−2c+c₀², thusλ_c=c−

c²−c²₀. Lemma 2.4. Letx >0.

− If|c| ≥c₀,thenE[Z_x] =e^λcx.

− If|c|< c₀,thenE[Z_x] = +∞.

Proof. We proceed similarly to the first part of Lemma2.2. Define the (optional) stopping lineτ of the individuals whose paths enter[x,∞)and all of whose ancestors’ paths have stayed inside(−∞, x). DefineI as in the proof of Lemma2.2. By the stopping line version of the many-to-one lemma we have

E[Zx] =E

u∈τ

1

=E

e^{mτ (ξI)}, I <∞ .

By Girsanov’s theorem, this equals W

e^{cx−(1/2)(c2−c20)Tx}, T_x<∞ ,

whereW is the law of standard Brownian motion started at 0 and Tx is the first hitting time ofx. The result now

follows from [8], p. 198, Formula 1.2.0.1.

2.3. Proof of Proposition2.1

By hypothesis,c > c0,E[L²]<∞and the BBM starts at the origin. Letx >0 and letτ=τxbe the stopping line of those individuals hitting the pointx for the first time. ThenZx= |τx|.

Leta <0 andn∈N. By the strong branching property,

P⁰(Z_x> n)≥P⁰(Z_x> n|Z_a,x≥1)P⁰(Z_a,x≥1)≥P^a(Z_x> n)P⁰(Z_a,x≥1).

IfP₋⁰ denotes the law of branching Brownian motion started at the point 0 with drift−c, then P^a(Z_x> n)=P₋⁰(Z_a−x> n)≥P₋⁰(Z_a−x,1> n).

In order to bound this quantity, we choosea=a_n in such a way thatn= ¹₂E₋⁰[Z_an−x,1].By Corollary2.3(a), (c) (applied with drift−c) and the Paley–Zygmund inequality, there is then a constantC₁>0, such that

P₋⁰(Z_an−x,1> n)≥1 4

E₋⁰[Z_an−x,1]² E₋⁰[Z_a²

n−x,1] ≥C₁ for largen.

Furthermore, by Corollary2.3(a) (applied with drift−c), we have 1

2C₁⁽¹⁾e^{−λc(an−x)}∼n, asn→ ∞,

and thereforea_n= −(1/λc)logn+O(1). Again by the Paley–Zygmund inequality and Corollary2.3(a), (b) (applied with driftc), there existsC₂>0, such that for largen,

P⁰(Z_an,x≥1)=P⁰(Z_an,x>0)≥E⁰[Za_n,x]² E⁰[Z_a²

n,x] ≥(C⁽¹⁾x )² 2C⁽²⁾x

e^λcan≥C2

n^d. This proves the proposition withC=C1C2.

3. The FKPP equation

As was already observed by Neveu [23], the translational invariance of Brownian motion and the strong branching property immediately imply that Z=(Zx)x≥0 is a homogeneous continuous-time Galton–Watson process (for an overview to these processes, see [4], Chapter III, or [14], Chapter V). There is therefore an infinitesimal generating function

a(s)=α _∞

n=0

p_nsⁿ−s

, α >0, p1=0, (3.1)

associated to it. It is a strictly convex function on[0,1], witha(0)≥0 anda(1)≤0. Its probabilistic interpretation is α= lim

x→0

1

xP (Z_x=1) and p_n=lim

x→0P (Z_x=n|Z_x=1),

hence q_n=αp_n for n∈N0\ {1}. Note that with no further conditions on c and L, the sum

n≥0p_n need not necessarily be 1, i.e. the rateαp_∞, wherep_∞=1−

n≥0pn, with which the process jumps to+∞, may be positive.

We further defineF_x(s)=E[s^Zx], which is linked toa(s)by Kolmogorov’s forward and backward equations ([4], p. 106, or [14], p. 102):

∂

∂xF_x(s)=a(s)∂

∂sF_x(s) (forward equation), (3.2)

∂

∂xF_x(s)=a F_x(s)

(backward equation). (3.3)

The forward equation implies that if a(1)=0 and φ(x)=E[Z_x] = _∂s^∂ F_x(1−), then φ(x)=a(1)φ(x), whence E[Z_x] =e^a(1)x. On the other hand, ifa(1) <0, then the process jumps to∞with positive rate, henceE[Z_x] = ∞ for allx >0.

