Coalescences in Continuous-State Branching Processes

Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes (with negative jumps) which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behavior and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy of the population, we define non-exchangeable Markovian coalescent processes obtained by sampling independent Poisson arrival times over the flow. These coalescent processes are called consecutive coalescents, as only consecutive blocks can merge. They are characterized in law by finite measures on $\mathbb{N}$ which can be thought as the offspring distributions of some inhomogeneous immortal Galton-Watson processes forward in time.


Introduction
Random population models can be divided in two classes, those with constant finite size and those whose size is varying randomly. It is known since the 2000s that populations with constant finite size, evolving by resampling, have genealogies given by exchangeable coalescents. These processes, defined by Möhle and Sagitov [MS01], Pitman [Pit99], Sagitov [Sag99] and Schweinsberg [Sch00], are generalisations of Kingman's coalescent for which multiple coalescences of ancestral lineages are allowed. They correspond to the genealogy backward in time of so-called generalized Fleming-Viot processes. Those processes, which can be seen as scaling limits of Moran models [Mor58], were defined and studied by Donnelly and Kurtz [DK99] (via a particle system called lookdown construction) and by Bertoin and Le Gall [BLG03] (via flows of exchangeable bridges). Both constructions are similar in many aspects and are summarized via the notion of flow of partitions, see Labbé [Lab14a,Lab14b] and Foucart [Fou12]. We refer to Bertoin's book [Ber06] for a comprehensive account on exchangeable coalescents.
The main objective of this work is to study coalescent processes induced by branching processes. We briefly explain how branching concepts have been developed from the sixties to the beginning of the twenty-first century. Continuous-state branching processes (CSBPs for short) are positive Markov processes representing the size of a continuous population. They have been defined by Jiřina [Jiř58] and Lamperti [Lam67a] and are known to be scaling limits of Galton-Watson Markov chains, see Grimvall [Gri74] and Lamperti [Lam67b]. The most famous CSBP is certainly the Feller's branching diffusion dX t = σ X t dB t + βX t dt which is the rescaled limit of binary branching processes, see Feller [Fel51] and Jiřina [Jiř69]. Feller's CSBP is the only CSBP with continuous paths, other ones have positive jumps which represent macroscopic reproduction events in the population.
At about the same time as the rise of exchangeable coalescents, considerable research was devoted to the study of the genealogy of branching processes forward in time. Galton-Watson processes have a natural lexicographical tree's genealogy. This representation leads Aldous [Ald93] and Duquesne and Le Gall [DLG02] to study scaling limits of discrete trees and establish remarkable convergences towards Brownian continuum tree in the Feller diffusion case and Lévy continuum tree in the case of a general CSBP. Another natural genealogy for a branching population is provided by Bertoin and Le Gall in their precursor article [BLG00] in terms of flows of subordinators. At any fixed times s < t, the population between time s and t is represented by a subordinator (a Lévy process with non-decreasing paths). Individuals are ordered in such a way that ancestors from time s are the jumps locations of the subordinator and each ancestor from time s has a family at time t whose size is the size of the jump.
Both representations with trees and subordinators are future-oriented and less attention has been paid to the description of coalescences in ancestral lineages of continuous-state branching processes. We briefly review some methods that have been developed recently in order to study the genealogy backwards in time of branching processes.
When reproduction laws are stable, branching and resampling population models can be related through renormalisation by the total size and random time-change. We refer to Berestycki et al [BBS07], Birkner et al. [BBC + 05], Foucart and Hénard [FH13] and Schweinsberg [Sch03]. The connection between exchangeable coalescents and CSBPs is particular to stable laws and the study of the genealogy of a general branching process requires a different method.
One approach consists of conditioning the process to be non-extinct at a given time, sampling two or more individuals uniformly in the population and study the time of coalescence of their ancestral lineages. This program is at the heart of the works of Athreya [Ath12], Duquesne and Labbé [DL14], Harris et al. [HJR17], Johnston [Joh17], Lambert [Lam03] and Le [Le14].
Starting from a different point of view, Bi and Delmas [BD16] and Chen and Delmas [CD12] have considered stationary subcritical branching population obtained as processes conditioned on the non-extinction. The genealogy is then studied via a Poisson representation of the population. We refer also to Evans and Ralph [ER10] for a study in the same spirit.
A third approach is to represent the backwards genealogy through point processes. Aldous and Popovic [AP05] and Popovic [Pop04] have shown how to encode the genealogy of a critical Feller diffusion with a Poisson point process on R + ×R + called Coalescent Point Process. Atoms of the coalescent point process represents times of coalescences between two "consecutive" individuals in the boundary of the Brownian tree. Such a description was further developed by Lambert and Popovic [LP13] for a Lévy continuum tree. In this general setting, multiple coalescences are possible and the authors build a point process with multiplicities, which records both the coalescence times and the number of involved mergers in the families of the current population. Their method requires in particular to work with the height process introduced by Le Gall and Le Jan in [LGLJ98].
In the present article, we choose a different route and seek for a dynamical description of the genealogy. We first observe that flows of subordinators provide a continuous branching population whose size is infinite at any time and whose ancestors are arbitrarily old. We then study the inverse flow which tracks backward in time the ancestral lineage of an individual in the current population. This process corresponds to the Siegmund dual of the CSBP.
In a second time, we construct random partitions by sampling arrival times of an independent Poisson process (with a fixed intensity) on the flow. We then describe how partitions coagulate when time's arrow points to the past and define new elementary non-exchangeable Markovian coalescents. We call these processes consecutive coalescents as only consecutive blocks will be allowed to merge. Our method follows closely that of Bertoin and Le Gall for exchangeable coalescents ([BLG03], [BLG05], [BLG06a]). Heuristically, the exchangeable bridges are replaced by subordinators and the uniform random variables by arrival times of a Poisson process.
Consecutive coalescents are simple dual objects of continuous-time Galton-Watson processes and allow one to simplify the description of the genealogy of general CSBPs given by the Coalescent Point Process as introduced in [LP13, Section 4]. We shall also answer an open question in [LP13,Remark 6] by showing how to define the complete genealogy of individuals standing in the current generation when the so-called Grey's condition is satisfied.
In the case of Neveu's CSBP (which does not fulfill Grey's condition), Bertoin and Le Gall in [BLG00] have shown that the genealogy of the CSBP, started from a fixed size (without renormalization nor time-change) is given by a Bolthausen-Sznitman coalescent. We will see that for this CSBP, the consecutive coalescents have simple explicit laws. This will enable us to recover results of Möhle [Möh15] and Möhle and Kukla [KM18] about the number of blocks in a Bolthausen-Sznitman coalescent.
We wish to mention that Grosjean and Huillet in [GH16] have studied a recursive balls-inboxes model which can be seen as a consecutive coalescent in discrete time. Moreover, Johnston and Lambert [JL18] have independently considered Poissonization techniques for studying the coalescent structure in branching processes.
The paper is organized as follows. In Section 1, we recall the definition of a continuous-state branching process and how Bochner's subordination can be used to provide a representation of the genealogical structure associated to CSBPs. In Section 2, we investigate the inverse flow by characterizing its semi-group and studying its long-term behavior. In Section 3, we provide a complete study of the inverse flow in the case of the Feller diffusion. We recover with an elementary approach the Coalescent Point Process of Popovic [Pop04]. In Section 4, we study the coalescences in the inverse flow of a general CSBP by defining the consecutive coalescents. We describe the genealogy of the whole population standing at the current generation under the Grey's condition (recalled in Section 1). In Section 5, we investigate the infinitesimal dynamics of the inverse flow. The process of the ancestral lineage of a fixed individual is characterized by its generator. In Section 6, we provide some examples for which calculations can be made explicitly.

Generalities on continuous-state branching processes
This section is divided in two parts. In the first one, we introduce the continuous-state branching processes as well as their fundamental properties. In the second one, we show how continuousstate branching processes can be constructed as flows of subordinators. Their main properties are also stated.

