CONTINUOUS-STATE BRANCHING PROCESSES WITH COMPETITION DUALITY AND REFLECTION AT INFINITY ´ CLEMENT FOUCART

Abstract. 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 ∞ to be accessible. We show that when ∞ is inaccessible, it is always an entrance boundary. In the case where ∞ is accessible, explosion can occur either by a single jump to ∞ (the process at z is killed at rate λz) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when ∞ is accessible and 0 ≤ 2λ < 1, c the extended process is reflected at ∞. In the case 2λ ≥ 1, ∞ is an exit of the extended process. When the c branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at ∞ 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.

1. Introduction Continuous-state branching processes (CSBPs for short) have been defined by Jiˇrina [Jiˇr58] and Lamperti [Lam67a] for modelling the size of a random continuous population whose individuals reproduce and die independently with a 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 L´evy-Khintchine function Ψ (called 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 c (1) dzt = γzt dt − zt2 dt, 2 where, informally speaking, the Malthusian growth γzt dt is replaced by the full dynamics of a continuous-state 2 branching process. For instance, when the mechanism Ψ of the CSBP reduces to Ψ(z) = σ2 z 2 − γz, the process (Zt , t ≥ 0) is the logistic Feller diffusion p c (2) dZt = σ Zt dBt + γZt dt − Zt2 dt. 2 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 [BFF15] 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 others, 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). 2010 Mathematics Subject Classification. 60J80, 60J70,92D25. Key words and phrases. Continuous-state branching process, generalized Feller diffusion, branching process with interaction, explosion, coming down from infinity, entrance boundary, reflecting boundary, Lamperti’s time change, duality. [email protected], Universit´ e Paris 13, Laboratoire Analyse, G´ eom´ etrie & Applications UMR 7539 Institut Galil´ ee. 1

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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 [Li16] 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 to 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 push back in [0, ∞) the population size instantaneously after explosion. This phenomenon is captured by the notion of regular reflecting boundary. By reflecting, we mean here that the Lebesgue measure of {t ≥ 0, Zt = ∞} is zero almost-surely. 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 get extinct after having exploded, 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 resolvant 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 (Zt , t ≥ 0), then for any x ≥ 0, z ∈ [0, ∞[ and t ≥ 0, Ez (e−xZt ) = Ex (e−zUt )

(?) where (Ut , t ≥ 0) is a solution to (3)

dUt =

p

cUt dBt − Ψ(Ut )dt,

U0 = x.

We shall see that the condition for ∞ to be inaccessible for (Zt , t ≥ 0) is precisely given by Feller’s test for 0 to be an exit of (Ut , t ≥ 0). We stress that Equation (3) has not always a unique solution as 0 can be regular for certain non-lipschitz mechanisms Ψ. This is precisely for such mechanisms that ∞ will be regular for logistic CSBPs. Heuristically, if (?) holds for some processes (Zt , t ≥ 0) and (Ut , t ≥ 0), then the entrance boundaries of (Zt , t ≥ 0) will be classified in terms of exit boundaries of (Ut , t ≥ 0). We refer to Cox and R¨osler [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 limit of non-explosive processes. We highlight that this construction is different from the classical Itˆ o’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. [KPRS16] 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 [KPRS16], a logistic CSBP can reach ∞ by accumulation of large jumps over a finite interval of time. We mention that the duality (?) has been 2 observed in a spatial context for the branching mechanism Ψ(u) = σ2 u2 − γu by Hutzenthaler and Wakolbinger [HW07]. Lastly, other competition mechanisms than the quadratic drift have been studied. We refer for instance to the monography of Pardoux [Par16] and Ba and Pardoux [BP15] for some generalisations of the logistic Feller diffusions. See Palau and Pardo [PP15] for a model with competition in random environment. It is worth noticing that the relation (?) does not hold for general competition mechanims.

