On the Eve property for CSBP

We consider the population model associated to continuous state branching processes and we are interested in the so-called Eve property that asserts the existence of an ancestor with an overwhelming progeny at large times, and more generally, in the possible behaviours of the frequencies among the population at large times. In this paper, we classify all the possible behaviours according to the branching mechanism of the continuous state branching process.


Introduction
Continuous State Branching Processes (CSBP for short) have been introduced by Jirina [18] and Lamperti [23,22,24].They are the scaling limits of Galton-Watson processes: see Grimvall [15] and Helland [16] for general functional limit theorems.They represent the random evolution of the size of a continuous population.Namely, if Z = (Z t ) t∈[0,∞) is a CSBP, the population at time t can be represented as the interval [0, Z t ].In this paper, we focus on the following question: as t → ∞, does the population concentrate on the progeny of a single ancestor e ∈ [0, Z 0 ] ?If this holds true, then we say that the population has an Eve.More generally, we discuss the asymptotic frequencies of settlers.A more formal definition is given further in the introduction.
The Eve terminology was first introduced by Bertoin and Le Gall [5] for the generalised Fleming-Viot process.Tribe [30] addressed a very similar question for super-Brownian motion with quadratic branching mechanism, while in Theorem 6.1 [10] Donnelly and Kurtz gave a particle system interpretation of the Eve property.In the CSBP setting, the question has been raised for a general branching mechanism in [21].Let us mention that Grey [14] and Bingham [7] introduced martingales techniques to study the asymptotic behaviours of CSBP under certain assumptions on the branching mechanism: to answer the above question in specific cases, we extend their results using slightly different tools.For related issues, we also refer to Bertoin, Fontbona and Martinez [3], Bertoin [2] and Abraham and Delmas [1].
In the finite variation cases, Ψ can be rewritten as follows: In these cases, note that D = lim λ→∞ Ψ(λ)/λ.We shall always avoid the cases of deterministic CSBP that correspond to linear branching mechanisms.Namely, we shall always assume that either β > 0 or π = 0.
The integral equation ( 9) also implies the following: If Ψ allows extinction in finite time, namely if it necessarily implies that Ψ satisfies (4), namely that Ψ is of infinite variation type.In this case, for any t, x ∈ (0, ∞), ) is one-to-one and decreasing.Thus, P x -a.s.
Let us give here the precise definition of the Eve property.To that end, we fix x ∈ (0, ∞), we denote by B([0, x]) the Borel subsets of [0, x].We also denote by M ([0, x]) the set of positive Borel-measures on [0, x] and by M 1 ([0, x]) the set of Borel probability measures.Let us think of m t ∈ M 1 ([0, x]), t ∈ [0, ∞), as the frequency distributions of a continuous population whose set of ancestors is [0, x] and that evolves through time t.Namely for any Borel set B ⊂ [0, x], m t (B) is the frequency of the individuals at time t whose ancestors belong to B. The relevant convergence mode is the total variation norm: Here, it is natural to assume that t → m t is cadlag in total variation norm.The Eve property can be defined as follows.
Definition 1.1 We denote by ℓ the Lebesgue measure on R (or its restriction to [0, x] according to the context).Let t ∈ (0, ∞) −→ m t ∈ M 1 ([0, x]) be cadlag with respect to • var and assume that there exists Here, a is called the dust, S is a countable subset of [0, x] that is the set of settlers and for any y ∈ S, m ∞ ({y}) is the asymptotic frequency of the settler y.
If a = 0, then we say that the population m := (m t ) t∈(0,∞) has no dust (although m t may have a diffuse part at any finite time t).If a = 0 and if S reduces to a single point e, then m ∞ = δ e and the population m is said to have an Eve that is e.Furthermore, if there exists t 0 ∈ (0, ∞) such that m t = δ e , for any t > t 0 , then we say that the population has an Eve in finite time.
The following theorem asserts the existence of a regular version of the frequency distributions associated with a CSBP.
Theorem 1 Let x ∈ (0, ∞).Let Ψ be a branching mechanism of the form (3). We assume that Ψ is not linear.Then, there exists a probability space (Ω, F , P) on which the two following processes are defined.
We call M the frequency distribution process of a CSBP(Ψ, x).If Ψ is of finite variation type, then M is • var -right continous at time 0, which is not the case if Ψ is of infinite variation type as explained in Section 2.3.The strong regularity of M requires specific arguments: in the infinite variation cases, we need a decomposition of CSBP into Poisson clusters, which is the purpose of Theorem 3 in Section 2.2.(see this section for more details and comments).
The main result of the paper concerns the asymptotic behaviour of M on the following three events.