Continuous-state branching processes
We give here an overview of continuous-state branching processes and their fundamental properties. Most statements in this section can be found for instance in [Li11,Chapter 3] or [Kyp14, Chapter 12].
Definition 1.1. A continuous-state branching process is a Feller process (X t , t ≥ 0), taking values in [0, ∞], with 0 and ∞ being absorbing states, whose semi-group satisfies the so-called branching property: ∀x, y ≥ 0, ∀t ≥ 0, X t (x + y) where (X t (x), t ≥ 0) and (X t (y), t ≥ 0) are two independent processes with the same law as (X t , t ≥ 0), started respectively from x and y.
(v ) Under condition (1.8), the following events coincide The event {X t (x) = 0 for some t ≥ 0} is called extinction, and {X t (x) = ∞ for some t ≥ 0} is called explosion. We refer to the integral conditions (1.7) and (1.8) as Grey's condition for extinction and explosion respectively. Lambert [Lam07] and Li [Li00] have studied the quasistationary distribution of subcritical CSBPs conditioned on the non-extinction. Li [Li00]). In the subcritical case, under Grey's condition for extinction´∞ du Ψ(u) < ∞, there exists a probability measure ν over (0, ∞] such that for any The Laplace transform of ν is given bŷ (1.9)

Flows of subordinators
Observe that on the one hand, by the the branching property of CSBP, the random variable X t (x) is a positive infinitely divisible random variable, parametrized by x. Therefore, for all t ≥ 0, the process x → X t (x) is a positive Lévy process, hence a subordinator. In particular, the map λ → v t (λ) is the Laplace exponent of this (possibly killed) subordinator, and can be written with κ t ≥ 0, d t ≥ 0 and t a Lévy measure on R + such that´∞ 0 (1 ∧ x) t (dx) < ∞.
Remark 1.2. Note that the quantities v t (∞) and v t (0) defined in Theorem 1.A can be rewritten, with the formula in (1.10) Therefore (1.7) is equivalent to the finiteness of the measure t for all t > 0, and (1.8) to the positivity of κ t for all t > 0.
On the other hand, the semigroup property entails that for any s, t ≥ 0, Therefore, writingX an independent copy of the CSBP X, we have This last observation leads Bertoin and Le Gall [BLG00] to consider representing a CSBP as a flow of subordinators, which we now define.
Definition 1.3. A flow of subordinators is a family (X s,t (x), s ≤ t, x ≥ 0) satisfying the following properties: (ii) For every t ∈ R, (X r,s , r ≤ s ≤ t) and (X r,s , t ≤ r ≤ s) are independent.
(iii) For every r ≤ s ≤ t, X r,t = X s,t • X r,s .
(iv ) For every s ∈ R and x ≥ 0, we have X s,s (x) = x = lim t→s X s,t (x) in probability.
Remark 1.4. The convergence in (iv ) also holds uniformly on compact sets by second Dini's theorem. It was proved by Bertoin and Le Gall [BLG00] that any CSBP can be constructed as a flow of subordinators. For the sake of completeness, we prove here that CSBP and flow of subordinators are in one-to-one map.
Lemma 1.5. Let (X s,t (x), s ≤ t, x ≥ 0) be a flow of subordinators as in Definition 1.3, there exists a branching mechanism Ψ such that for all s ∈ R and x ≥ 0, (X s,s+t (x), t ≥ 0) is a CSBP(Ψ) starting from x. Reciprocally, for each branching mechanism Ψ, there exists a flow of subordinators such that for all s ∈ R and x ≥ 0, (X s,s+t (x), t ≥ 0) is a CSBP(Ψ) starting from x.
Proof. Let (X s,t (x), s ≤ t, x ≥ 0) be a flow of subordinators. By Definition 1.3(ii) and (iii), we have that t → X s,s+t (x) is a Markov process for all x ≥ 0 and s ∈ R. Moreover, Definition 1.3(iv) implies this Markov process to be continuous in probability, therefore Feller, and by Definition 1.3(i), we conclude that this Markov process is homogeneous in time, and satisfies the branching property, as X s,s+t (x + y) = X s,s+t (x) + (X s,s+t (x + y) − X s,s+t (x)) , and X s,s+t (x + y) − X s,s+t (x) is independent of X s,s+t (x) and has same law as X s,s+t (y). Reciprocally, by [BLG00, Proposition 1], given a branching mechanism Ψ, there exists a process (S (s,t) (a), s ≤ t, a ≥ 0) such that almost surely (i) for all s ≤ t, a → S (s,t) (a) is a càdlàg subordinator with Lévy-Khintchine exponent (iv ) the finite dimensional distributions of t → S (s,s+t) (a) are the ones of a CSBP(Ψ).
One readily observe that points (i)-(iii) imply Definition 1.3(i)-(iii). Moreover, by the fourth point, S (s,s+t) (a) has the law of a CSBP(Ψ) X t starting from X 0 = a. As X is a Feller process, we have lim t→0 X t = a in probability, thus (iv) yields lim t→0 S (s,s+t) (a) = a in probability, completing the proof.
A noteworthy consequence of the above lemma is that if (X s,t (x), s ≤ t, x ≥ 0) is a flow of subordinators associated to the branching mechanism Ψ, we have that for all s ≤ t and x ≥ 0, where v t−s (λ) is the function defined in (1.3). One can think of this flow of subordinators as a way to couple on the same probability space every Markov property and every branching property (1.1), for all values of t, x, y simultaneously in one process. The flow of subordinators provides a genuine continuous-space branching population model. More precisely, the interval [0, X s,t (x)] can be interpreted as the set of descendants at time t of the population that was represented at time s as the interval [0, x]. With this interpretation, the genealogy forward in time of the population is defined as follows. If X s,t (y−) < X s,t (y), we say that for all z ∈ (X s,t (y−), X s,t (y)], the individual z at time t is a descendant of the individual y living at time s. If X s,t (y−) = X s,t (y) (i.e. X s,t is continuous at y), we then say that individual z = X s,t (y) at time t is the descendant of the individual y living at time s if and only if y = inf{x > 0 : X s,t (x) = z}. One can observe that the cocycle property ensures that this construction indeed defines a genealogy. If z at time t is a descendant of y at time s, which is a descendant of x at time r, we have X s,t (y−) < z ≤ X s,t (y) and X r,s (x−) < y ≤ X r,s (x).
By the cocycle property (X r,t = X s,t • X r,s ) and as X s,t is non-decreasing then X r,t (x−) = X s,t (X r,s (x−)) ≤ X s,t (y−) < z and X r,t (x) = X s,t (X r,s (x)) ≥ X s,t (y) ≥ z, thus z at time t is a descendant of x at time r. Similar computations can be written if X s,t is continuous at point y and/or X r,s is continuous at point x.
Recall the condition (1.6) for the sample paths of the CSBP(Ψ) to have infinite variations. This condition ensures the subordinator X s,t to be driftless, i.e. d r = 0 for all r ≥ 0 in (1.10). As a result, under (1.6), the range X s,t ([0, ∞)) of the subordinator has zero Lebesgue measure, ensuring that almost every individual x at time t belong to one of the infinite families of ancestors at time s. This, assumption (1.6) often simplifies the interpretation of results obtained in this article. Under this assumption, we denote by J s,t = {x ≥ 0 : X s,t (x) = X s,t (x−)} the set of jumps of X s,t . By definition of the genealogy, almost surely the population at time t, indexed by R + , can be partitioned according to their ancestor at time s by {(X s,t (y−), X s,t (y)], y ∈ J s,t }.
Recall that according to Theorem 1.A-(ii), Grey's condition´∞ du Ψ(u) < ∞ entails that for any t > 0, t ([0, ∞]) < ∞. Under this condition, the subordinators X s,t are therefore compound Poisson processes. In particular, the set J s,t is the set of arrival times of a Poisson process with intensity v t (∞). Note that the partition {(X s,t (y−), X s,t (y)], y ∈ J s,t } consists of a family of consecutive intervals. This justifies the introduction of consecutive coalescents on N in Section 4.

