Continuous-state branching processes with competition: duality and reflection at Infinity

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $\lambda z$ for some $\lambda>0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty$ is accessible and $0\leq \frac{2\lambda}{c}<1$, the extended process is reflected at $\infty$. In the case $\frac{2\lambda}{c}\geq 1$, $\infty$ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty$ get extinct almost-surely. Moreover absorption at $0$ is almost-sure if and only if Grey's condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.


Introduction
Continuous-state branching processes (CSBPs for short) have been defined by Jiřina [Jiř58] and Lamperti [Lam67a] for modelling the size of a random continuous population whose individuals reproduce and die independently with the same law. Lamperti [Lam67b] and Grimvall [Gri74] have shown that these processes arise as scaling limits of Galton-Watson Markov chains. Their laws are characterised in terms of a Lévy-Khintchine function Ψ (called a branching mechanism). A shortcoming of CSBPs for modelling population lies in their degenerate longterm behavior. In the long run, a CSBP either tends to 0 or to ∞. On the event of extinction, the process can decay indefinitely or be absorbed at 0 in finite time. Similarly, on the event of non-extinction, the CSBP can grow indefinitely or be absorbed at ∞ in finite time. The latter event is called explosion and occur typically when the process performs infinitely many large jumps in a finite time with positive probability. Since the sixties, several generalizations of CSBPs have been defined to overcome various unrealistic properties of pure branching processes. Lambert [Lam05] has introduced a generalization of these processes by incorporating pairwise interactions between individuals. These processes, called logistic continuous-state branching processes, are the random analogues of the logistic equation where, informally speaking, the Malthusian growth γz t dt is replaced by the full dynamics of a continuous-state branching process. For instance, when the mechanism Ψ of the CSBP reduces to Ψ(z) = σ 2 2 z 2 − γz, the process (Z t , t ≥ 0) is the logistic Feller diffusion The negative quadratic drift represents additional deaths occurring due to pairwise fights among individuals. Intuitively, these fights can be interpreted as competition (for resources for instance). We refer to Le, Pardoux and Wakolbinger [LPW13] and Berestycki, Fittipaldi and Fontbona [BFF17] for a study of the competition at the level of the genealogy. In a logistic CSBP, individuals and their progenies are not independent between each other, and the branching property, from which all properties of CSBPs can be deduced, is lost. One of the main interest of logistic CSBPs is to provide a model of population with a possible self-limiting growth. The objective of this article is to study these processes with most general mechanisms and to understand precisely how the competition regulates the growth. We shall study the nature of the boundaries 0 (extinction of the population) and ∞ (explosion of the population).
Throughout this article, we follow the terminology of Feller, introduced in [Fel54], for classifying boundaries of a diffusion (see Section 2 for their meaning). The state of the art is as follows. In the continuous case (2), Feller tests provide that ∞ is an entrance and 0 an exit. For a general mechanism Ψ, the logistic CSBP has (typically unbounded) positive jumps and such general tests do not exist. Lambert in [Lam05] has found a set of sufficient conditions over the mechanism Ψ for ∞ to be an entrance boundary. Under these conditions, it is also shown in [Lam05] that the competition alone has no impact on the extinction of the population. In other words, the boundary 0 is an exit if and only if the branching mechanism Ψ satisfies Grey's condition. The entrance property from ∞ coincides with the notion of coming down from infinity observed in many stochastic models. We refer for instance to Cattiaux et al. [CCL + 09] and Li [Li18] for recent related works and to Donnelly [Don91] for a classical result in coalescent theory. We shall follow a different route than [Lam05] and study directly the semigroup of logistic CSBPs. A rather surprising first phenomenon is that the competition does not always prevent explosion. Some reproduction laws have large enough tails for ∞ to be accessible (meaning for explosion to occur). We provide a necessary and sufficient condition for ∞ to be inaccessible and show that under this condition the boundary ∞ is an entrance. Since the competition pressure increases with the size of the population, one may wonder if some compensation occurs near the boundary ∞ for a general mechanism Ψ. The main contribution of this article is to answer the following question. Is it possible for a logistic continuous-state branching process to leave and return to ∞ in finite time? We shall indeed see that the reproduction can be strong enough for explosion to occur and the quadratic competition strong enough to instantaneously push the population size back into [0, ∞) after explosion. This phenomenon is captured by the notion of regular instantaneous reflecting boundary. By reflecting, we mean here that the Lebesgue measure of {t ≥ 0, Z t = ∞} is zero almost-surely. The boundary is instantaneous in the sense that starting from ∞ the process enters immediately (0, ∞). Only in some cases, for which explosion is made by a single jump, the boundary ∞ is an exit. We stress also that it may well occur that the population goes extinct after exploding, so that ∞ is not always recurrent.