Theorem 2
We assume that Ψ is a non-linear branching mechanism.Let x ∈ (0, ∞) and let M and Z be as in Theorem 1.Then, P-a.s.lim t→∞ M t −M ∞ var = 0, where M ∞ is of the form (16).Moreover, the following holds true P-almost surely.
(i) On the event A = {ζ < ∞}, M has an Eve in finite time.
First observe that the theorem covers all the possible cases, except the deterministic ones that are trivial.On the absorption event A = {ζ < ∞}, the result is easy to explain: the descendent population of a single ancestor either explodes strictly before the others, or gets extinct strictly after the others, and there is an Eve in finite time.
The cases where there is no Eve -namely, Theorem 2 (ii-b), (ii-c) and (iii-b) -are simple to explain: the size of the descendent populations of the ancestors grow or decrease in the same (deterministic) scale and the limiting measure is that of a normalised subordinator as specified in Proposition 3.1, Lemma 3.2, Proposition 3.3, Lemma 3.4, and also in the proof Section 3.2.Let us mention that in Theorem 2 (iii-b1) and (iii-b2), the dust of M ∞ comes only from the dust of the M t , t ∈ (0, ∞): it is not due to limiting aggregations of atoms of the measures M t as t → ∞.
Theorem 2 (ii-a) and (iii-a) are the main motivation of the paper: in these cases, the descendent populations of the ancestors grow or decrease in distinct scales and one dominates the others, which implies the Eve property in infinite time.This is the case of the Neveu branching mechanism Ψ(λ) = λ log λ, that is related to the Bolthausen-Sznitman coalescent: see Bolthausen and Sznitman [8], and Bertoin and Le Gall [4].
Let us first make some comments in connection with the Galton-Watson processes.The asymptotic behaviours displayed in Theorem 2 (ii) find their counterparts at the discrete level: the results of Seneta [27,28] and Heyde [17] implicitly entail that the Eve property is verified by a supercritical Galton-Watson process on the event of explosion iff the mean is infinite.However neither the extinction nor the dust find relevant counterparts at the discrete level so that Theorem 2 (i) and (iii) are specific to the continuous setting.
CSBP present many similarities with generalised Fleming-Viot processes, see for instance the monograph of Etheridge [13]: however for this class of measure-valued processes Bertoin and Le Gall [5] proved that the population has an Eve without assumption on the parameter of the model (the measure Λ which is the counterpart of the branching mechanism Ψ).We also mention that when the CSBP has an Eve, one can define a recursive sequence of Eves on which the residual populations concentrate, see [21].Observe that this property is no longer true for generalised Fleming-Viot processes, see [20].
The paper is organized as follows.In Section 2.1, we gather several basic properties and estimates on CSBP that are needed for the construction of the cluster measure done in Section 2.2.These preliminary results are also used to provide a regular version of M which is the purpose of Section 2.3.Section 3 is devoted to the proof of Theorem 2: in Section 3.1 we state specific results on Grey martingales associated with CSBP in the cases where Grey martingales evolve in comparable deterministic scales: these results entail Theorem 2 (ii-b), (ii-c) and (iii-b), as explained in Section 3.2.Section 3.3 is devoted to the proof of Theorem 2 (ii-a) and (iii-a): these cases are more difficult to handle and the proof is divided into several steps; in particular it relies on Lemma 3.9, whose proof is postponed to Section 3.3.4.
2 Construction of M.
Recall that for any x ∈ [0, ∞], P x stands for the law on D([0, ∞), [0, ∞]) of a CSBP(Ψ, x) and recall that Z stands for the canonical process.It is easy to deduce from (1) the following monotone property: Lemma 2.2 Assume that Ψ is not linear.Then, for all t, x, y ∈ (0, ∞), P x Z t > y > 0 .
The following lemmas are used in Section 2.2 for the construction of the cluster measure.