The next lemma is an extension of a result which is stated, but not proven, in [23], Eq. (1.1). According to Neveu, it is due to A. Joffe. To the knowledge of the author, no proof of this result exists in the current literature, which is why we prove it here.

Lemma 3.1. Let(Yt)t≥0be a homogeneous Galton–Watson process started at1,which may explode and may jump to+∞with positive rate.Letu(s)be its infinitesimal generating function andF_t(s)=E[s^Yt].Letq be the smallest zero ofu(s)in[0,1].

1. Ifq <1,then there existst₋∈R∪ {−∞}and a strictly decreasing smooth functionψ₋:(t₋,+∞)→(q,1)with limt→t₋ψ₋(t)=1andlimt→∞ψ₋(t)=q,such that on(q,1)we haveu=ψ₋ ◦ψ₋⁻¹,F_t(s)=ψ₋(ψ₋⁻¹(s)+t ).

2. Ifq >0,then there existst₊∈R∪ {−∞}and a strictly increasing smooth functionψ₊:(t₊,+∞)→(0, q)with limt→t₊ψ₊(t)=0andlimt→∞ψ₊(t)=q,such that on(0, q)we haveu=ψ₊ ◦ψ₊⁻¹,F_t(s)=ψ₊(ψ₊⁻¹(s)+t ).

The functionsψ₋andψ₊are unique up to translation.

Moreover,the following statements are equivalent:

− For allt >0,Y_t<∞a.s.

− q=1ort₋= −∞.

Proof. We first note that u(s) >0 on (0, q) and u(s) <0 on (q,1), since u(s) is strictly convex, u(0)≥0 and u(1)≤0. Since F0(s)=s, Kolmogorov’s forward equation (3.2) implies that F_t(s) is strictly increasing in t for s∈(0, q)and strictly decreasing int fors∈(q,1). The backward equation (3.3) implies thatF_t(s)converges toq as t → ∞for everys∈ [0,1). Repeated application of (3.3) yields that F_t(s)is a smooth function oft for every s∈ [0,1].

Now assume thatq <1. Forn∈Nsets_n=1−2⁻ⁿ(1−q), such thatq < s₁<1,s_n< s_n+1ands_n→1 asn→ ∞.

Sett₁=0 and definet_nrecursively by

t_n+1=t_n−t, wheret>0 is such thatF_t(s_n+1)=s_n.

Then(tn)n∈Nis a decreasing sequence and thus has a limitt₋∈R∪ {−∞}. We now define fort∈(t₋,+∞), ψ₋(t)=F_t−tn(s_n), ift≥t_n.

The functionψ₋is well defined, since for everyn∈Nandt≥t_n, F_t−tn(s_n)=F_t−tn

F_tn−tn+1(s_n+1)

=F_t−tn+1(s_n+1),

by the branching property. The same argument shows us that ifs∈(q,1),sn> sandt>0 such thatF_t(sn)=s, then Ft(s)=F_t+t(sn)=ψ₋(t+t+tn)for allt≥0. In particular,ψ₋(t+tn)=s, henceFt(s)=ψ₋(ψ₋⁻¹(s)+t ). The backward equation (3.3) now gives

u(s)= ∂

∂tFt(s) t=0

=ψ₋

ψ₋⁻¹(s) .

The second part concerningψ₊is proven completely analogously. Uniqueness up to translation ofψ₋ andψ₊ is obvious from the requirementψ (ψ⁻¹(s)+t )=F_t(s), whereψis eitherψ₋orψ₊.

For the last statement, note thatP (Y_t<∞)=1 for allt >0 if and only ifF_t(1−)=1 for allt >0. But this is the

case exactly ifq=1 ort₋= −∞.