The inverse flow
We start this section by a preliminary observation on the genealogy backward in time of a CSBP. Consider the Poisson point process on R + × (0, ∞) with some renormalisation constant a t > 0 for all t > 0. Recall ρ the largest positive root of Ψ and ν the quasi-stationary distribution (6.2) of a subcritical CSBP conditioned on the non-extinction.
Remark 2.2. In the supercritical case, flows of CSBPs can be renormalized to converge almostsurely. We refer to Duquesne and Labbé [DL14], Grey [Gre74], and Foucart and Ma [FM16]. Since for any time t, X −t,0 and X 0,t have the same law, we could therefore renormalize in law the size of the descendants at time 0 of x from time −t. Typically, ∆X −t,0 (x) is of order exponential in the finite mean case (|Ψ (0+)| < ∞), and double exponential in the infinite mean case (|Ψ (0+)| = ∞).
Let us describe in details the meaning of the above convergence, for supercritical, critical and subcritical CSBPs. Observe that E t encodes information on the individuals at time −t having a large family of descendants at time 0. Thus, (2.2) gives information on the origin of the earliest ancestors of the population at time 0. Depending on the sign of Ψ (0+), we have three different behaviours: (i) If Ψ (0+) < 0, a unique ancestor from time −∞, located at an exponential random variable with parameter ρ, which generates all individuals at time 0. This individual is the ancestor of the process.
(ii) If Ψ (0+) = 0, then a t := v t (∞) −→ t→∞ 0 and the whole population at time 0 has a common ancestor, but the backward lineage of this ancestor converges in law as t → ∞ towards ∞.
(iii) If Ψ (0+) > 0, then the population at time 0 is split into distinct families, each of which coming down from a different ancestor at time −∞.
In the (sub)critical case, individuals from generation −t with descendance at time 0 are located at distance O(1/v t (∞)) from 0. Proposition 2.1 motivates a more complete study of the ancestral lineages of individuals alive in the population at time 0. Our main aim is to provide an almostsure description of how the (E t , t ≥ 0) evolves and to get precise information on the sizes of the families. We now introduce the inverse flow of the flow of subordinators (X s,t , s ≤ t) and study some of its properties. We first define, for s ≤ t and y ≥ 0 The process X −1 s,t is the right-continuous inverse of the càdlàg process X s,t . Note that the individual X −1 s,t (y) is the ancestor alive at time s of the individual y considered at time t ≥ s. It is therefore a natural process to introduce in order to study the genealogy of a CSBP backwards in time. We call inverse flow the process (X s,t (y), s ≤ t, y ≥ 0) defined for all s ≤ t, y ≥ 0 as followsX We first list some straightforward properties of inverse flows.
(ii) For every t ≥ 0, (X r,s , r ≤ s ≤ t) and (X r,s , t ≤ r ≤ s) are independent.
(iii) Almost surely, for every s ≤ t ≤ u,X s,u =X t,u •X s,t .
Remark 2.4. The convergence in (iv ) also holds uniformly on compact sets.
Proof. These results are an immediate consequence of Lemma A.1, which describes well-known properties of right-continuous inverses, and the definition of flow of subordinators. More precisely, the first point is a consequence of Lemma A.1(ii), the third one of Lemma A.1(iii) and the fourth one follows from Lemma A.1(iv) and Definition 1.3(iv). Finally, the second point follows simply from the fact that for all a ≤ b ≤ t,X a,b is measurable with respect to (X r,s , −t ≤ r ≤ s). Hence, by Definition 1.3(ii), we conclude that (ii) holds.
We shall denote (X t (y), y ≥ 0, t ≥ 0) the flow of inverse subordinators (X 0,t (y), y ≥ 0, t ≥ 0). As noted above, it tracks backwards in time the ancestral lineages of the population at time 0. Since individuals are ordered,X t (y) can also be interpreted as the random size of the population at time −t whose descendance at time 0 has size y. Observe that by Lemma 2.3(i) and Definition 1.3(i), we have ∀s ≤ t, ∀x, y ≥ 0, P(X s,t (x) > y) = P(X s,t (y) < x). (2.5) The relation (2.5) is known as Siegmund duality. We refer the reader for instance to Siegmund [Sie76] and Clifford and Sudbury [CS85].
Theorem 2.5. Fix y > 0. The process (X t (y), t ≥ 0) is a Markov process in (0, ∞). Its semigroup (Q t , t ≥ 0) satisfies for any bounded measurable function f and any t ≥ 0 where e q and e vt(q) are exponential random variable with parameter q and v t (q) and e q is independent of (X t (y), t ≥ 0, y ≥ 0).
Observe that (2.6) characterizes the semigroup Q t , by identification of the Laplace transforms, as it can be rewritten as: for all q ≥ 0, ∞ 0 qe −qy Q t f (y)dy =ˆv t (q)e −vt(q)y f (y)dy, therefore Q t f is the inverse Laplace transform of q → vt(q) q´e −vt(q)y f (y)dy.
Proof. We observe that the cocycle property and the independence, obtained in points (ii) and (iii) of Proposition 2.3 readily entail that t →X t (y) has the Markov property. Moreover, if X 0,t (y) = 0 then X −t,0 (0) = y > 0, which is impossible, as X −t,0 (0) is the value at time t of a CSBP starting from mass 0, and 0 is an absorbing point for a CSBP. Similarly,X 0,t (y) = ∞ yields that lim z→∞ X −t,0 (z) ≤ y, which is impossible as well as X −t,0 (z) is a non-null subordinator.
Finally, we now turn to the computation of the semigroup ofX(y), which is obtained through the Siegmund duality. Let e q be an independent exponential random variable with parameter q, we have which implies that (2.6) holds.
The above theorem shows that the semigroup of (X t ) can be expressed in simple terms when applied to exponential distributions. This will motivate later on the study of the action of the flowX on Poisson point processes.
We now observe that the Markov process t →X 0,t (y) can be straightforwardly extended as a Markov process on [0, ∞].
(i) The boundary 0 is an entrance boundary of (X t , t ≥ 0) if and only if´∞ du Remark 2.7. The Markov processes (X 0,t (0)) and (X 0,t (∞)) have the following interpretations, in terms of the CSBP (i) The process (X 0,t (0), t ≥ 0), starting from 0 at time 0, represents the smallest individual at generation −t to have descendants at time 0. If´∞ du |Ψ(u)| < ∞, there is extinction in finite time for the CSBP X (i.e. with positive probability, X −t,0 (x) = 0). In that caseX 0,t (0) is a non-trivial Markov process. If´∞ du |Ψ(u)| = ∞, there is no extinction in finite time for the CSBP, thus all individuals at time t have descendants at time 0, (ii) The process (X t (∞), t ≥ 0), starting from ∞, represents the smallest individual at generation t with an infinite progeny at time 0. If´0 du |Ψ(u)| < ∞, there is explosion in finite time for the CSBP X (i.e. with positive probability, X −t,0 (x) = ∞). In that case,X 0,t (∞) is a non-trivial Markov process. If´0 du |Ψ(u)| = ∞, there is no explosion in finite time and all individuals at time t have finitely many descendants at time 0.
The first point for the boundary 0 is obtained as follows. By the duality relation, we see that By letting y to 0, we have According to Theorem 1.A-(ii),´∞ du Ψ(u) < ∞ is a necessary and sufficient condition for v t (∞) < ∞. It remains to justify the formula for Q t f (0). By using Theorem 2.5 and the facts that in probability, lim q→∞ e q = 0 and lim by dominated convergence. We deduce the formula for Q t f (0). We now prove that the semigroup property holds at 0. By definition of Q t f (0), we have that Therefore, as v t+s = v t • v s , we complete the proof of (i).
The proof of (ii) follows very similar lines to the proof of (i), and is based on the fact that is found using that lim q→0 e q = ∞ in probability. Finally, to prove that the semigroup Q t extended to [0, ∞] is Feller, we observe that the random map y →X t (y) jumps only on constant stretch of X −t,0 (being its right-continuous inverse). There is no fixed value in (0, ∞) at which X −t,0 is constant and therefore y ∈ (0, ∞) →X t (y) has no fixed discontinuities. This entails that for any continuous function f over [0, ∞], Q t f is continuous at any point y ∈ (0, ∞). By and one only needs to show the pointwise continuity at 0 of Q t f , which follows from Proposition 2.3(iv).
We study now the long term behaviour of (X t , t ≥ 0) in the critical and subcritical case. By transience, we mean thatX t (x) −→ t→∞ ∞ a.s. for any x ∈ (0, ∞).
Proposition 2.8. Let Ψ be a branching mechanism. We observe that Remark 2.9. Intuitively, in the subcritical case, for any fixed a > 0, individuals below level a living at arbitrarily large time in the past will have no progeny at time 0. Therefore the ancestral lineage of an individual x living at time 0, goes above any fixed level a as time goes to ∞. This explains the transience. In the critical case, large oscillations can occur when´0 x Ψ(x) dx = ∞. This latter condition is known see Duhalde et al. [DFM14] to entails that first entrance times of the CSBP have infinite mean, in such case the process (X t , t ≥ 0) is null recurrent. Note that if Proof. We first prove (i). Let y ∈ (0, ∞). By duality (2.5) and Theorem 1.A-(i) Assume now Ψ subcritical or critical. For any Borelian set B and any p > 0, set By monotone convergence In the subcritical case Ψ (0+) > 0, therefore´q 0 x Ψ(x) dx < ∞ and for almost every x ∈]0, ∞[, one has 0 < U 0 (x, B) < ∞. Since for any x ≤ y,X t (x) ≤X t (y) then therefore U 0 (x, B) ≥ U 0 (y, B) and then 0 < U 0 (x, B) < ∞ for all x. We may now invoke Proposition 2.2-(iv') in Getoor [Get80], by taking the increasing sequence B n :=]0, n[. This entails that the process (X t , t ≥ 0) is transient. In the critical case, if´q 0 x Ψ(x) dx < ∞ then the process is transient. If now´q 0 x Ψ(x) dx = ∞ then by (2.7) for any set B with positive Lebesgue measure, U 0 (x, B) = ∞ for all x. By Proposition 2.4-(i) in [Get80], we conclude that (X t , t ≥ 0) is recurrent.