Lemma 2.4
Assume that Ψ is of infinite variation type.Then, for all ε ∈ (0, 1) and all s, t ∈ (0, ∞) such that s < t, Proof The equality follows from (21).Next observe that ν Since ν s does not vanish on (0, ∞), Lemma 2.2 entails that the first member is strictly positive.
We shall need the following simple result in the construction of M in Section 2.3.
Let θ and C as in (23).First note that P r (T ≤ a) ≤ P r (Z a > θy)+P r (T ≤ a; Z a ≤ θy).Then, by the Markov property at T and ( 23), we get We next state a more precise inequality that is used in the construction of the cluster measure of CSBP.
By (25), δ > 0. Let a ∈ (0, 1 4 t 0 ) be such that (26) holds true.We then fix x ∈ [0, η], b ∈ [0, a] and c ∈ [ 1 2 t 0 , t 0 ] and we introduce the stopping time where B := P x T ≤ b ; Z b ≤ η ; Z c > ε is bounded as follows: by the Markov property at time b and by (20), we first get Recall that p t (x, dy) = P x (Z t ∈ dy) stands for the transition kernels of Z.The Markov property at time T then entails Next observe that P x -a.s.b−T ≤ a and Z T > 2η, which implies p b−T Z T , [0, η] ≤ δ by (26).Thus, where we use (20) in the last inequality.Thus, by the Markov property at time T and the previous inequalities, we finally get which implies the desired result by (29).
Then, there exists (Z * t ) t∈[0,∞) , a CSBP(Ψ * , x) on (Ω, F, P) such that Proof Without loss of generality, we assume that there exists a Lévy process X defined on (Ω, F, P) such that Z is derived from X by the Lamperti time-change (30).We then set X * t = X t + Dt that is a subordinator with Laplace exponent −Ψ * and with initial value x.Since D is positive, we have X t ≤ X * t for all t ∈ [0, ∞).Observe that τ * = ∞.Let L * and Z * be derived from X * as L and Z are derived from X in (30).Then, Z * is a CSBP(Ψ * , x) and observe that which easily implies the desired result since Z * is non-decreasing.