In order to classify the boundaries as explained above, we need to define an extension of the minimal process in [0, ∞]. This requires in general a deep study of the minimal semi-group. However, since processes with competition do not satisfy the branching property, most arguments for CSBPs are not applicable. The resolvent of logistic CSBPs is rather involved and we will not discuss all possibilities of extensions in this article but only construct a natural one by approximation. We first establish a duality relationship between non-explosive logistic continuous-state branching processes and some generalizations of the logistic Feller diffusion process (2). Namely we will show that when ∞ is inaccessible for the process (Z t , t ≥ 0), then for any We shall see that the condition for ∞ to be inaccessible for (Z t , t ≥ 0) is precisely given by Feller's test for 0 to be an exit of (U t , t ≥ 0). We stress that Equation (3) has not always a unique solution as 0 can be regular for certain non-lipschitz mechanisms Ψ. It is precisely for such mechanisms that ∞ will be regular for logistic CSBPs. Heuristically, if ( ) holds for some processes (Z t , t ≥ 0) and (U t , t ≥ 0), then the entrance boundaries of (Z t , t ≥ 0) will be classified in terms of exit boundaries of (U t , t ≥ 0). We refer to Cox and Rösler [CR84] and Liggett [Lig05] for a study of boundaries by duality of semi-groups. The identity ( ) provides a representation of the semi-group of any non-explosive process with competition and will allow us to construct an extended process over [0, ∞] with ∞ reflecting as a limit of non-explosive processes. We highlight that this construction is different from the classical Itô's concatenation procedure for building recurrent extensions. In particular, our approach is not based on a measure theoretical description of the excursions from ∞ but on a direct description of the extended semi-group.
A very similar phenomenon of reflection at ∞ has been recently observed by Kyprianou et al. [KPRS17] for a certain exchangeable fragmentation-coalescence process. We shall observe the same phase transition between the reflecting boundary case and the exit boundary one. Lambert [Lam05] and Berestycki [Ber04] have noticed that discrete logistic branching processes share many properties with the number of fragments in some exchangeable coalescence-fragmentation processes. Discrete logistic branching processes are interesting in their own and will be studied elsewhere. We highlight that contrary to the process studied in [KPRS17], a logistic CSBP can reach ∞ by accumulation of large jumps over a finite interval of time. We mention that the duality ( ) has been observed in a spatial context for the branching mechanism Ψ(u) = σ 2 2 u 2 − γu by Horridge and Tribe [HT04] for the logistic SPDE, see also Hutzenthaler and Wakolbinger [HW07]. Lastly, other competition mechanisms than the quadratic drift have been studied. We refer for instance to the monograph of Pardoux [Par16] and Ba and Pardoux [BP15] for some generalisations of the logistic Feller diffusions. It is worth noticing that the relation ( ) does not hold for general competition mechanims.
The paper is organised as follows. In Section 2, we recall some known facts about CSBPs and define minimal logistic CSBPs through a martingale problem. We state our main results in Section 3 and describe some examples. In Section 4, we show how to solve the martingale problem by time-changing an Ornstein-Uhlenbeck type process. Some first properties of the minimal process, such as a criterion for its explosion, are derived from this time-change. In Section 5, we gather the possible behaviors of the diffusion (U t , t ≥ 0) at its boundaries. Then we establish the duality under the non-explosion assumption and deduce the entrance property. In Section 6, we define and study an extension of the minimal process. Lastly, in Appendix, we provide the calculations needed for classifying the boundaries of (U t , t ≥ 0) according to Ψ and the parameter c.

Preliminaries
As we will use Feller's terminology repeatedly, we briefly recall how to classify boundaries. Consider a process valued in an interval (a, b) with a, b ∈R and a < b, -the boundary b is said to be accessible if there is a positive probability that it will be reached in finite time (the process can enter into b). If b is accessible, then either the process cannot get out from b and the boundary b is said to be an exit or the process can get out from b and the boundary b is called a regular boundary. -If the boundary b is inaccessible, then either the process cannot get out from b, and the boundary b is said to be natural or the process can get out from b and the boundary b is said to be an entrance.
Notation. We denote by [0, ∞] the extended half-line and by For any interval I ⊂ R, we denote by C 2 c (I) the space of continuous functions over I with compact support that have continuous first two derivatives.
We recall the definition and some basic properties of continuous-state branching processes without competition. Most of the sequel can be found in Chapter 12 of Kyprianou's book [Kyp14]. A CSBP is a Feller process (X t , t ≥ 0) valued in [0, ∞] satisfying the branching property: for any z, z ≥ 0, t ≥ 0 and x > 0 The branching and Markov properties ensure the existence of a map (x, t) → u t (x) such that for all x > 0 and all t, s ≥ 0, u t (x) ∈ (0, ∞), Silverstein in [Sil68, Theorem 4, page 1046] has shown that the map t → u t (x) is the unique solution to a non-linear ordinary differential equation where Ψ is a Lévy-Khintchine function of the form with λ ≥ 0, γ ∈ R, σ ≥ 0, and π a Borel measure carried on R + satisfying Any branching mechanism Ψ is Lipschitz on compact subsets of (0, ∞) and thus the deterministic equation (5) admits a unique solution. As in Silverstein [Sil68], we interpret the killing term with parameter λ as the possibility for the process to jump to ∞ in finite time. Since for any t ≥ 0 and any x > 0, u t (x) > 0 then according to the semi-group equation (4), ∞ and 0 are either natural or exit boundaries. Grey [Gre74] classifies further the boundaries ∞ and 0 of a CSBP as follows.