The Feller flow
In this section, we investigate the genealogy backwards in time of Feller CSBPs. These are continuous CSBPs with quadratic branching mechanisms of the form Ψ : q → σ 2 2 q 2 − βq, with β ∈ R and σ 2 ≥ 0. For any fixed x, the Feller CSBP (X t (x), t ≥ 0) with mechanism Ψ can be constructed as the solution of the stochastic differential equation where (B t , t ≥ 0) is a Brownian motion. We study here in detail the flow (X s,t (x), t ≥ s, x ≥ 0) of CSBPs with branching mechanism Ψ and the inverse flow (X s,t (x), t ≥ s, x ≥ 0). Many calculations can be made explicit in this setting, see for instance Pardoux [Par08] for a study of the flow (X s,t (x), t ≥ s, x ≥ 0). Note that Ψ is subcritical if β < 0, critical if β = 0 and supercritical if β > 0. Moreover, in the latter case we have ρ = 2β σ 2 . Observe also that the differential equation (1.3) can be rewritten Remark 3.1. Observe that above, we often make a distinction between β = 0 and β = 0, but the functions v t , t orβ t that we defined are continuous at β = 0.
We now study the law of the inverse Feller flow (X s,t (y), s ≤ t, y ≥ 0), in particular characterizing its marginal distributions as a process in the variable t or y.
we have (i) for any fixed y ≥ 0, (X t (y), t ≥ 0) is a Markov process with semigroup given by (ii) For any fixed t, (X t (y), y ≥ 0) is a subordinator with Laplace exponentv t started from the positive random variableX t (0) whose Laplace transform is is a flow of continuous-state branching processes with immigration with mechanismsΨ and linear immigrationΦ(q) := σ 2 2 q. In particular, we obtain that this is in fact a compound Poisson process, with jump rateβ t e βt and exponential jump distribution with parameterβ t . Therefore, writing (N (t) x , x ≥ 0) which are i.i.d. exponential random variables, and reciprocally we write M (t) i ≤ y} for all y ≥ 0, which is the Poisson process with inter-arrival times (x (t) j , j ≥ 1). We observe that by (3.1),X 0,t being the right-continuous inverse of X −t,0 , we havê Note that we haveX 0,t (0) > 0, contrarily to X 0,t (0) = 0, but thatX 0,t is also a compound Poisson process with exponential jump rate. The construction of X −t,0 andX 0,t are represented on Figure 3.
Figure 1: Inverse of compound Poisson process Note that by (3.2), we have that and moreover, for all y ≥ 0 by straightforward Poisson computations. It remains to verify that ,which concludes the proof.
The inverse Feller flow being itself a flow of subordinators with explicit law, many quantities can be computed explicitly, such as the most recent common ancestor of a population. Picking two individuals x ≤ y at time 0, the age T x,y of the most recent common ancestor of x and y is the first time t such that there exists an individual z at generation −t that gave birth to both x and y, or more precisely This definition of most recent common ancestor can naturally be generalized as follows: given A a subset of R + , we set However, as the partition of Therefore, obtaining the law of T x,y will be enough to study the genealogy of the Feller flow.
Proposition 3.4. For any 0 ≤ x ≤ y ≤ z, we have and T x,y and T y,z are independent. In particular, we have Among other things, this proposition proves that the population comes down from a single ancestor in critical or supercritical cases (β ≥ 0), while in the subcritical case, for β < 0, the population at time 0 can be separated into families with different ancestors at time −∞.
Proof. This result is a consequence of the inverse flow representation of Theorem 3.2. Indeed, for all x ≤ y and λ ≥ 0, we have thus, letting λ → ∞ we obtain P(T x,y ≤ t) = e −(y−x)vt(∞) . Moreover, we observe thatv t (∞) = β t , proving the first equation.
By (3.3), and given that (X 0,t (y) −X 0,t (x), t ≥ 0) and (X 0,t (z) −X 0,t (y), t ≥ 0) are independent Feller CSBP with mechanismΨ, starting from y − x and z − y respectively, we conclude that T x,y and T y,z are independent.
To obtain P(T x,y = ∞) we compute concluding the proof.
Remark 3.5. A straightforward consequence of the above coalescent is that for any choice of {x 1 , . . . x n } of individuals at generation 0, the coalescent tree of this family of individuals will only consist in binary merging. Indeed, for every pair (x i , x i+1 ) of consecutive individuals, their time of coalescence is independent from the time of coalescence of any other pair of consecutive individuals in the population, and has density with respect to the Lebesgue measure. Therefore, almost surely the first coalescing time will consists in the merging of only two neighbours. Proposition 3.4 readily entails the representation of the genealogical tree of the population at time 0 as a functional of a Poisson point process. The following construction is reminiscent of the comb representation by Lambert and Uribe Bravo [LUB17]. Proof. We observe from Proposition 3.4 that for all x ≤ y ≤ z, T x,y and T y,z are independent and T x,z L = max(T x,y , T y,z ). Moreover, note by definition that T x,z ≥ max(T x,y , T y,z ) a.s. This yields that for all x ≤ y ≤ z, We consider the event of probability one for which the above equation is true simultaneously for all x, y, z ∈ Q + . As a result, the field T x,y is decreasing in x and increasing in y. Therefore, there exists a càdlàg modification of the field satisfying (3.4) simultaneously for all x, y, z ∈ R + . As a result, we can construct a simple point process G on each atom in the point process has mass one). Moreover, note that where N is a Poisson point process with intensity dx ⊗ (β t dt +β ∞ δ ∞ (dt)). Hence, by monotone classes theorem, for all measurable relatively compact set B ⊂ R + × (R + ∪ {∞}), we have As a result, by [Kal02,Theorem 10.9], we have N L = G, which concludes the proof.
Pitman and Yor [PY82, Sections 3 and 4], see also [DL14] for a more general setting, have shown that any flow of Feller's branching diffusions can be represented through a Poisson point process on ]0, ∞[×C, where C denotes the space of continuous paths on R + . In our setting, the flow (X t (x) −X t (0), t ≥ 0) can be represented as follows: for all t > 0, and n is the so-called cluster measure (see [DL14]). The atoms (X i , i ∈ I) can be interpreted as the ancestral lineages of the initial individuals (x i , i ∈ I). They are independent Feller diffusions with mechanismΨ starting from infinitesimal masses. For any i ∈ I, denote by ζ i := inf{t ≥ 0;X i t = 0}. The time ζ i represents a binary coalescence time between two "consecutive" individuals. By definition of n, for any t > 0, n(ζ > t) =v t (∞) and therefore i∈I δ (x i ,ζ i ) has the same law as N . We represent the ancestral lineages and their coalescences in Figure 2. Recall also from Remark 2.7 thatX t (0) is the first individual from generation −t to have descendants at time 0.
subcritical casê Xt (0) past 0 (super)critical case is a flow of supercritical CSBPs. Following Bertoin et al. [BFM08] (see also Pardoux [Par08, Section 7]), one can define the random sequence (x n , n ≥ 1) recursively as follows: The random sequence (x n , n ≥ 1) is known as the initial prolific individuals of the flow of supercritical CSBPs (X t (x) −X t (0), x ≥ 0, t ≥ 0) and corresponds to the jumps times of a Poisson process with intensity − 2β σ 2 . Within the framework of inverse flow, the random partition of R + : ..) corresponds to current families with distinct common ancestors. Note that the sequence (x n , n ≥ 1) is also the sequence of atoms of the point process N (· × {∞}), defined in Lemma 3.6.
We observed in this section that the law of the flowX is explicit when X is a Feller flow. When the branching mechanism Ψ is not of the quadratic form, multiple births occur in the population. Thus, when time runs backward, coalescences of multiple lineages should arise. The law of the inverse flowX becomes then more involved. In the next section, we construct a simple class of Markovian coalescents which will allow us to encode easily multiple coalescences in lineages backward in time. The law of the lineage's location (X t (x), t ≥ 0) for a fixed individual x ≥ 0 is studied further in Section 5.

Consecutive coalescents
In this section we study the genealogy of branching processes both forward and backward in time, using random partitions of consecutive integers. We shall see how to define a coalescent process in this framework and that the associated coalescent theory is elementary. In a second time, we apply these results to the genealogy of a population in a continuous-state branching process sampled according to a Poisson point process with intensity λ. In a third time, by making the parameter λ increase to ∞, we obtain a full description of the genealogical tree of individuals in a CSBP under Grey's condition.

Consecutive coalescents in continuous-time Galton-Watson processes
In this section, we construct a class of simple Markovian coalescents arising when studying the genealogy backward in time of continuous-time Galton-Watson processes. We begin by introducing the classical notation for coalescent processes on the space of partitions. For a more precise description of that framework, in the context of exchangeable coalescents, we refer to Bertoin's book [Ber06, Chapter 4], from which we borrow our definitions and notation.
Let n ∈ N ∪ {∞}, we denote by [n] = {j ∈ N : j ≤ n} the set of integer smaller or equal to n. We call consecutive partition of [n] a collection C of disjoint subsets {C 1 , C 2 , . . .} with consecutive integers (i.e. intervals of [n]), such that C i = [n]. Without loss of generality, we will always assume that the subsets of the collection C are ranked in the increasing order of their elements. We denote by C n the set of consecutive partitions of [n]. Note that any C ∈ C n is characterized by the ranked sequence of its blocks sizes (#C 1 , #C 2 , . . .), as ∀j ∈ N, C j = {k ∈ N : #C 1 + · · · + #C j−1 < k ≤ #C 1 + · · · + #C j } . We introduce some classical operations on C n . For each k ≤ n and C ∈ C n , we denote by We define a distance on C ∞ the set of consecutive partitions of N by setting Note that the metric space (C ∞ , d) is compact. We next introduce the coagulation operation. For any C ∈ C n and C ∈ C n such that #C ≤ n , we define the partition Coag(C, C ) by It is straightforward that Coag(C, C ) ∈ C n , as each of its blocks are the union of a consecutive sequence of consecutive blocks. Thus, Coag defines an internal composition law on C ∞ . Moreover for any C, C such that #C ≤ #C and n ≥ 1 The operator Coag is therefore Lipschitz continuous with respect to d and we easily see that it is associative. For any partition C ∈ C n , Coag(C, 0 [n] ) = C and Coag(C, 1 [n] ) = 1 [n] . We are interested in random consecutive partitions such that blocks sizes (#C j , j ≥ 1) are i.i.d. random variables in N ∪ {∞}. We observe that if C and C are two independent random consecutive partitions with i.i.d. block sizes, then hence Coag(C, C ) is a random consecutive partition with blocks coarser than those of C, and with i.i.d. sizes. In view of the very particular form of a consecutive partition, it is legitimate to question whether the framework of partitions is needed. However, the use of the operator Coag enables us to encode easily multiple coalescences and to follow closely the theory of exchangeable coalescents and its terminology. This encoding simplifies the main formulas we obtain when studying the genealogy of a continuous-state branching population.
Definition 4.1. A Markov process (C(t), t ≥ 0) with values in C N is called consecutive coalescent if its semigroup is given as follows: the conditional law of C(t + s) given C(t) = C is the law of Coag(C, C ) where C is some random consecutive partition with i.i.d blocks sizes and whose law may depend on t and s. A consecutive coalescent is said to be homogeneous if C depends only on s and standard if C(0) = 0 [∞] .
We now recall further well-known material on continuous-time Galton-Watson processes. We refer to Athreya and Ney [AN04, Chapter III] for more details on these processes. Consider a finite measure µ on Z + such that µ(1) = 0. A continuous-time Galton-Watson process (Z t (n), t ≥ 0) with reproduction measure µ, is a Markov process counting the number of individuals in a random population with n ancestors where all individuals behave independently, and each individual has an exponential lifetime ζ with parameter µ(Z + ) and begets at its death a random number of children with probability distribution µ/µ(Z + ). The process (Z t (n), t ≥ 0) is characterized in law by µ and thus by the function x ∈ [0, 1].