The cluster measure of CSBP with infinite variation.
Recall that D([0, ∞), [0, ∞]) stands for the space of [0, ∞]-valued cadlag functions.Recall that Z stands for the canonical process.For any t ∈ [0, ∞), we denote by F t the canonical filtration.Recall from (11) the definition of the times of absorption ζ 0 , ζ ∞ and ζ.Also recall from the beginning of Section 2.1 the definition of the measure ν t on (0, ∞]. Theorem 3 Let Ψ be of infinite variation type.Then, there exists a unique sigma-finite measure N Ψ on D([0, ∞), [0, ∞]) that satisfies the following properties.
for any nonnegative functionals F, G and for any t ∈ (0, ∞).
The measure N Ψ is called the cluster measure of CSBP(Ψ).
Comment 2.1 The existence of N Ψ is not really new: for sub-critical Ψ, N Ψ can be derived from the excursion measure of the height process of the Lévy trees and the corresponding super-processes as introduced in [11] (see also [12] for a different approach on super-processes).Let us also mention the notes of lecture [26], with an indirect proof that works when Ψ ′ (0+) = −∞.Here, we provide a brief and self-contained proof of the existence of the cluster measure for CSBP that works in all cases.
Proof The only technical point to clear is (a): namely, the right-continuity at time 0. For any s, t ∈ (0, ∞) such that s ≤ t and for any ε ∈ (0, 1), we define a measure for any functional F .By Lemma 2.4, (31) makes sense and it defines a probability measure on the space D([0, ∞), [0, ∞]).The Markov property for CSBP and Lemma 2.4 easily imply that for any s ≤ s 0 ≤ t, We first prove that for t and ε fixed, the laws Q s t,ε are tight as s → 0. By (32), it is clear that we only need to control the paths in a neighbourhood of time 0. By a standard criterion for Skorohod topology (see for instance Theorem 16.8 [6] p. 175), the laws Q s t,ε are tight as s → 0 if the following claim holds true: for any η, δ ∈ (0, 1), there exists a 1 ∈ (0, 1  4 t) such that To prove (33), we first prove that for any η, δ ∈ (0, 1), there exists a 0 ∈ (0, t) such that Proof of (34).Recall that 1 [1,∞] (y) ≤ C(1 − e −y ), for any y ∈ [0, ∞], where C = (1 − e −1 ) −1 .Fix η, δ ∈ (0, 1) and s, b ∈ (0, t) such that s ≤ b.Then, (32), with b = s 0 , implies that By developping the product in the integral of the last right member of the inequality, we get We then define a 0 such that sup b∈(0,a 0 ] f (b) < 1 3 δ, which implies (34).Proof of (33).We fix η, δ ∈ (0, 1).Let a ∈ (0, 1  4 t) such that (24) in Lemma 2.6 holds true with t 0 = t.Let a 0 as in (34).We next set a 1 = a∧a 0 .We fix s ∈ (0, a 1 ] and we then get the following inequalities: Here we use (34) in the second line, (24) in the third line and (34) in the fourth one.
We have proved that for t, ε fixed, the laws Q s t,ε are tight as s → 0. Let Q t,ε stand for a possible limiting law.By a simple argument, Q t,ε has no fixed jump at time s 0 and basic continuity results entail that (32) holds true with Q t,ε instead of Q s t,ε , which fixes the finite-dimensional marginal laws of Q t,ε on (0, ∞).Next observe that for η, δ ∈ (0, 1) and a 1 ∈ (0, 1  4 t) as in (33), the set {sup (0,a 1 ) Z > 2η} is an open set of D([0, ∞), [0, ∞]).Then, by (33) and the Portmanteau Theorem, Q t,ε (sup (0,a 1 ) Z > 2η) ≤ δ.This easily implies that Q t,ε -a.s.Z 0 = 0, which completely fixes the finite-dimensional marginal laws of Q t,ε on [0, ∞).This proves that there is only one limiting distribution and that decreases to 0. We define a measure N t by setting By the first equality, N t is a well-defined sigma-finite measure; the second equality shows that the definition of N t does not depend on the sequence (ε p ) p∈N , which implies for any ε ∈ (0, 1).Consequently, we get N t − N t ′ = N t ( • ; Z t ′ = 0), for any t ′ > t > 0. Fix t q ∈ (0, 1), q ∈ N, that decreases to 0. We define N Ψ by setting The first equality shows that N Ψ is a well-defined measure and the second one that its definition does not depend on the sequence (t q ) q∈N , which implies This easily entails that for any nonnegative functional Recall that ζ is the time of absorption in {0, ∞}.Since N tq,εp (ζ = 0) = 0, we get N Ψ (ζ = 0) = 0 and thus, N Ψ ({O}) = 0, where O stands for the null function.
Properties (b) and (c) are easily derived from (36), ( 35) and standard limit-procedures: the details are left to the reader.