-The boundary ∞ is accessible if and only if 0+ du |Ψ(u)| < +∞. -The boundary 0 is accessible if and only if ∞ du Ψ(u) < ∞ (Grey's condition) The integral conditions above ensure respectively the existence of a non-degenerate solution of (5) started from x = 0+ and x = ∞. It is important to note that λ = 0 is necessary for ∞ to be inaccessible but not sufficient. Indeed, the process can explode continuously by having unbounded paths over finite time intervals. A basic example is provided by the stable mechanism Ψ(z) = −z α for α ∈ (0, 1) which satisfies 0+ du |Ψ(u)| < ∞. We now recall the longterm behavior of CSBPs. We refer to Theorem 12.5 in [Kyp14] for a complete classification. Denote by ρ the largest positive root of Ψ. For any z ∈ [0, ∞], When −Ψ is the Laplace exponent of a subordinator then ρ = ∞ and the process goes to ∞ almost-surely. When −Ψ is not the Laplace exponent of a subordinator then ρ < ∞, and the process goes to 0 with positive probability. If moreover ∞ du Ψ(u) = ∞ then X t −→ t→∞ 0 with positive probability albeit X t > 0 for all t ≥ 0 almost-surely. In the latter case, we say that 0 is attracting. A classical construction of a CSBP with mechanism Ψ is by time-changing a spectrally positive Lévy process with Laplace exponent −Ψ (see for instance Lamperti [Lam67a], Caballero, Lambert and Uribe-Bravo [CLUB09]). In particular, the sample paths of a càdlàg CSBP have no negative jump and are non-decreasing when −Ψ is the Laplace exponent of a subordinator. This time-change leads to the following form for the extended generator of (X t , t ≥ 0).
To incorporate quadratic competition, one considers an additional negative quadratic drift in the extended generator above and set We define a minimal logistic continuous-state branching process as a càdlàg Markov process (Z min t , t ≥ 0) on [0, ∞] with 0 and ∞ absorbing, satisfying the following martingale problem (MP). For any function f ∈ C 2 c ((0, ∞)), the process is a martingale under each P z , with ζ := inf{t ≥ 0; Z t / ∈ (0, ∞)}. By minimal process, we mean that the process remains at ∞ from its first (and only) explosion time ζ ∞ := inf{t ≥ 0, Z min t = ∞}. As already observed by Lambert [Lam05], one way to construct a minimal logistic CSBP is by time-changing an Ornstein-Uhlenbeck type process. The problem of explosion is not discussed in [Lam05] and we shall give out some details in Section 4. In the sequel, we say that a process (Z t , t ≥ 0) extends the minimal process if (Z t , t ≥ 0) takes its values in [0, ∞] and (Z t∧ζ∞ , t ≥ 0) L = (Z min t , t ≥ 0). Note that elementary return processes restarting after explosion from states in (0, ∞) are ruled out from our study. We will only be interested in the existence of a continuous extension (Z t , t ≥ 0), for which Z t −→ t→0 ∞, P ∞ -almost-surely. As explained in the introduction, the semi-group of logistic CSBPs will be represented in terms of a certain diffusion. For any mechanism Ψ of the form (6), we call Ψ-generalized Feller diffusion, the minimal diffusion (U t , t < τ ) solving where (B t , t ≥ 0) is a Brownian motion and τ := inf{t; U t / ∈ (0, ∞)}. As u → √ u is 1/2-Hölder and Ψ is locally Lipschitz, standard results (see e.g. [RY99, Section 3, Chapter IX]) ensure the existence and uniqueness of a strong solution to Equation (9) up to time τ . Note that (9) coincides with (5) when c = 0. The duality between the Ψ-generalized Feller diffusion and the logistic CSBP can be easily seen at the level of generators, see the forthcoming Lemma 5.1.1. Duality of semigroups requires more work and is part of the main results. Remark. The integrals 0 |Ψ(u)| u du and ∞ log(u)π(du) have the same nature. In
Theorem 3.2 (Infinity as entrance boundary). Assume E = ∞. The Markov process (Z min t , t ≥ 0) can be extended in [0, ∞] to a Feller process (Z t , t ≥ 0) with ∞ as an entrance boundary. The boundary 0 is an exit of the diffusion (U t , t ≥ 0) solution to (9), and the semi-group of (Z t , t ≥ 0) satisfies for all For any Lévy measure π and any x ≥ 0, setπ(x) := π([x, ∞)). Given Ψ of the form (6) and k ≥ 1, define a branching mechanism Ψ k by Plainly |Ψ k (0+)| < ∞, for any k ≥ 1, and by Theorem 3.1, the minimal logistic CSBP with mechanism Ψ k does not explode. Call (Z (k) t , t ≥ 0) the càdlàg logistic CSBP, provided by Theorem 3.2, with mechanism Ψ k and ∞ as entrance boundary.