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The process (Z t (n), t ≥ 0) satisfies the branching property for any n, m ∈ Z + (4.1) where (Z t (m), t ≥ 0) is a continuous-time Galton-Watson process independent of (Z t (n), t ≥ 0), and with the same law as (Z t (m), t ≥ 0). This entails that the generating function of Z t (n) for any t ≥ 0 has the form where for all t ≥ 0, u t (s) is the solution of´s ut(s) dz ψ(z) = t for any t ≥ 0. When µ has no mass at 0, the process is called immortal. Each individual has at least two children and (Z t (n), t ≥ 0) is non-decreasing in time.
With the same procedure as in Definition 1.3, we can represent the family of continuous-time branching processes by considering a flow of random walks (Z s,t (n), t ≥ s, n ≥ 1) satisfying the following properties: (i) for any s ≤ t, (Z s,t (n), n ≥ 0) is a continuous-time random walk whose jump law has support included in N and generating function u t−s .
(ii) For every t 1 < t 2 < ... < t p , the random walks (Z t i ,t i+1 , i < p) are independent and satisfy i.e. We sum up the main straightforward properties of C in the following proposition. (iii) The random variables (#C i (s, t), i ≥ 1) are valued in N and i.i.d.
(v ) The random variable C(s, t) has the same law as C(0, t − s).
The Markov process (C(t), t ≥ 0) defined by C(t) := C(0, t) for any t ≥ 0 is an homogeneous standard consecutive coalescent in the sense of Definition 4.1. Note that by (i), for any s, t ≥ 0, C(t + s) = Coag(C(t), C(t, t + s)) namely for any j ≥ 1, so that consecutive blocks are merging as time runs.
Remark 4.3. One can readily check from (4.2) that for any i ≥ 1, s ≥ 0 and t ≥ 0 This consistency property ensures that the family of jump rates of (C |[n] (t), t ≥ 0) characterizes the law of (C(t), t ≥ 0). In the next lemma, the coagulation rate of a consecutive coalescent restricted to [n] is provided. in , C j out } associated to the first random clock that rings.
Proof. For any n ≥ 1, any t ≥ 0, Moreover, by associativity of the operator Coag, the restricted process (C |[n] (t), t ≥ 0) starting from C, whose number of blocks is m, has the same law as the process (Coag(C, Therefore, we only need to focus on the jump rates of the standard coalescent. For any j, the rate at which the process jumps from 0 [m] to C j out := ({1}, ..., {j, ..., m}) is therefore given by lim s→0+ 1 s P(#C j (t, t + s) ≥ m − j + 1).
Since #C j (t, t + s) has the same law as the random variable Z s (1) where (Z t (1), t ≥ 0) is a continuous-time Galton-Watson process with reproduction measure µ, then the latter limit is µ(m − j + 1).
Similarly, for any k ≤ m − j − 1, the rate at which the process jumps from 0 [m] to C j,k in is There is no simultaneous births forward in time and therefore no simultaneous coalescences.
By letting n and m to ∞ in Lemma 4.4, we see that the coalescences in the consecutive coalescent process C valued in C ∞ can be described in the following way: to each block j of C is associated a family (e j,k , k ≥ 2) of exponential clocks, that ring at rate (µ(k), k ≥ 2). Each time a clock e j,k rings, the consecutive blocks j, j + 1, ... , j + k − 1 coalesce into one. Note that these clocks could also be used to construct the immortal Galton-Watson process forward in time: each time the clock e j,k rings, the jth individual produces k children.
We now take interest in the number of blocks of a consecutive coalescent. Similarly to the continuous-state space, the dual processẐ defined for any n ∈ N and any t ≥ 0 bŷ Z t (n) := min{k ∈ N : Z −t,0 (k) ≥ n}.
The processẐ is a Markov process, and for all n ∈ N,Ẑ t (n) is the ancestor at time −t of the individual n considered at time 0. Remark 4.6. Consecutive coalescents can be defined for a measure µ with a mass at 0 from the relation (4.4). However the process (C(t), t ≥ 0) in this case is inhomogeneous in time. We mention that the process (Ẑ t (n), t ≥ 0) is studied by Li et al. in [LPLG08].

Consecutive coalescents in CSBPs through Poisson sampling
We now explain how consecutive coalescents arise in the study of the backward genealogy of CSBPs. Loosely speaking, exchangeable bridges in the theory of exchangeable coalescents, [BLG03], are replaced by subordinators and the sequence of uniform random variables by the arrival times of a Poisson process with intensity λ. The following typical random consecutive partitions will play a similar role as paintboxes for exchangeable coalescents.
Definition 4.7. We call (λ, φ)-Poisson box a random consecutive partition C obtained by setting where X is a subordinator with Laplace exponent φ and (J j , j ≥ 1) are the ranked atoms of an independent Poisson process with intensity λ.
The (λ, φ)-Poisson boxes will occur as typical random partitions in genealogical trees of CSBPs. More precisely, in the coalescent process describing the genealogy of individuals sampled according to a Poisson point process, the partitions will be distributed as (λ, φ)-Poisson boxes. The following Lemma is proved in Appendix A.2, and can be thought of as a revisiting of Pitman's discretization of subordinators [Pit97]. and (J k , k ≥ 1) the arrival times of an independent Poisson process with intensity λ. Let C be the (λ, φ)-Poisson box constructed with X and (J k , k ≥ 1) and set for any i ≥ 1, J i := X −1 (J k ) for k ∈ C i . Then (i) C is a random consecutive partition with i.i.d blocks sizes and for any k ≥ 1 for all s ∈ [0, 1].
(ii) The sequence (J i , i ≥ 1) are the arrival times of a Poisson process of intensity φ(λ).
(iii) (J i , i ≥ 1) and C are independent.
Remark 4.9. We shall also consider killed subordinators with a Laplace exponent that satisfies φ(0) = κ > 0, or equivalently ({∞}) = κ. The above Lemma can be extended to this case. See Corollary A.4. The associated (λ, φ)-Poisson box has finitely many blocks. Formulas in (i) hold true and additionally each block has probability φ(0)/φ(λ) to be infinite. The sequence (J i , 1 ≤ i ≤ #C) forms the first arrival times of a Poisson process with intensity φ(λ).
We now construct consecutive coalescent processes related to the genealogy of the flow of subordinators (X s,t (x), s ≤ t, x ≥ 0). Denote by (J λ i , i ≥ 1) the sequence of arrival times of an independent Poisson process with intensity λ. For any t ≥ 0, we define C λ (t) as The next theorem describes the law of the process (C λ (t), t ≥ 0).