Poisson decomposition of CSBP.
From now on, we fix (Ω, F , P), a probability space on which are defined all the random variables that we mention, unless the contrary is explicitly specified.We also fix x ∈ (0, ∞) and we recall that ℓ stands for the Lebesgue measure on R or on [0, x], according to the context.We first briefly recall Palm formula for Poisson point measures: let E be a Polish space equipped with its Borel sigma-field E .Let A n ∈ E , n ∈ N, be a partition of E. We denote by M pt (E) the set of point measures m on E such that m(A n ) < ∞ for any n ∈ N; we equip M pt (E) with the sigma-field generated by the applications m → m(A), where A ranges in E. Let N = i∈I δ z i , be a Poisson point measure on E whose intensity measure µ satisfies µ(A n ) < ∞ for every n ∈ N. We shall refer to the following as to the Palm formula: for any measurable We next introduce the Poisson point measures that are used to define the population associated with a CSBP.

Infinite variation cases.
We assume that Ψ is of infinite variation type.Let , where N Ψ is the cluster measure associated with Ψ as specified in Theorem 3.Then, for any t ∈ (0, ∞), we define the following random point measures on [0, x]: We also set

Finite variation cases.
We assume that Ψ is of finite variation type and not linear.Recall from (6) the definition of π(dr) P r (dZ) , where P r is the canonical law of a CSBP(Ψ, r) and where π is the Lévy measure of Ψ.Then, for any t ∈ (0, ∞), we define the following random measures on [0, x]: We also set In both cases, for any t ∈ [0, ∞) and any B ∈ B([0, x]), Z t (B) and Z t− (B) are [0, ∞]-valued F -measurable random variables.The finite dimensional marginals of (Z t (B)) t∈[0,∞) are those of a CSBP(Ψ, ℓ(B)): in the infinite variation cases, it is a simple consequence of Theorem 3 (c); in the finite variation cases, it comes from direct computations: we leave the details to the reader.Moreover, if B 1 , . . ., B n are disjoint Borel subsets of [0, x], note that the processes To simplify notation, we also set that has the finite dimensional marginals of a CSBP(Ψ, x).

Regularity of Z.
Since we deal with possibly infinite measures, we introduce the following specific notions.We fix a metric d on [0, ∞] that generates its topology.For any positive Borel measures µ and ν on [0, x], we define their variation distance by setting The following proposition deals with the regularity of Z on (0, ∞), which is sufficient for our purpose.The regularity at time 0 is briefly discussed later.
Proposition 2.8 Let Z be as in ( 39) or (41).Then, Proof We first prove the infinite variation cases.We proceed by approximation.Let us fix s 0 ∈ (0, ∞).
Note that in the finite variation cases, Z is d var -right continuous at 0. In the infinite variation cases, this cannot be so: indeed, set B = [0, x]\{x i ; i ∈ I}, then Z t (B) = 0 for any t ∈ (0, ∞) but Z 0 (B) = ℓ(B) = x.However, we have the following lemma.Lemma 2.9 Assume that Ψ is of infinite variation type.Let Z be defined on (Ω, F , P) by ( 39).Then This implies that P-a.s.Z t → Z 0 weakly as t → 0+.