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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 timechange. In Section 5, we gather the possible behaviors of the diffusion (Ut , 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 (Ut , t ≥ 0) according to Ψ and the parameter c. 2. Preliminaries As we will use Feller’s terminology repeatedly, we briefly recall how to classify boundaries. Consider a process ¯ and a < b, valued in an interval (a, b) with a, b ∈ R - 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 Cb ([0, ∞]) the space of continuous real-valued functions defined over [0, ∞]. Since [0, ∞] is compact, any function f ∈ Cb (|0, ∞]) is bounded. We set D([0, ∞]) the space of c` adl` ag functions from R+ to [0, ∞]. For any interval I ⊂ R, we denote by Cc2 (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 (Xt , t ≥ 0) valued in [0, ∞] satisfying the branching property: for any z, z 0 ≥ 0, t ≥ 0 and x > 0 Ez+z0 [e−xXt ] = Ez [e−xXt ]Ez0 [e−xXt ]. The branching and Markov properties ensure the existence of a map (x, t) 7→ ut (x) such that for all x > 0 and all t, s ≥ 0 (4)

Ez [e−xXt ] = e−zut (x) and us+t (x) = us ◦ ut (x).

Silverstein in [Sil68] has shown that the map t 7→ ut (x) is the unique solution to a non-linear ordinary differential equation d ut (x) = −Ψ(ut (x)) for all x ∈ (0, ∞) dt where Ψ is a L´evy-Khintchine function of the form Z +∞
σ2 2 (6) Ψ(z) = −λ + z + γz + e−zx − 1 + zx1{x≤1} π(dx) 2 0

(5)

with λ ≥ 0, γ ∈ R, σ ≥ 0, and π a Borel measure carried on R+ satisfying Z +∞ (1 ∧ x2 )π(dx) < +∞. 0

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 ut (x) > 0 for any t ≥ 0 and any 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. R du - The boundary ∞ is accessible if and only if 0+ |Ψ(u)| < +∞. R ∞ du - The boundary 0 is accessible if and only if < ∞ (Grey’s condition) Ψ(u) 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 A basic R intervals. du example is provided by the stable mechanism Ψ(z) = −z α for α ∈ (0, 1) which satisfies 0+ |Ψ(u)| < ∞. We

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now recall the longterm behavior of CSBPs. We refer to Theorem 12.5 in [Kyp14] for a complete classification. Denote by ρ the largest root of Ψ. For any z ∈ [0, ∞], Pz (Xt −→ 0) = e−zρ and Pz (Xt −→ ∞) = 1 − e−zρ . t→∞

t→∞

When −Ψ is the Laplace exponent of a subordinator then ρ = ∞ and the process goes to ∞ almost-surely. When −Ψ is not theR Laplace exponent of a subordinator then ρ < ∞, and the process goes to 0 with positive probability. ∞ du −→ 0 with positive probability albeit Xt > 0 for all t ≥ 0 almost-surely. In If moreover Ψ(u) = ∞ then Xt t→∞ the latter case, we say that 0 is attracting. A classical construction of a CSBP with mechanism Ψ is by timechanging a spectrally positive L´evy process with Laplace exponent −Ψ (see for instance Lamperti [Lam67a], Caballero, Lambert and Uribe-Bravo [CLUB09]). In particular, the sample paths of a c`adl`ag 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 (Xt , t ≥ 0). For any f ∈ Cc2 ((0, ∞)) 1 Z ∞ σ 2 00 0 G f (z) := −λzf (z) + zf (z) − γzf (z) + z (f (z + u) − f (z) − u1[0,1] (u)f 0 (z))π(du). 2 0 To incorporate quadratic competition, one considers an additional negative quadratic drift in the extended generator above and set c (7) L f (z) := G f (z) − z 2 f 0 (z). 2 We define a minimal logistic continuous-state branching process as a c`adl`ag Markov process (Ztmin , t ≥ 0) on [0, ∞] with 0 and ∞ absorbing, satisfying the following martingale problem (MP). For any function f ∈ Cc2 ((0, ∞)), the process Z t