(4.6)
There is no simultaneous coalescences and for any k ≥ 2, the rate at time t at which k given consecutive blocks coalesce is In the supercritical case, by choosing for intensity λ = ρ, the process (C ρ (t), t ≥ 0) becomes time-homogeneous. Corollary 4.11 is obtained by a direct application of Theorem 4.10 since v t (ρ) = ρ for any t ≥ 0.
Corollary 4.11. Assume Ψ supercritical and take λ = ρ. The coalescent process (C ρ (t), t ≥ 0) is homogeneous in time and the coagulation rate of k given consecutive blocks is Remark 4.12. Bertoin et al. [BFM08] have shown that in any flow of supercritical CSBPs one can embedd an immortal continuous-time Galton-Watson process counting the so-called prolific individuals, whose lines of descent are infinite. The prolific individuals are located in R + as the arrival times of a Poisson process with intensity ρ at any time. Moreover, this continuous-time Galton-Watson process has reproduction measure µ ρ . The consecutive coalescent (C ρ (t), t ≥ 0) represents its genealogy backward in time.
We prove Theorem 4.10. We stress that by definition, from (4.5), C λ (t) is a (λ, v t )-Poisson box and C λ (0) = 0 [∞] sinceX 0 = Id. Our first lemma proves that the partition-valued process (C λ (t), t ≥ 0) is Markovian in its own filtration and is a consecutive coalescent (possibly inhomogeneous in time) in the sense of Definition 4.1.
Lemma 4.13. For any s, t ≥ 0 Proof. For any s, t ≥ 0 and all l ≥ 1, set J λ l (t) :=X t (J λ i ) for all i ∈ C λ l (t). Let C λ (t, t + s) the random consecutive partition defined by l C λ (t,t+s) ∼ k if and only ifX t,t+s (J λ l (t)) =X t,t+s (J λ k (t)). Then by the key lemma 4.8- Recall the cocycle propertyX t+s =X t,t+s •X t (Theorem 2.5-i)). Let i, j ∈ N. Set k and l such that i ∈ C λ k (t) and j ∈ C λ l (t). By the cocycle property,X t,t+s (J λ k (t)) =X t,t+s (J λ l (t)) holds if and only if i The generating function of the block's size at time t, given in (4.6), is obtained by a direct application of the Key lemma 4.8 since C λ (t) is a (λ, v t )-Poisson box. We now show that the semigroup satisfies the Feller property.
Lemma 4.14. The process (C λ (t), t ≥ 0) is Feller and for any t ≥ 0, C λ (t, t + s) −→ Proof. The Feller property corresponds to the continuity of the map for any continuous function ϕ from C ∞ to R + . This is clear since Coag is Lipschitz continuous. We now show the weak continuity of the semigroup. By definition J λ i (t) =X t (J k ) for any k ∈ C λ (t) and for any i = j, J λ i (t) = J λ j (t). By Lemma 2.3-(ii) and independence between (J i , i ≥ 1) andX, we see that (J λ i (t), i ≥ 1) is independent ofX t,t+s . By Lemma 2.3-(iv), sincê X t,t+s (x) −→ s→0 x uniformly on compact sets, in probability, then for any n, We now seek for the coagulation rate (4.7).
Lemma 4.15. For any z ∈ (0, 1), Proof. Let z ∈ (0, 1), since by Lemma 4.8, the random variables (J λ l (t), l ≥ 1) are the arrival times of an independent Poisson process with intensity v t (λ), then . (4.9) Thus The latter convergence holds since By letting θ = v t (λ) in the next technical lemma, we see that for any t ≥ 0, the measures µ λ t on N defined in (4.7) have generating function ϕ λ t .
We now explain how coalescences take place in the process (C λ (t), t ≥ 0). By construction, the laws of (C λ |[n] (t), t ≥ 0) for n ≥ 1 are consistent and as in Lemma 4.4 the family of jump rates of (C λ |[n] (t), t ≥ 0) characterizes the law of (C λ (t), t ≥ 0). The following lemma is obtained along the same lines as Lemma 4.4 but in an inhomogeneous time setting. We provide now some basic properties of the consecutive coalescent (C λ (t), t ≥ 0).
Proof. Recall that for any The process (#C λ 1 (t), t ≥ 0) is non-decreasing and thus converges almost-surely in N. Therefore By monotonicity, #C λ 1 (t) −→ t→∞ #C λ 1 (∞) a.s. Therefore for large enough time t 1 , for t ≥ t 1 , C λ 1 (t) = C λ 1 (∞). Since there is no coalescence between blocks C λ 1 and C λ 2 after time t 1 , the process (#C λ 2 (t), t ≥ t 1 ) is non-decreasing and converges almost-surely towards #C λ 2 (∞). By induction, for any n 0 , there exists t n 0 such that for any t ≥ t n 0 , #C λ The following proposition is a direct consequence of the strong law of large numbers. vt(λ) . This represents the proportion of ancestors that have not been involved in coalescences by time t.

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We have seen in Theorem 1.A-(iv) and Proposition 2.6 that when´0 dx |Ψ(x)| < ∞, the CSBP explodes and ∞ is an entrance boundary of (X t , t ≥ 0). Recall that for any t > 0,X t (∞) is the first individual from generation t to have an infinite progeny at time 0.
Proposition 4.20 (Coming down from infinity). For any t > 0, #C λ (t) < ∞ a.s if and only if where e 1/λ is an exponential random variable with parameter 1/λ.

Backward genealogy of the whole population
In the previous section, we have defined some coalescent processes arising from sampling initial individuals along a Poisson process with an arbitrary intensity λ. The consecutive coalescents obtained by this procedure are only approximating the backward genealogy. They give the genealogy of a random sample of the population. The objective of this subsection is to observe that when the Grey condition holds, one can define a consecutive coalescent matching with the complete genealogy of the population from any positive time. In all this section, assume the Grey's conditionˆ∞ dx Ψ(x) < ∞.
Heuristically, we make λ → ∞ in Theorem 4.10, to study the genealogy of the whole population. The limiting process would indeed characterize the genealogy of the CSBP as in this case, an everywhere dense sub-population would be sampled and its genealogy given, which is enough to deduce the genealogical relationship between any pair of individuals. However, this method cannot work directly as one would have jump rates that may explode.
Fix a time s > 0. The subordinator (X −s,0 (x), x ≥ 0) is a compound Poisson process with Lévy measure s (dx) independent of (X −t,−s (x), x ≥ 0, t ≥ s). Let (J vs(∞) i , i ≥ 1) be the jump times of (X −s,0 (x), x ≥ 0). They are arrival times of a Poisson process with intensity v s (∞) = s ((0, ∞)), independent of (X s,t , t ≥ s). Consider (C(s, t), t ≥ s) the partition-valued process defined by The process (C(s, t), t > s) provides a dynamical description of the genealogy of initial individuals whose most recent common ancestors are found at time s > 0. The following theorem is a direct application of Theorem 4.10.
Theorem 4.21. For any s > 0, the flow of random consecutive partitions (C(u, t), t ≥ u ≥ s) satisfies for any t ≥ u ≥ s, C(s, t) = Coag(C(s, u), C(u, t)) a.s (4.10) where C(u, t) is a Poisson box with parameters (v u (∞), v t−u ) independent of C(s, u). Moreover for any i ≥ 1, t ≥ s and z ∈ (0, 1), and the consecutive coalescent (C(s, t), t > s) has coagulation rates (µ ∞ t , t > s) with We see in the next corollary that by reversing time in any block of the consecutive coalescent (4.12) Moreover, denoting by γ T i , the time of its first jump, one has for any t ∈ [0, T [ .
Proof. The law of Z T i (t) for fixed t is obtained by a direct application of Theorem 4.21. Only remains to show the branching property. By (4.10) for any s and t such that 0 ≤ t + s < T Remark 4.23. The process (Z T 1 (t), 0 ≤ t < T ) corresponds to the reduced Galton-Watson process obtained by Duquesne and Le Gall [DLG02, Theorem 2.7.1] in the (sub)critical case. We refer also to Fekete et al. [FFK17] for an approach with stochastic differential equations. In the supercritical case, since for any t ≥ 0, v T −t (∞) −→ T →∞ ρ, we see in (4.12) that (Z T 1 (t), t ≥ 0) converges, as T goes to infinity, in the finite-dimensional sense, towards a Markov process (Z ∞ (t), t ≥ 0) whose semigroup satisfies for any z ∈ (0, 1) Namely, (Z ∞ (t), t ≥ 0) is a continuous-time Galton-Watson process, homogeneous in time, with reproduction measure µ ρ . Heuristically, individuals from time −t with descendants at time T will correspond at the limit with prolific individuals.
The coalescent process (C(s, t), t ≥ s) only describes coalescence in families from time s > 0. We define now a coalescent process from time 0 by using the flow of subordinators. Denote by C R + the space of partitions of ]0, ∞[ into consecutive half-closed intervals. That is to say, partitions of the form C = (]0, x 1 ], ]x 1 , x 2 ], ...) for some non-decreasing sequence of positive real numbers (x i , i ≥ 1). The space of consecutive partitions of N, (C ∞ , Coag), acts as follows on C R + : for any C ∈ C R + and C ∈ C ∞ , for any i ≥ 1 where C j =]x j−1 , x j ] and x 0 = 0. The following theorem achieves one of our goals and has to be compared with our preliminary observation in Proposition 2.1. It describes completely the genealogy backwards in time as well as the sizes of asymptotic families.
Theorem 4.24. Define the process (C (t), t > 0) valued in C R + as follows: The process (C (t), t > 0) is a time-inhomogeneous Markov process such that for any t ≥ s > 0, C(s, t)) a.s.
In the critical or supercritical case, a.s and the length of a typical interval at the limit has for law the quasi-stationary distribution of the CSBP conditioned on the non-extinction: Proof. Recall that X −t,0 = X −s,0 • X −t,−s andX t =X s,t •X s . This entails that for any (4.13) For any t > 0 and any j ≥ 1, set C j (t) = (x j−1 (t), ] with x 0 (t) = 0. By definition of C(s, t) and (4.13), we have Proposition 4.18 ensures that (C(s, t), t ≥ s) converges almost-surely as t goes to ∞. This entails the almost-sure convergence of C (t), t > 0). In the supercritical or critical case, C(s, t) −→ Note moreover that C (∞) = Coag(C (s), C(s, ∞)) for any s > 0.
The process (C (t), t > 0) sheds some light on this deterministic equation since µ ∞ t (k) is the rate in (C (t), t > 0) at which k given intervals coagulate and by the strong law of large numbers, a.s for any t > 0, where |C i (t)| denotes the length of the i th interval in C (t). We refer the reader to Iyer et al. [ILP15], [ILP18] for recent works on Equation 4.14.
When the CSBP explodes, the individuals in the current generation have finitely many ancestors. The following proposition is the analogue of Proposition 4.20.
Proof. For any t > 0, the lengths of the intervals in C (t) are i.i.d random variable with law t(dx) vt(∞) . Under the assumption,´0 du |Ψ(u)| < ∞, t ({∞}) = v t (0) > 0 and therefore the number of intervals in C (t) has a geometric law with parameter vt(0) vt(∞) . The convergence in law is proved by a similar calculation as in Proposition 4.20.
We saw in Proposition 4.5 that the number of blocks in (C |[n] (t), t ≥ 0), the coalescent process associated to a continuous-time Galton-Watson process, corresponds to the inverse flow of random walks (Ẑ t (n), t ≥ 0) at a fixed level n. Recall that in continuous-state space the process (X t (x), t ≥ 0) can then be interpreted as the size of the ancestral population whose descendants at time 0 form a family of size x. The study is more involved than in the discrete setting and is the aim of the next section.