The boundary behaviors found in Theorems 3.2, 3.3 and 3.4 and Corollary 3.1 can be summarized as follows Table 1. Boundaries of Z and boundaries of U Corollary 3.2 (Stationarity). Assume Ψ(z) < 0 for all z > 0, then −Ψ is the Laplace exponent of a subordinator and takes the form is not satisfied then (Z t , t ≥ 0) converges in law towards the distribution carried over ( 2δ c , ∞) whose Laplace transform is Remark. The condition in Corollary 3.2 for the existence of a non-degenerate stationary distribution can be rephrased as follows. The condition (A) is not satisfied if and only if at least one of the following holds lim u→∞ Ψ(u) . This already appears with λ = 0 and the log-moment assumption in [Lam05, Theorem 3.4]. One can easily see from the Laplace transform that λ = 0 and ∞ log(x)π(dx) < ∞ are necessary and sufficient conditions for the stationary distribution to admit a first moment.
. Symbolic representation of the two behaviors at 0 in the non-subordinator case with ∞ entrance or reflecting.
We provide now several examples for which different behaviors at infinity occur.
Remark. Roughly speaking, the latter example can be seen as the continuous-state analogue of the number of fragments in a fast-fragmentation-coalescence process as defined in [KPRS17]. Note in particular that the same phase transition between ∞ reflecting and exit boundary occurs. However, contrary to the process in [KPRS17], the process (Z t , t ≥ 0) has no stationary distribution over (0, ∞).

The conditions
∞ log(x)π(dx) < ∞ and λ > 0 are not necessary for having respectively E = ∞ and E < ∞. In the following example λ = 0, ∞ log(x)π(dx) = ∞ and a phase transition occurs between entrance and regular.
The following proposition allows us to study easily Example 3 and to construct more general explicit Lévy measures for which ∞ is regular or entrance.
Proposition 3.1. Assume λ = 0 and set Moreover there exists a universal 2 constant κ > 0 and C 1 , C 2 two non-negative constants such that

Minimal process and time-change
We will prove in this section Theorem 1 and show that the martingale problem (MP) for the minimal logistic CSBP is well-posed. As described in Definition 3.2 in [Lam05], one way to construct a logistic CSBP is to start from an Ornstein-Uhlenbeck type process and to time change it in Lamperti's manner. The problem of explosion was not addressed in [Lam05] and lies at the heart of our study. We provide therefore some details. We start by recalling some known results about Ornstein-Uhlenbeck type processes. Consider (Y t , t ≥ 0) a spectrally positive Lévy process with Laplace exponent −Ψ, killed at ∞ at an independent exponential random variable e λ with There is a unique process (R t , t ≥ 0) satisfying (10), see Sato [Sat13, Chapter 3, Section 17 page 104]. By definition it is called an Ornstein-Uhlenbeck type process with Lévy process (Y t , t ≥ 0) and parameter c/2. Unkilled Ornstein-Uhlenbeck type processes have been deeply studied by Hadjiev [Had85], Sato and Yamazato [SY84] and Shiga [Shi90]. From Lemma 17.1 in Sato [Sat13], one has for any θ > 0 and any In particular, by letting θ to 0, we see that the process (R t , t ≥ 0) will never reach ∞ in finite time if it is not killed. In the unkilled case, it is shown in . Moreover, the process can be positive recurrent, null-recurrent or transient.
If λ = −Ψ(0) > 0, then one can easily see that E < ∞, so that explosion by a single jump can be seen as a particular case of transience. We work in the sequel with the process (R t , t ≥ 0) stopped on first entry into (−∞, 0). Define A first consequence of this definition is that the process (Z min t , t ≥ 0) hit its boundaries if and only if θ ∞ < ∞. On the one hand, if σ 0 < ∞ then ζ ∞ = ∞ and Note that if λ > 0, then R s = ∞ for any s ≥ e λ and the last integral is finite.
Recall (8) and the martingale problem (MP) defining the minimal logistic CSBP.
Lemma 4.1. The process (Z min t , t ≥ 0) is a minimal logistic continuous-state branching process.
Proof. Notice first that if the process (Z min t , t ≥ 0), as defined above, hits 0 or ∞, then it is absorbed. Denote by L Y the generator of the (possibly killed) Lévy process (Y t , t ≥ 0) and L R the generator of (R t , t ≥ 0), which acts on C 2 c ([0, ∞)) as follows By Itô's formula (or by applying [SY84, Theorem 3.1]), one can see that the pro- is a local martingale. By definition, for any z ≥ 0, L f (z) = zL R f (z) and since f has compact support then L f is bounded. Therefore the above local martingale has paths which are bounded on time-intervals [0, t], so that it is a true martingale and (Z min Lemma 4.2. There exists a unique minimal logistic CSBP.
Proof. We have seen above how to construct a solution to the martingale problem.