min t ∈ [0, ζ) 7→ f Zt − L f Zsmin ds 0

is a martingale under each Pz , with ζ := inf{t ≥ 0; Zt ∈ / (0, ∞)}. As for CSBPs, we consider the boundary 0 as a trap, so that Ztmin = 0 for any t ≥ ζ0 with ζ0 := inf{t ≥ 0, Ztmin = 0}. By minimal process, we mean that the process remains at ∞ from its first (and only) explosion time ζ∞ := inf{t ≥ 0, Ztmin = ∞}. 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 (Zt , t ≥ 0) extends the minimal process if (Zt , t ≥ 0) takes its values L in [0, ∞] and (Zt∧ζ∞ , t ≥ 0) = (Ztmin , 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. 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 (Ut , t < τ ) solving p (8) dUt = cUt dBt − Ψ(Ut )dt, U0 = x √ where (Bt , t ≥ 0) is a Brownian motion and τ := inf{t; Ut ∈ / (0, ∞)}. As u 7→ u is 1/2-H¨older 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 (8) up to time τ . Note that (8) 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 8. Duality of semigroups requires more work and is part of the main results. 3. Main results Theorem 1 (Accessibility of ∞). Assume c > 0. The boundary ∞ is inaccessible for (Ztmin , t ≥ 0) if and only if ! Z θ Z 1 2 θ Ψ(u) E := exp du dx = ∞, for some (and then for all) θ > 0. c x u 0 x Remark. The integrals implies E = ∞.

R 0

|Ψ(u)| u du

and

R∞

log(u)π(du) have the same nature. In particular

R∞

log(u)π(du) < ∞

The next theorems introduce extensions in [0, ∞] of the minimal process. The usual convention 0.∞ = ∞.0 = 0 is taken. In particular, note that e−0.z = 1 for all z ∈ [0, ∞]. 1the space of twice continuously differentiable functions vanishing outside a compact subset of (0, ∞).

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Theorem 2 (Infinity as entrance boundary). Assume E = ∞. The Markov process (Ztmin , t ≥ 0) can be extended in [0, ∞] to a Feller process (Zt , t ≥ 0) with ∞ as an entrance boundary. The boundary 0 is an exit of the diffusion (Ut , t ≥ 0) solution to (8), and the semi-group of (Zt , t ≥ 0) satisfies for all t ≥ 0, all z ∈ [0, ∞], all x ∈ [0, ∞) Ez (e−xZt ) = Ex (e−zUt ). For any L´evy measure π and any x ≥ 0, set π ¯ (x) := π([x, ∞)). Given Ψ of the form (6) and k ≥ 1, define a branching mechanism Ψk by Z ∞
σ2 2 Ψk (z) := z + γz + e−zx − 1 + zx1x∈(0,1) πk (dx), with πk = π|]0,k[ + (¯ π (k) + λ)δk . 2 0 Plainly |Ψ0k (0+)| < ∞, for any k ≥ 1, and by Theorem 1, the minimal logistic CSBP with mechanism Ψk does (k) not explode. Call (Zt , t ≥ 0) the c` adl` ag logistic CSBP, provided by Theorem 2, with mechanism Ψk and ∞ as entrance boundary. (k)

Theorem 3 (Infinity as regular reflecting boundary). Assume E < ∞ and 0 ≤ 2λ c < 1. The processes (Zt , t ≥ 0) converges weakly in D([0, ∞]) towards a Feller process (Zt , t ≥ 0), extending (Ztmin , t ≥ 0), with ∞ regular reflecting. The semi-group of (Zt , t ≥ 0) satisfies for all t ≥ 0, all z ∈ [0, ∞] and x ∈ [0, ∞), 0

Ez (e−xZt ) = Ex (e−zUt ) where (Ut0 , t ≥ 0) is solution to (8) with 0 regular absorbing. (k)

Theorem 4 (Infinity as exit boundary). Assume 2λ c ≥ 1. The processes (Zt , t ≥ 0) converges weakly in D([0, ∞]) towards a Feller process (Zt , t ≥ 0), extending (Ztmin , t ≥ 0), with ∞ exit. The boundary 0 is an entrance of the diffusion (Ut , t ≥ 0) solution to (8) and the semi-group of (Zt , t ≥ 0) satisfies for all t ≥ 0, all z ∈ [0, ∞] and x ∈ (0, ∞), Ez (e−xZt ) = Ex (e−zUt ).

Zt

Zt

(a)

(b)

Zt

Zt

(c)

(d)

x E =∞

t

E < ∞, λ = 0 t

0 < 2λ/c < 1

t

2λ/c ≥ 1

t

Figure 1. Symbolic representation of the four behaviors at ∞. Corollary 1 (Zero as exit or natural boundary). R ∞ dz i) Assume |Ψ(z)| < ∞ then 0 is an exit boundary of (Zt , t ≥ 0) and ∞ is an entrance boundary of (Ut , t ≥ 0). R ∞ dz ii) Assume |Ψ(z)| = ∞ then 0 is a natural boundary of (Zt , t ≥ 0) and ∞ is a natural boundary of (Ut , t ≥ 0). The boundaries behaviors found in Theorems 2, 3 and 4 and Corollary 1 can be summarized as follows Boundary of Z ∞ entrance ∞ regular reflecting ∞ exit 0 exit 0 natural