A martingale problem for the inverse flow
We investigate the infinitesimal dynamics of (X t , t ≥ 0) through its extended generatorL. Recall that we write L the generator of the CSBP with mechanism Ψ. As we consider the flow of subordinators over [0, ∞], it is natural to express L as follows for all C 2 bounded function G: LG Then for any y > 0, (X t (y), t ≥ 0) solves the following well-posed martingale problem F (X s (y))ds, t ≥ 0 is a martingale for any function F in D : Remark 5.2. Note that ψ h,u = ∆ −1 h,u is the right-continuous inverse function of ∆ h,u . For any y ≥ 0, if individual u makes at time t a progeny of size h, then ψ h,u (y) at time t− is the infinitesimal parent of individual y at time t: if y < u, then y has no parent but himself, if y ∈ [u, u + h], the parent of y is ψ h,u (y) = u, if y > u + h then its parent is ψ h,u (y) = y − h. If y 1 = y 2 then ψ h,u (y 1 ) = ψ h,u (y 2 ) if and only if y 1 , y 2 ∈ [u, u + h]. Proof. Let F ∈ C 2 b , for any y > 0 ˆ∞ where π(x) :=´∞ x π(du) for any x > 0. For any y > 0, u > 0 and h > 0, we have Therefore, if h < 1, we have that On the one hand, On the other hand Thus, for any h ≥ 0 with a certain constant C independent of y and h. The integral with respect to π(dh) inLF is therefore convergent andL well-defined.
We now follow the same method as Bertoin and Le Gall in [BLG05, Theorem 5] to show that L is an extended generator, i.e. that F (X t (u)) −´t 0L F (X s (u))ds, t ≥ 0 is a martingale for all F in D. Let g be a continuous function over [0, ∞[ and f a function in C 2 0 . Set G(t) =´t 0 g(u)du and F (t) =´∞ t f (x)dx. Note that Moreover, one classically has that Recall that by (2.5), we have P(X s (u) < x) = P(X s (x) > u) for all x, u ≥ 0. Then, integrating this equality with respect to f (x)g(u)dxdu provideŝ Therefore, a first step in the search forL is computing the right-hand side of (5.3).
Lemma 5.4. Let λ > 0 and g(x) = e −λx for any x ∈ R + then for any is in the domain of the generator L of the CSBP (X t , t ≥ 0) (defined in (5.1)) and therefore where we write for all twice derivable function H, which denote respectively the continuous and discontinuous parts of the generator L. We start by studying the discontinuous part. For any s ≥ 0, we can rewritê We first compute By Lemma A.1(ii) and Remark 5.2, one has ∆ h,u (x) > y if and only if x > ψ h,u (y), therefore Integrating with respect to f (x)dx we obtain (5.6) We now compute the compensated part of the discontinuous generator L d , by integration by part we havê (5.7) Moreover, we observe that Therefore, as P t G(∞) = G(∞), (5.7) becomeŝ Using the above result and (5.6), (5.5) yieldŝ Above, (5.8) and (5.9) follow from applying Fubini's theorem, which we now justify. For any t and x, Since v t (λ)e −xvt(λ) ≤ 1 x then sup [x,x+h] |(P t G) (z)| ≤ vt(λ) λx , and by Taylor's inequality Since f is integrable then the upper bound is integrable with respect to f (x)dx1 [0,s] (t)dtπ(dh)du, which justifies the application of Fubini's theorem in (5.8).
We now explain why (5.9) holds. Recall first that f (z) = −F (z). By Theorem 2.5, for any x. This, with the bound (5.2) allows us to conclude that Υ(h, u, t, v) is integrable with respect to In a second time, we deal with the continuous part of the generator L c . Applying Fubini's theorem, one hasˆ∞ Since by assumption F ∈ D, then lim x→∞ φ(x) = φ(∞) = 0. We now compute´∞ 0 dxf (x)L c P t G(x). By two integration by partŝ We can now conclude as follows. One haŝ Lemma 5.5. For any F ∈ D and any y ≥ 0, F (X t (y)) −ˆt 0L F (X s (y))ds, t ≥ 0 is a martingale.
Proof. Recall that g(v) = e −λv . We will show that (5.10) entails that for any v and any s: From the Feller property of (X t , t ≥ 0) and the continuity ofLF for any function F in D, the map v → E[LF (X t (v))] is continuous. For any a > 0, and any v ≤ a,X t (v) ≤X t (a) a.s. therefore using the bound (5.2), we see that the function defined on [0, a] by is continuous, and since a is arbitrary, the mapping is continuous on [0, ∞[. This corresponds to the continuity of v →´s 0 dtE L d F (X t (v)) . On the other hand, one can check the continuity of v →´s 0 dtE L c F (X t (v)) and by injectivity of the Laplace transform, (5.10) entails (5.11). This provides the martingale problem, as the following routine calculation shows. Let t ≥ 0 and s ≥ 0. Denote by (F s , s ≥ 0) the natural filtration associated to (X s (x), s ≥ 0, x ≥ 0), In the following Lemma, we rewrite the generatorL of the one-point motion in its Courrège form. We refer to Kolokoltsov [Kol11] for a general study of generators of stochastically monotone Markov processes.
Lemma 5.6. For any f ∈ C 2 b , Moreover, the martingale problem (MP) is well-posed.

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For the second integral II: Summing both expressions, I + II equals to: and we obtain We now verify uniqueness of the solution to (MP) by applying Theorem 5.1 of Kolokoltsov [Kol11]. Assumptions (i) and (ii) of the theorem can be readily checked. The third assumption (iii) is that for any where for the first inequality we use the fact that −z´z 1 hπ(dh) +´z 1 h 2 π(dh) ≤ 0 and we choose a large enough c for the second inequality.
Proof of Theorem 5.1. It follows directly by combination of Lemmas 5.5 and 5.6.

Examples
In this section, we apply the results obtained in the previous ones to the two following important examples: the stable CSBP and the Neveu CSBP. These CSBPs arise in many different frameworks and are known for instance to be closely related to the class of exchangeable coalescents called Beta-coalescents.

Feller and stable CSBPs
A stable CSBP is a continuous-state branching process with branching mechanism given by Ψ : u → c α u α − βu, for some α ∈ (1, 2], c α > 0 and β ∈ R. Note in particular that the Feller flow, whose inverse flow was studied in details in Section 3 is a stable CSBP with α = 2. As a direct application of Theorem 4.24, we obtain that the Markovian coalescent (C (t), t > 0) associated to the Feller flow has coagulation rates δ 2 (k).
In particular, in the subcritical case (β < 0), C (t) converges almost-surely as t → ∞ towards intervals with i.i.d. exponentially distributed lengths with parameterρ = 2β/c 2 . This corresponds to the partition of R + into random intervals (]0, x 1 [, ]x 1 , x 2 [, ...) corresponding to different ancestors at time −∞ found in Section 3. We now assume that Ψ(u) = c α u α − βu for some α ∈ (1, 2), with c α := Γ(2−α) α(α−1) (which corresponds to a simple time dilatation). By assumption α > 1 and Grey's condition holdś ∞ du Ψ(u) < ∞. Solving the differential equation (1.3) satisfied by v t (λ), we have in particular that For the stable CSBP, the coagulation rates of its associated Markovian coalescent (C (t), t > 0) are given by . The normalized associated probability measure is which is time-independent. This probability distribution corresponds to the reproduction measure of prolific individuals in supercritical stable CSBP, see Example 3 in [BFM08]. It also appears in the study of reduced α-stable trees and Beta(2 − α, α)-exchangeable coalescents, see respectively Duquesne and Le Gall , t), t > s) representing the genealogy of any stable CSBP from time s > 0 is obtained by a deterministic time-change of the homogeneous consecutive coalescent (Č(t), t ≥ 0) with coagulation rates µ α via the transformation: for any t ≥ s, According to Theorem 4.24, in the subcritical case (C (t), t > 0) converges almost-surely as t → ∞ towards a partition of intervals with i.i.d. lengths with law ν α such that We now turn to the martingale problem satisfied by the inverse flow of the stable CSBP. One easily computes the drift and the jump measure from Remark 5.7.
Proposition 6.1. The process (X t , t ≥ 0) is characterized by the martingale problem associated toL, acting on C 2 b , given in Lemma 5.6, with In the critical case, one can identify the law ofX through some random-time change.
Proposition 6.2. If β = 0, the process (X t , t ≥ 0) is a positive self-similar Markov process with index a := α − 1. Namely for any k > 0 and any y > 0, where ϕ x (t) := inf{s > 0;´s 0 e (α−1)Lu du > t} and L is a spectrally negative Lévy process started from log x with Laplace exponent with ν α (dz) = e z (1 − e z ) −1−α + 1 α (1 − e z ) −α e z dz and d α = Proof. Recall that the critical CSBP (X t , t ≥ 0) is itself selfsimilar with index a := α − 1. See for instance Kyprianou and Pardo [KP08]. For any k > 0 and any x > 0, kX k −a t (x) L = X t (kx). Thus for any y > 0 By Lamperti's representation of positive self-similar Markov process, see e.g. [Kyp14, Chapter 13],X t (x) is of the form exp(L ϕ x (t) ) for some Lévy process L where t → ϕ x (t) the timechange given in the statement. To identify the Laplace exponent κ of L, note that by (α − 1)self-similarity, one has κ(q) = x −q+α−1L p q (x) with p q (x) = x q . The result follows from simple computations.