Only uniqueness has to be justified. Consider any solution (Z t , t < ζ) to the martingale problem (MP). Set C t := t 0 Z s ds for t < ζ and C t := C ζ for all t ≥ ζ. Let θ t := inf{s ≥ 0 : C s > t} and R t := Z θt for any time t ∈ [0, C ζ ). By definition, R Ct = Z t and thus C ζ = inf{t ≥ 0; R t / ∈ (0, ∞)}. As in Lemma 4.1, but in the opposite direction, one sees that the process (R t , t < C ζ ) solves the same martingale problem as an Ornstein-Uhlenbeck type process (with parametersΨ = Ψ + λ and c/2) stopped on first entry into (−∞, 0). The Ornstein-Uhlenbeck type process is uniquely defined in law (see [Sat13, Chapter 3, section 17] where existence and pathwise uniqueness of solution to (10) are established). Moreover, one can readily check that C t = inf{s ≥ 0, s 0 du Ru > t}, which entails that the law of (Z t , t ≥ 0) is uniquely determined by the law of (R t , t ≥ 0).
We now gather some path properties of minimal logistic CSBPs obtained directly by time-change.
Lemma 4.3. Assume that −Ψ is not the Laplace exponent of a subordinator. If E = ∞, then for any z > 0 According to Patie [Pat05, Proposition 3], for any z > a ≥ 0 and µ > 0 In order to make the paper selfcontained, a simple proof of (12) is provided in the Appendix (see Lemma 7.5). One can easily check that if E = ∞, then Lemma 4.4. Assume that −Ψ is not the Laplace exponent of a subordinator. Set ζ a := inf{t ≥ 0; Z min t ≤ a}. For any z > a ≥ 0 and µ > 0, one has Proof. By the time-change σ a = ζa 0 Z min s ds a.s and the statement follows directly by (12).
The next lemma establishes Theorem 3.1. Rs ds < ∞. Assume first E = ∞. Let a > 0 and consider the successive excursions of (R t , t ≥ 0) under a. Since the process is recurrent, there is an infinite number of such excursions. Let b 0 := 0 and for any n ≥ 1, a n := inf{t > b n−1 , R t ≤ a}, b n := inf{t > a n , R t > a}. We see that Since E < ∞, the last integral is finite for any θ > 0. Let b > 0. By Tonelli, one has The upper bound is finite since θ We deduce then that on the event {σ 0 = ∞}, Proof. We have seen in the proof of Lemma 4.5 that when E < ∞, the following two events coincide {ζ ∞ = ∞} = {σ 0 < ∞}. In the non-subordinator case, one has P z (σ 0 = ∞) < 1 since the unstopped process (R t , t ≥ 0) is irreducible in (−∞, ∞]. Assume that −Ψ is the Laplace exponent of a subordinator with drift δ ≥ 0 (possibly killed at rate λ). We show that σ 0 = ∞ a.s. Let (Y t , t ≥ 0) denote the subordinator with Laplace exponent −Ψ. Since Y t ≥ z + δt for all t ≥ 0 P z -a.s, a comparison argument in (10) entails that R t ≥ r t for all t ≥ 0, P z -a.s, with (r t , t ≥ 0) the solution to dr t = δdt − c 2 r t dt with r 0 = z. We deduce that R t ≥ e − c 2 t z + 2δ c (1 − e − c 2 t ) > 0 for all t ≥ 0, P z -a.s. This entails P z (σ 0 = ∞) = 1 for any z > 0.
Remark. When −Ψ is the Laplace exponent of a subordinator, the Ornstein-Uhlenbeck type process is irreducible in ( 2δ c , ∞). Namely, for any z > 2δ c , the process starting from z hit any value in ( 2δ c , ∞) with positive probability. By time-change, (Z min t , 0 ≤ t < ζ ∞ ) is therefore also irreducible in ( 2δ c , ∞). Lemma 4.7. If λ > 0, then the minimal process always explodes by a jump to ∞. In other words, the two types of explosion cannot occur for a given process.
Proof. Let (Z min t , t ≥ 0) be a minimal logistic CSBP with λ > 0. By the time-change, (R t , t ≥ 0) := (Z θt , t ≥ 0) is an Ornstein-Uhlenbeck type process killed at some exponential random variable e λ and stopped at its first entry in (−∞, 0). Since for any s < e λ , R s < ∞, then Z t = R Ct < ∞ for any t < θ e λ . Therefore, the process cannot explode before θ e λ and on the event {σ 0 = ∞}, explosion is made by a single jump which occurs at time θ e λ = e λ 0 ds Rs .
Remark. We have seen that when the Ornstein-Uhlenbeck type process (R t , t ≥ 0) is transient, the logistic CSBP explodes. Therefore, a logistic CSBP cannot grow indefinitely without exploding. This is a striking difference with CSBPs where indefinite growth with no explosion can occur when the Lévy process (Y t , t ≥ 0) drifts "slowly" towards ∞.

Infinity as an entrance boundary
The goal of this section is to show Theorem 3.2. The proof will follow from Lemmas 5.1.2 (part 1), 5.2.1 and 5.2.3.
where (B t , t ≥ 0) is a Brownian motion. The following observation is our starting point in the study of logistic continuous-state branching processes by duality.