The following proposition shows that the functionsψ₋andψ₊corresponding to(Z_x)_x≥0are so-calledtravelling wavesolutions of a reaction-diffusion equation called the Fisher–Kolmogorov–Petrovskii–Piskounov (FKPP) equa- tion. This should not be regarded as a new result, since Neveu ([23], Proposition 3) proved it already for the case c≥c₀ andL=2 a.s. (dyadic branching). However, his proof relied on a path decomposition result for Brownian motion, whereas we show that it follows from simple renewal argument valid for branching diffusions in general.

Recall thatf (s)=E[s^L]denotes the generating function ofL. Letq be the unique fixed point off in[0,1) (which exists, sincef(1)=m+1>1), and letqbe the smallest zero ofa(s)in[0,1].

Proposition 3.2. Assumec∈R.The functionsψ₋andψ₊from Lemma3.1corresponding to(Z_x)_x≥0are solutions to the following differential equation on(t₋,+∞)and(t₊,+∞),respectively.

1

2ψ−cψ=ψ−f◦ψ. (3.4)

Moreover,we have the following three cases:

1. Ifc≥c₀,thenq=q,t₋= −∞,a(1)=0,a(1)=λ_c,E[Z_x] =e^λcxfor allx >0.

2. If|c|< c₀,thenq=q,t₋∈R,a(1) <0,a(1)=2c,P (Z_x= ∞) >0for allx >0.

3. Ifc≤ −c₀,thenq=1,a(1)=0,a(1)=λ_c,E[Z_x] =e^λcxfor allx >0.

Proof. Lets∈(0,1)and define the functionψ_s(x)=F_x(s)=E[s^Zx]forx≥0. By symmetry,Z_xhas the same law as the number of individualsN absorbed at the origin in a branching Brownian motion started atx and with drift

−c. By a standard renewal argument (LemmaA.1), the functionψ_s is therefore a solution of (3.4) on(0,∞)with ψ_s(0+)=s. This proves the first statement, in view of the representation ofF_x in terms ofψ−andψ₊ given by Lemma3.1.

Lets∈(0,1)\ {q}and letψ (s)=ψ₋(s)ifs > qandψ (s)=ψ₊(s)otherwise. By (3.4), a(s)=ψ◦ψ⁻¹(s)

ψ◦ψ⁻¹(s) =2c+2ψ◦ψ⁻¹(s)−f◦ψ◦ψ⁻¹(s)

ψ◦ψ⁻¹(s) =2c+2s−f (s) a(s) ,

whence, by convexity, a(s)a(s)=2ca(s)+2

s−f (s)

, s∈ [0,1]. (3.5)

Assume|c| ≥c₀. By Lemma2.4,E[Z_x] =e^λcx, hencea(1)=0 anda(1)=λ_c, in particular,a(1) >0 forc≥c₀ anda(1) <0 forc≤ −c₀. By convexity,q <1 forc≥c₀andq=1 forc≤ −c₀. The last statement of Lemma3.1 now implies thatt₋= −∞ifc≥c0.

Now assume|c|< c0. By Lemma2.4,E[Zx] = +∞for allx >0, hence eithera(1) <0 ora(1)=0 anda(1)= +∞, in particular,q <1 by convexity. However, ifa(1)=0, then by (3.5),a(1)=2c−2m/a(1), whence the second case cannot occur. Thus,a(1) <0 anda(1)=2cby (3.5).

It remains to show thatq=qifq <1. Assumeq=q. Thena(q)=0 by the (strict) convexity ofaanda(q)= 2cby (3.5). In particular,a(q)≥a(1), which is a contradiction toabeing strictly convex.

4. Proof of Theorem1.1

We start with the following Abelian-type lemma:

Lemma 4.1. LetXbe a random variable concentrated onN0and letϕ(s)=E[s^X]be its generating function.Assume thatE[X(log⁺X)^γ]<∞for someγ >0.Then,ass→0,

ϕ(1)−ϕ(1−s)=O

log1 s

₋γ

and ϕ(1)s+ϕ(1−s)−1=O

s

log1 s

₋γ .

Proof. Lets₀>0 be such that the functions→s(log¹_s)^γ is increasing on[0, s0]. Lets∈(0, s0). Then, withp_k= P (X=k),

ϕ(1)−ϕ(1−s) log1

s γ

= ∞ k=1

kpk

1−(1−s)^k−1 log1

s γ

.