Proof of Theorem 1 and of Theorem 2 (i).
Recall the notation Z t = Z t ([0, x]).By Proposition 2.8, Z is cadlag on (0, ∞) and by arguing as in Lemma 2.9, without loss of generality, we can assume that Z is right continuous at time 0: it is therefore a cadlag CSBP(Ψ, x).Recall from (11) the definition of the absorption times ζ 0 , ζ ∞ and ζ of Z.We first Observe that M has the desired regularity on [0, ζ) by Proposition 2.8 and Lemma 2.9.Moreover M satisfies property (17).It only remains to define M for the times t ≥ ζ on the event {ζ < ∞}.
Let us first assume that P(ζ 0 < ∞) > 0, which can only happen if Ψ satisfies (15).Note that in this case, Ψ is of infinite variation type.Now recall P from (38) and Z from (39).Thus, ζ 0 = sup i∈I ζ i 0 , where ζ i 0 stands for the extinction time of Z i .Then, , that is the function defined right after (15) which satisfies ∞ v(t) dr/Ψ(r) = t.Since v is C 1 , the law (restricted to (0, ∞)) of the extinction time ζ 0 under N Ψ is diffuse.This implies that P-a.s. on {ζ 0 < ∞} there exists a unique i 0 ∈ I such that ζ 0 = ζ i 0 0 .Then, we set ξ 0 := sup{ζ i 0 ; i ∈ I\{i 0 }}, e = x i 0 and we get M t = δ e for any t ∈ (ξ 0 , ζ 0 ).Thus, on the event {ζ 0 < ∞} and for any t > ζ 0 , we set M t = δ e and M has the desired regularity on the event {ζ 0 < ∞}.An easy argument on Poisson point measures entails that conditional on {ζ 0 < ∞}, e is uniformly distributed on [0, x].
Let us next assume that P(ζ ∞ < ∞) > 0, which can only happen if Ψ satisfies (13).We first consider the infinite variation cases: note that ζ ∞ = inf i∈I ζ i ∞ , where ζ i ∞ stands for the explosion time of Z i .Then, that is the function defined right after (13) which satisfies κ(t) 0 dr/(Ψ(r)) − = t.Since κ is C 1 , the law (restricted to (0, ∞)) of the explosion time ζ ∞ under N Ψ is diffuse.This implies that P-a.s. on {ζ ∞ < ∞} there exists a unique i ∞ .Then, on {ζ ∞ < ∞}, we set e = x i 1 and M t = δ e , for any t ≥ ζ ∞ .Then, we get lim t→ζ∞− M t −δ e var = 0 and an easy argument on Poisson point measures entails that conditional on {ζ ∞ < ∞}, e is uniformly distributed on [0, x].This completes the proof when Ψ is of infinite variation type.In the finite variation cases, we argue in the same way: namely, by simple computations, one shows that for any t ∈ (0, ∞), #{j ∈ J : t j ≤ t, Z j t−t j = ∞} is a Poisson r.v. with mean xκ(t); it is therefore finite and the times of explosion of the population have diffuse laws: this proves that the descendent population of exactly one ancestor explodes strictly before the others, and it implies the desired result in the finite variation cases: the details are left to the reader.Remark 2.1 Note that the above construction of M entails Theorem 2 (i).
We complete this result by the following lemma.
We next consider the behaviour of finite variation sub-critical CSBP.
We complete this result by the following lemma.

Proof
The proof Lemma 3.2 works verbatim, except that in (50) which is easy to prove since e −Dt φ θ (e Dt λ) → d θ λ as t → ∞.

Proof of Theorem 2 (ii-b), (ii-c) and (iii-b).
We now consider the cases where there is no Eve property.Recall that x ∈ (0, ∞) is fixed and that ℓ stands for Lebesgue measure on R or on [0, x] according to the context.Recall that Ψ is not linear and recall the notation Z t := Z t ([0, x]).We first need the following elementary lemma.
If γ < ∞, then Proposition 3.1 entails that W θ is a compound Poisson process: in this case and on B, there are finitely many settlers and conditionally on B, the number of settlers is distributed as a Poisson r.v. with parameter xγ conditionned to be non zero, which completes the proof of Theorem 2 (ii-b).If γ = ∞, then the same proposition shows that W θ has a dense set of jumps.Therefore, a.s. on B there are a dense countable set of settlers, which completes the proof of Theorem 2 (ii-c).In both cases, the asymptotic frequencies are described by Proposition 3.1 and Lemma 3.2