Proof. One can readily check that for all x and z in ]0, ∞[, Intuitively, integrating each side in (15) should provide a duality at the level of semigroups of the form: The study of the boundaries 0 and ∞ of (U t , t ≥ 0) would then provide the nature of boundaries ∞ and 0 of (Z min t , t ≥ 0). However, there is not a unique semi-group associated to A as several boundary conditions are possible. Some precautions are then needed while showing the above duality. We gather in this section, the boundary conditions of the diffusion. The proofs of the following statements are rather technical and postponed in the Appendix. When E < ∞, 0 is regular and there are several possibilities for extending the minimal diffusion after τ . In the next lemma, we denote by (U 0 t , t ≥ 0) the diffusion (14) with either 0 regular absorbing or exit.

Duality and entrance law.
In this section, we assume E = ∞. Recall that it ensures the inaccessibility of ∞ for the process (Z min t , t ≥ 0) and that 0 is an exit for the diffusion (U t , t ≥ 0). Proof. Recall (R t , t ≥ 0) the Ornstein-Uhlenbeck type process. For any x > 0, by Itô's formula, one sees that the process is a local martingale. By time-changing it, we obtain that is a local martingale. Since x > 0, then z → L e x (z) is bounded and (M Z t , t ≥ 0) is a martingale. Consider now (U t , t ≥ 0) a Ψ-generalized Feller diffusion independent of (Z min t , t ≥ 0). By applying Itô's formula, we have that for any z ≥ 0, and > 0; From the last equality, it comes where we have obtained the last equality using the martingale (M Z t , t ≥ 0) conditionally given τ since τ is independent of (Z min t , t ≥ 0). By letting to 0, τ −→ →0 τ 0 a.s and the last equality provides We know that under the condition E = ∞ the process (Z min t , t ≥ 0) does not explode. Therefore the limit above is 0 and E z where in the second inequality we have used uz 2 e −uz ≤ 2 u and in the third one, ze − z ≤ 1 . We now argue by comparison in order to show that sup s≤T U 2 s∧τ is integrable. When u ≥ , we have Ψ(u) ≥ Ψ( ) u ≥ −γ u for some γ > 0. Recall that Ψ is locally Lipschitz on (0, ∞). Applying the results of [RY99, Section 3, Chapter IX], one can then construct, with the same Brownian motion (B t , t ≥ 0), the process (U t , t ≥ 0) as a strong solution to (14) with 0 exit and the process (V t , t ≥ 0) as a strong solution to dV t = cV t dB t + γ V t dt, V 0 = x. Both processes are adapted to the natural filtration of (B t , t ≥ 0). Applying the comparison theorem [RY99, Theorem IX.3.7] up to the stopping time τ , one has that almost-surely for any 0 ≤ s ≤ τ , U s ≤ V s . Note that (V t , t ≥ 0) is a supercritical Feller diffusion with branching mechanism Φ(u) = c 2 u 2 + γ u. It is easily checked that for any t ≥ 0, V t has a second moment. Moreover, the process (V s , s ≥ 0) is a submartingale and by Doob's inequality Since for any > 0, sup Let P min t , t ≥ 0 be the semigroup of (Z min t , t ≥ 0). Lemma 4.5 ensures that when E = ∞, ∞ is inaccessible. To see that ∞ is an entrance boundary, we show in the following lemmas how to define a Feller semigroup coinciding with P min t , t ≥ 0 over [0, ∞), with an entrance law from ∞.
Proof. By taking limits as z → ∞ in the duality formula in Lemma 5.2.1, one has: Since 0 is an exit thanks to the assumption E = ∞, P x (U t = 0) = P x (τ 0 ≤ t) > 0. By Lévy 's continuity theorem, x → P x (τ 0 ≤ t) is the Laplace transform of a certain finite measure η t which is the weak limit of the law of Z min t under P z as z → ∞. Moreover, lim x→0 P x (τ 0 ≤ t) = P 0+ (τ 0 ≤ t) = 1 and η t is a probability measure over [0, ∞). By continuity of the paths of (U t , t ≥ 0), if x > 0, then lim t→0 P x (U t = 0) = 0, and if x = 0 then lim t→0 P x (U t = 0) = 1. This entails that η t → δ ∞ weakly as t → 0.
. We show now that (P t , t ≥ 0) is a semigroup. Since it coincides with the semigroup (P min t , t ≥ 0) on [0, ∞[, then for any s, t ≥ 0, any function f ∈ C b ([0, ∞]) and any z ∈ [0, ∞) P t+s f (z) = P t P s f (z). For z = ∞, we have The last equality above holds since P s f ∈ C b ([0, ∞]). This provides P t+s f (∞) = P t P s f (∞). It remains to justify the continuity of (P t , t ≥ 0) at 0. That is to say sion (with continuous paths and with infinite life-time) then t → P t e x (z) = E x e −zUt is continuous, in particular continuous at 0. Hence, t → P t f (z) is continuous at zero for any z ∈ [0, ∞).