Neveu CSBP
We now turn in this section to the Neveu CSBP. This CSBP has branching mechanism Ψ(q) = q log(q). Recall its Lévy-Khintchine form where γ =´∞ 1 e −y y −2 dy is the Euler-Mascheroni constant. Note that Grey's condition is not verified by this process. Solving the differential equation (1.3) yields v t (λ) = λ e −t for any t ≥ 0 and λ ∈ (0, ∞). For any fixed t, the subordinator (X t (x), x ≥ 0) is stable with parameter e −t . For Neveu CSBP, the consecutive coalescent process C λ defined in Section 4 happens to be homogeneous in time, and not to depend on λ.
Proof. By Theorem 4.10, and applying the change of variable u = v t (λ)x, we see that for any k ≥ 2, the other statements can be obtained by a direct application of Theorem 4.10. See also the calculations around Lemma 7 in Pitman [Pit97].
Lemma 6.4. Consider a consecutive coalescent (C(t), t ≥ 0) with coagulation rate µ(k) = 1 k(k−1) for any k ≥ 2 then, as n goes to ∞ in finite-dimensional sense in time and in the Skorokhod topology in x.

A.1 Right-continuous inverse
In this section, we recall and compile some elementary properties on right continuous inverse of càdlàg non-decreasing functions. As multiple competing definitions of generalized inverse coexist, it can be challenging to find a single reference for the results we need. Therefore we give a short proof of these well-known facts, in order to be self-contained. Let f be a càdlàg non-decreasing function on R + , we denote by its right continuous inverse.
(i) The function f −1 is non-decreasing and càdlàg.
Remark A.2. Dini's theorems imply that both convergences in (iv) hold uniformly on compact sets.
Let x, y ≥ 0, we first assume that f −1 (y) < x. Then by definition of f −1 , there exists u ∈ [f −1 (y), x) such that f (u) > y. As f is non-decreasing, we deduce that f (x) ≥ f (u) > y.
We now assume that f −1 (y) ≤ x. As f is non-decreasing, we observe that f (x) ≥ f (f −1 (y)). Therefore, the only thing left to prove is that ∀y ≥ 0, f (f −1 (y)) ≥ y (A.2) We write z = f −1 (y). By definition of f −1 (y), for all > 0, there exists u < z + such that f (u) > y. Then, as f is right-continuous, we have f (z) = inf u>z f (u), thus for all η > 0, there exists > 0 such that if u < z + , then f (u) < f (z) + η. As a result, for all η > 0, there exists u < z + such that y < f (u) < f (z) + η. This inequality being true for all η > 0, we therefore conclude that f (z) ≥ y, completing the proof (A.2). We thus deduce that f (x) ≥ y, completing the proof of (ii).
We finally prove the last point. Let (f n ) be a sequence of non-decreasing càdlàg functions such that lim n→∞ f n = Id pointwise. We prove that for all y ≥ 0, lim n→∞ f −1 n (y) = y. Let > 0, by point (ii), we have that f −1 n (y) < y + ⇐⇒ f n (y + ) > y.
As lim n→∞ f n (y + ) = y + , we conclude that for all n large enough, f −1 n (y) < y + . Similarly, we also have f −1 n (y) ≥ y − ⇐⇒ f n (y − ) ≥ y. therefore f −1 n (y) ≥ y − for all n large enough by pointwise convergence of f n at point y − . This concludes the proof of (iv).

A.2 Discretization of subordinators
In this section, we introduce the key lemma allowing to study the genealogical structure of CSBPs. Namely, we prove that the pullback measure of a Poisson process by a subordinator is a Poisson process decorated by i.i.d. integer-valued random variables. Lemma A.3. Let λ ≥ 0 and (X(x), x ≥ 0) be a subordinator with Lévy-Khinchine exponent φ : µ → dµ +ˆ 1 − e −µx (dx).
We denote by N an independent Poisson point process with intensity λ, and we write (J j , j ≥ 1) the positions of the atoms of N , ranked in the decreasing order. Then, setting M = ∞ j=1 δ X −1 (J j ) the image measure of N by X −1 , we have where (J j , j ≥ 1) are the atoms of a Poisson point process with intensity φ(λ) and (Z j , j ≥ 1) are i.i.d. random variables, independent of (J j , j ≥ 1), such that i.e. E(z Z 1 ) = 1 − φ(λ(1−z)) φ(λ) for all z ∈ [0, 1].
Proof. The proof is based on a joint construction by the same "master" Poisson point process of the subordinator X and the Poisson point process N , in such a way that M becomes a simple functional of that master point process. To see why such a construction is possible, we write with d ≥ 0 the drift and the Lévy measure of X on R + . By the Lévy-Itô décomposition, one can write ∀t ≥ 0, with (t, x t ) t≥0 being the atoms of a Poisson point process with intensity dt ⊗ (dx). The proof being slightly simpler for d = 0, we focus here on the case d > 0.
Recalling that D denote the set of càdlàg non-decreasing functions on R + , we introduce the point process R = i∈I δ (t i ,x i ,N (i) ) on R + × R + × D with intensity dt ⊗ dx ⊗ P λ (dN ), with P λ begin the law of a Poisson point process with intensity λ on R + . We also set N (0) an independent Poisson point process with intensity λ. We then define which is a subordinator with Lévy-Khinchine exponent φ. Then, denoting (J (i) j , j ≥ 1) the atoms of the Poisson point process N (i) , we set Heuristically, the point process N can be thought of as follows: R + is divided in intervals ∪ i∈I [X t i − , X t i ] corresponding to jumps in the subordinator X and the remaining space I corresponding to points with an antecedent by X. Atoms are added to the interval [X t i − , X t i ] according to the point process N (i) , and to the set I with the point process N 0 . It should then be heuristically clear that N is a Poisson point process with intensity λ independent of X. To verify it, we compute its conditional Laplace transform against a smooth locally compact test function f . By construction, (N (i) , i ∈ I ∪ {0}) are i.i.d. Poisson point process with intensity λ, which are further independent from X, thus by change of variable, using that X t = d at all continuity points t of X.

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As a result, the couple (X, N ) has same law as (X, N ) given in the lemma. Moreover, we have immediately by construction that and computing the law of that point process is straightforward by the definition of R. Indeed, by independence, for any continuous function f with compact support, we have Then, using Campbell's formula, we have both But as under law P λ , N ([0, x]) is a Poisson random variable with parameter λx, the last inequality can be written, by Fubini theorem We deduce that the Laplace transform of M is the same as the one of M given in the lemma, which concludes the proof.
This result can be straightforwardly extended to killed subordinators, by constructing it as a limit of non-killed subordinators. For the sake of completeness, we add a proof of the following result.
With the same notation as in the previous lemma, we have M = ∞ j=1 Z j δ J j , where (J j , j ≥ 1) are the atoms of a Poisson point process with intensity φ(λ), (Z j , j ≥ 1) are i.i.d. random variables, independent of (J j , j ≥ 1), such that and Proof. Let Y be a subordinator with Laplace exponent λ → dλ +´1 − e −λx (dx), and R an independent Poisson process with intensity κ. Observe that for all r > 0, the process defined by Y r (t) = Y (t) + rR(t), t ≥ 0, is a Lévy process, and that X = lim r→∞ Y r is a Lévy process with Laplace exponent φ. We set (J j , j ≥ 1), (J r j , j ≥ 1) and (Z r j , j ≥ 1) the quantities obtained by applying Lemma A.3, and T = inf{t > 0 : R t = 1}.
Observe that for all j such that J r j < T , the quantities J r j and Z r j do not depend on r. On the contrary, for all j such that J j > T , as r → ∞, all value (Y r ) −1 (J j ) converge toward T , and the associated value of Z r to the atom at position T converges toward ∞.
Explicit formulas for the law of Z ∞ are straightforward Poisson computations.