Lemma 3.7
We assume that Ψ is not linear and conservative.Then, there exists a random probability measure M ∞ on [0, x] such that P-a.s.lim t→∞ M t = M ∞ with respect to the topology of the weak convergence.
Proof By Lemma 3.6, it is sufficient to prove that for any q ∈ Q ∩ [0, x], P-a.s.lim t→∞ M t ([0, q]) exists.
For any v ∈ [0, 1) and any t ∈ (0, ∞], we set Let U, V : Ω → [0, 1) be two independent uniform r.v. that are also independent of the Poisson point measures P and Q.Then, for any t, s ∈ (0, ∞], the conditional law of (R −1 t (U ), R −1 s (V )) given P and Q is M t ⊗M s .Moreover, Lemma 3.7 and standard arguments entail For any t ∈ (0, ∞), we recall the definition of the function v It is increasing and one-to-one, which implies that lim λ→v(t) u(−t, λ) = ∞.
Proof We first prove the lemma when Ψ is of infinite variation type: recall from (38) the definition of P and note that on {Z t+s > 0}, To simplify notation, we denote by A the left member in (60).By Palm's formula (37) we then get, .
The proof in the finite variation cases is similar except that Z and M are derived from the Poisson point measure Q defined by (40).Note that Ψ is persistent.We moreover assume it to be conservative: thus, Z t ∈ (0, ∞), for any t ∈ [0, ∞).Let A stand for the left member in (60).Then, A = A 1 + A 2 where A 1 corresponds to the event where U falls on a jump of R t , while A 2 deals with the event where it falls on the dust.The latter gives , where for any λ 1 , λ 2 ∈ (0, ∞) we have set Here we apply Palm formula to derive the second line from the first one.The first expectation in (62) yields E Z t+s e −λ 1 Zt e −λ 2 Z t+s = ∂ λ u(s, λ 2 )∂ λ u(t, λ 1 + u(s, λ 2 )) x e −xu(t,λ 1 +u(s,λ 2 )) .
The second term of the product in (62) gives Here, to derive the second line from the first one, we use To derive the third one from the second one, we use the identity ∂ λ u (t, λ) = −Ψ(λ) −1 ∂ t u(t, λ) and we do an integration by part.Recall B(λ 1 , λ 2 ) from (61).By the previous computations we get Recall that we already proved that x 2 ∞ 0 ∞ 0 dλ 1 dλ 2 B(λ 1 , λ 2 )−B(λ 1 , λ 2 +θ) equals the right member of (60).So, to complete the proof, we set and calculations similar as in the infinite variation case yield x 2 (F (0) − F (θ)) = −A 2 , which entails the desired result in the finite variation cases.
To complete the proof of Theorem 2, we need the following technical lemma whose proof is postponed.Lemma 3.9 We assume that Ψ is not linear.Then, P-a.s. for all y ∈ [0, x], lim t→∞ M t ({y}) exists.

Proof of Theorem 2 (iii-a).
We assume that Ψ is persistent, of infinite variation type and such that γ < ∞.Observe that P under P( • | lim t→∞ Z t = 0) is a Poisson point measure associated with the branching mechanism Ψ(• + γ) that is sub-critical (and therefore conservative).So the proof of Theorem 2 (iii-a) reduces to the cases of sub-critical persistent branching mechanisms and without loss of generality, we now assume that Ψ is so.Thus, lim θ→∞ u(t, θ) = v(t) = ∞.By letting θ go to ∞ in Lemma 3.8, we get Since Ψ is sub-critical and persistent for all w ∈ (0, ∞), u(−t, w) increases to ∞ as t ↑ ∞.Moreover, since Ψ is of infinite variation type, λ/Ψ(λ) decreases to 0 as λ ↑ ∞, which implies that lim t→∞ B(t) = 0.By (65) and Lemma 3.9, we get P-a.s.M t ({e}) → 1, and thus M t −δ e var → 0 by Lemma 3.5, as t → ∞, which completes the proof of Theorem 2 (iii-a).

Proof of Lemma 3.9.
To complete the proof of Theorem 2, it only remains to prove Lemma 3.9.We shall proceed by approximation, in several steps.Recall from (38) and (40) the definition of the Poisson point measures P and Q.For any t ∈ (0, ∞), we define the following: and Q t = j∈J 1 {t j ≤t} δ (x j ,t j ,Z j •∧(t−t j ) ) . (66) We then define G t as the sigma-field generated either by P t if Ψ is of infinite variation type, or by Q t if Ψ is of finite variation type.
there is no dust and M has finitely many settlers whose number, under P( • |B), is distributed as a Poisson r.v. with mean xγ conditionned to be non zero;