We give in the next lemma, an alternative proof for the property of entrance at ∞, based on arguments that are not involving duality. Proof. Recall G the generator of a CSBP with mechanism Ψ. Let h(z) = 1 z . One has By the optional stopping theorem, E z [M ζa∧Tm ] = h(z) and we obtain, letting m to ∞ We conclude that E z (ζ a ) ≤ 4 Proof. For all z ∈ R + , Therefore Proof. According to Lemma 5.1.3, (U t , t ≥ 0) exits (0, ∞) either by 0 or by ∞. Thus, for any z ∈ (0, ∞] and x ≥ 0, A direct application of Lemma 5.1.3 provides the two first convergences. The support of the stationary measure is ( 2δ c , ∞) since (Z min t , t ≥ 0) is irreducible in ( 2δ c , ∞), see the Remark below Lemma 4.6.
The next Lemma establishes part 1) of Theorem 3.5 under the additional condition E = ∞. Later we get the same part proved under E < ∞.

Infinity as regular reflecting or exit boundary
In this subsection, we assume that E is finite and will prove Theorems 3.3 and 3.4. Recall that if 0 ≤ 2λ/c < 1 then 0 is a regular boundary of the Ψ-generalized Feller diffusion and if 2λ/c ≥ 1, then 0 is an entrance boundary.
If 0 ≤ 2λ c < 1, then as k goes to ∞, the sequence of processes (U (k) t , t ≥ 0) converges pointwise almost-surely towards a process (U 0 t , t ≥ 0) which is the solution to (14) absorbed at 0. If 2λ c ≥ 1, then as k goes to ∞, the sequence of processes (U (k) t , t ≥ 0) converges almost-surely towards (U t , t ≥ 0) the unique solution to (14).
The next lemmas will provide proofs of Theorem 3.3 and Theorem 3.4. For any k ≥ 1, denote by (Z where (U 0 t , t ≥ 0) is the Ψ-generalized Feller diffusion satisfying (14) with 0 regular absorbing.
Proof. Denote by (P t , t ≥ 0) is defined in Lemma 6.1. Since, for any t ≥ 0, U (k) t converges almost-surely towards U (∞) t as k goes to infinity, then lim t (z, ·) converges weakly as k goes to ∞ towards some probability p t (z, ·) over [0, ∞] satisfying is a contraction. The upper bound goes to 0 as k goes to ∞ by the uniform convergence. Let f ∈ C b ([0, ∞]) and apply the last convergence to g k := P (k) s f and g = P s f , one has, by the semigroup property of (P (k) . As in Lemma 5.2.3, we see that it is continuous at 0. Lastly, since the convergence of semigroups is uniform in C b ([0, ∞]), one invokes Theorem 2.5 p167 in [EK86] to claim that the sequence of processes (Z (k) t , t ≥ 0) converges weakly (in the Skorokhod topology) towards a càdlàg Markov process (Z t , t ≥ 0) with semigroup (P t , t ≥ 0). Lemma 6.3. Assume 2λ c ≥ 1, the sequence ((Z (k) t ) t≥0 , k ≥ 1) converges weakly towards a càdlàg Feller process (Z t , t ≥ 0) valued in [0, ∞] such that for all z ∈ [0, ∞], all t ≥ 0, and all x ∈ (0, ∞) Proof. The only difference with the proof above lies in the fact that we have to separate the cases x > 0 and x = 0, since 0 is an entrance boundary. By Lemma 6.1, (U t ] = 1 and so P t e 0 (z) = 1 for any z ∈ [0, ∞]. The rest of the proof is similar to the one above. By replacing τ ∞ by ∞, since 0 is inaccessible, and using the fact that P x (τ k ≤ t) −→ k→∞ 0 for any fixed t, we see that The next lemma ensures that the processes (Z t , t ≥ 0) defined above are extensions of the minimal logistic CSBP.
Lemma 6.4 (Extension). The limiting processes (Z t , t ≥ 0) in Lemma 6.2 and Lemma 6.3 stopped at ζ ∞ have the same law as (Z min t , t ≥ 0).
Proof. By definition, ∞ is absorbing for (Z t∧ζ∞ , t ≥ 0). By the semigroup representation obtained in Lemma 6.2 and Lemma 6.3, we see that 0 is absorbing for the process (Z t , t ≥ 0). Therefore it is absorbing for the process (Z t∧ζ∞ , t ≥ 0). It remains only to see if (Z t∧ζ∞ , t ≥ 0) satisfies the martingale problem (MP). If this holds true, Lemma 4.2 will ensure that (Z t∧ζ∞ , t ≥ 0) has the same law as (Z min t , t ≥ 0). Denote by L (k) the generator of (Z (k) t , t ≥ 0) as defined in Equation (7). For any function f ∈ C 2 c ((0, ∞)), one has For any z ∈ (0, ∞) and any k ≥ 1, under P z , the process f (Z s )ds, t ≥ 0 is a martingale. The convergence above being uniform, Lemma 5.1 page 196 in [EK86] entails that the process is a martingale. By [RY99,Corollary III.3.6], the latter process stopped at time ζ ∞ , is a martingale. Now, since f has a compact support then for any t ≥ 0, Indeed, in the first equality above, either t ≤ ζ ∞ and the second integral vanishes or t > ζ ∞ and Lf (Z s∧ζ∞ ) = 0 for any s > ζ ∞ , since f has a compact support. We deduce that for any f ∈ C 2 c ((0, ∞)), the process is a martingale. Therefore the process (Z t∧ζ∞ , t ≥ 0) has the same law as (Z min t , t ≥ 0). Note that by Lemma 4.5, since E < ∞, then (Z min t , t ≥ 0) explodes with a positive probability. This ensures that ∞ is accessible for the process (Z t , t ≥ 0). Lemma 6.5 (Reflecting boundary). Assume E < ∞ and 0 ≤ 2λ c < 1. The boundary ∞ of the Feller process (Z t , t ≥ 0) is instantaneous regular reflecting. Moreover, P ∞ -almost-surely, (Z t , t ≥ 0) enters (0, ∞) continuously.