If k≥s⁻¹, then (1−(1−s)^k−1)(log¹_s)^γ ≤(logk)^γ. If s₀⁻¹ ≤k < s⁻¹, then s(log¹_s)^γ < ¹_k(logk)^γ and thus (1−(1−s)^k−1)(log¹_s)^γ< ks(log¹_s)^γ≤(logk)^γ.Hence,

∞ k=s₀⁻¹

kpk

1−(1−s)^k−1 log1

s γ

≤ ∞ k=s₀⁻¹

pkk(logk)^γ ≤E X

log⁺Xγ .

Furthermore, we have fors∈(0,1),

s⁻₀¹ k=1

kp_k

1−(1−s)^k−1 log1

s γ

≤

s

log1 s

γs⁻₀¹ k=1

k²p_k≤C

for someC >0. Collecting these results, we have, for everys∈(0,1), ϕ(1)−ϕ(1−s)

log1 s

γ

≤C+E X

log⁺Xγ

<∞,

by hypothesis. This yields the first equality. Settingg(s)=ϕ(1)s+ϕ(1−s)−1, we note thatg(0)=0 and g(s)=ϕ(1)−ϕ(1−s)=O

log1

s ₋γ

,

by the first equality. Since(log¹_s)^−γ is slowly varying, g(s)=

_s

0

g(r)dr=O

s

log1 s

₋γ ,

by standard theorems on the integration of slowly varying functions (see e.g. [11], Section VIII.9, Theorem 1).

Proof of Theorem1.1. We havec=c0by hypothesis. Letψ₋be the travelling wave from Proposition3.2, which is defined onR, sincet₋= −∞. Letφ(x)=1−ψ₋(−x), such thatφ(−∞)=1−q,φ(+∞)=0 and

1

2φ(x)+c₀φ(x)=f

1−φ(x)

−

1−φ(x)

, (4.1)

by (3.4). Furthermore,a(1−s)=φ(φ⁻¹(s))andF_x(1−s)=1−φ(φ⁻¹(s)−x).

Under the hypothesis E[L(logL)^2+ε]<∞, it is known [20] that there exists K ∈(0,∞), such that φ(x)∼ Kxe^−c0x as x → ∞. Sincea(1)=0 anda(1)=c₀ by Proposition3.2, this entails that φ(x)=a(1−φ(x))∼

−c₀Kxe^−c0x, asx→ ∞.

Setϕ₁=φandϕ₂=φ. By (4.1), d

dx

ϕ₁(x) ϕ₂(x)

=

φ(x) φ(x)

=

−2c0φ(x)+2[f (1−φ(x))−(1−φ(x))] φ(x)

.

Settingg(s)=c²₀s+2[f (1−s)−(1−s)] =2[f(1)s+f (1−s)−1], this gives d

dx

ϕ₁(x) ϕ₂(x)

=M ϕ₁(x)

ϕ₂(x)

+

g(ϕ₂(x)) 0

withM=

−2c0 −c₀²

1 0

. (4.2)

The Jordan decomposition ofMis given by J=A⁻¹MA=

−c₀ 1 0 −c₀

, A=

−c₀ 1−c₀

1 1

. (4.3)

Setting_ϕ1

ϕ2

=A_ξ1

ξ2

, we get withξ=_ξ1

ξ2

:

ξ(x)=J ξ(x)+

−g(φ(x)) g(φ(x))

,

which, in integrated form, becomes ξ(x)=e^xJξ(0)+e^xJ

_x

0

e^−yJ

−g(φ(y)) g(φ(y))

dy. (4.4)

Note that e^xJ =

e^−c0x xe^−c0x 0 e^−c0x

. (4.5)

With the above asymptotic ofφ, we haveg(φ(x))=O(e^−c0x/x^1+ε), asx→ ∞, by Lemma4.1and the hypothesis onL. Eqs (4.4) and (4.5) now imply that

ξ2(x)∼e^−c0x

ξ2(0)+ _∞

0

e^c0yg(φ(y))dy

,

and

(ξ1+ξ2)(x)∼xe^−c0x

ξ2(0)+ _∞

0

e^c0yg(φ(y))dy

,