Proof. Recall that we assume E < ∞ and 2λ c < 1. Accordingly to Lemma 6.4, ∞ is accessible. Moreover, for every t ≥ 0, Thus, ∞ is a regular boundary. We verify now that ∞ is reflecting. Since (U 0 t , t ≥ 0) has 0 regular absorbing then This ensures that the Lebesgue measure of the set of times at which the process is at ∞ is zero. In other words, ∞ is reflecting. We show now that ∞ is instantaneous. Let T = inf{t ≥ 0, Z t < ∞}. For any t > 0, P ∞ (T ≤ t) ≥ P ∞ (Z t < ∞) = 1, then by letting t to 0, one has P ∞ (T = 0) = 1. By right-continuity of (Z t , t ≥ 0), we have P ∞ (lim t→0 Z t = ∞) = 1.
Proof. Firstly, by Lemma 6.4, we know that the process (Z t , t ≥ 0) explodes with positive probability (note that λ > 0, so that it will explode by a jump with positive probability). Moreover, by Lemma 6.2, letting z → ∞, we obtain E ∞ [e −xZt ] = P x (U t = 0) = 0 since 0 is an entrance boundary of U . Therefore, (Z t , t ≥ 0) get absorbed at ∞. Note that for any t > 0, The proof of Theorem 3.3 now follows by combining Lemmas 6.2, 6.4, 6.5. Theorem 3.4 follows from Lemmas 6.3, 6.4 and 6.6. In order to understand the long-term behavior of the extended process (Z t , t ≥ 0), we establish now Corollary 3.1, Corollary 3.2 and Theorem 3.5 in the case E < ∞. The arguments for Corollary 3.1 and Corollary 3.2 are exactly the same as those in Lemma 5.3.1 and Lemma 5.3.2 for the case E = ∞ (but with λ ≥ 0). Indeed, according to Lemma 5.1.3, the dual diffusion (U 0 t , t ≥ 0) can leave the interval (0, ∞) by ∞ only when the condition (A) is not satisfied. We show Theorem 3.5 in the case E < ∞.
Proof. Let a > 0. Define ζ  ∞ , t ≥ 0) are independent and with the same law as the minimal process starting from a. Set 0} for any n ≥ 1. By Lemma 4.3, P z (E n ) = P a (σ 0 < ∞) > 0.
It remains to study the process when 2λ c ≥ 1. Lemma 6.8. Assume 2λ c ≥ 1. a) If −Ψ is the Laplace exponent of a subordinator then the process is absorbed at ∞ almost-surely. b) If −Ψ is not the Laplace exponent of a subordinator then the process tends to 0 with probability: cv dv du ∈ (0, 1).
On this event, the process get absorbed at 0 if and only if ∞ du Ψ(u) < ∞. Proof. By Lemma 6.6, the process (Z t , t ≥ 0) has 0 and ∞ has absorbing boundary. By Lemma 6.4, it satisfies (MP), therefore (Z t , t ≥ 0) has the same law as (Z min t , t ≥ 0). Proof of a). Assume that −Ψ is the Laplace exponent of a subordinator, then by Lemma 4.6, explosion is almost-sure. Proof of b). If now −Ψ is not the Laplace exponent of a subordinator, then by Lemma 4.3 The process (Z t , t ≥ 0) hit 0 with positive probability if and only if Lemma 6.8-a) gives part 2) of Theorem 3.5, and b) gives part 3). As mentioned in the introduction, our extension of the minimal process has been done without using excursions theory. In particular, we have not discussed the existence of a local time at ∞. We end this article by showing that when ∞ is regular reflecting, the process immediately returns to ∞ after leaving it. Such a boundary is said to be regular for itself and standard theory, see for instance Bertoin [Ber96,Chapter IV], would then ensure the existence of a local time.
The last proposition leads to the natural question of characterizing the inverse local time. A related question is to see if one can continue Table 1 to the case where the boundary ∞ of Z is regular absorbing. This requires us to work with the reflected diffusion and seems to require a deeper analysis of the duality at the level of generators. We intend to study these questions in future work.