Boundary of U 0 exit 0 regular absorbing 0 entrance ∞ entrance ∞ natural

Table 1. Boundaries of Z and boundaries of U

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Corollary 2 (Stationarity). Assume Ψ(z) < 0 for all z > 0, then −Ψ is the Laplace exponent of a subordinator and takes the form Z ∞ Ψ(z) = −λ − δz − (1 − e−zu )π(du) 0 R∞ with λ ≥ 0, δ ≥ 0 and 0 (1 ∧ u)π(du) < ∞. Assume 0 ≤ 2λ c < 1 and define the condition (A) as follows (A) : (δ = 0 and π ¯ (0) + λ ≤ c/2). - If (A) is satisfied then (Zt , t ≥ 0) converges in probability to 0. - If (A) is not satisfied then (Zt , t ≥ 0) converges in law towards the distribution carried over ( 2δ c , ∞) whose Laplace transform is  R R∞ y 2Ψ(z) dz dy exp cz x θ  . R x ∈ R+ 7→ E[e−xZ∞ ] := R ∞ y exp θ 2Ψ(z) cz dz dy 0 Remark. The condition in Corollary 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) u

= −δ 6= 0, π((0, 1)) = ∞, π ¯ (0) + λ > 2c .

This already appears with λ = 0 and the log-moment assumption in [Lam05, Theorem 3.4]. One can easily see R∞ 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. Theorem 5 (Long-term behaviors). Consider (Zt , t ≥ 0) the process started from z ∈ (0, ∞). < 1 and Ψ(z) ≥ 0 for some z > 0 then 1) If 0 ≤ 2λ Rc ∞ du 1-1) If −→ 0 a.s. Ψ(u) = ∞, then Zt > 0 for any t ≥ 0 a.s. and Zt t→∞ R ∞ du 1-2) If Ψ(u) < ∞, then (Zt , t ≥ 0) get absorbed at 0 in finite time almost-surely. 2λ 2) If c ≥ 1 and Ψ(z) < 0 for all z > 0 then (Zt , t ≥ 0) get absorbed at ∞ in finite time almost-surely. 3) If 2λ c ≥ 1 and Ψ(z) ≥ 0 for some z > 0 then  R  R ∞ −zu 1 u 2Ψ(v) exp − dv du e u cv 0 θ  R  ∈ (0, 1). Pz (Zt −→ 0) = 1 − Pz (ζ∞ < ∞) = R ∞ u 2Ψ(v) t→∞ 1 exp − dv du cv 0 u θ R ∞ du R ∞ du If Ψ(u) = ∞, then Zt > 0 for any t ≥ 0 a.s and if Ψ(u) < ∞, then {Zt −→ 0} = {ζ0 < ζ∞ }. t→∞

Zt

Zt

RE∞= 1∞ Ψ 0 and set π(du) =

α u(log u)β+1

1{u≥2} du.

i) If β = 1 and 2α c ≤ 1/2 then E = ∞ and ∞ is an entrance boundary (case (a)). ii) If β = 1 and 2α c > 1/2 then E < ∞ and ∞ is a regular reflecting boundary (case (b)). iii) If β ∈]0, 1[, then E < ∞ and ∞ is a regular reflecting boundary (case (b)). We refer to the calculations of Sato and Yamazato [SY84, Section 7]. The following proposition allows us to construct explicit L´evy measures for which ∞ is regular or entrance. Proposition 1. Assume λ = 0. The integral E has the same nature as   Z θ Z 1 2 ∞ −xv π ¯ (v) E0 = exp − dv dx. e c 1 v 0 x Moreover there exists a universal2 constant κ > 0 and C1 , C2 two non-negative constants such that ! ! Z θ Z Z 1/x Z θ 1 2κ 1/x π ¯ (u) 2 π ¯ (u) 1 C1 exp − du dx ≤ E ≤ C2 exp − du dx c 1 u cκ 1 u 0 x 0 x 4. Minimal process and time-change As described in Definition 3.2 in [Lam05], one way to construct a logistic CSBP is to start from an OrnsteinUhlenbeck 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 (Yt , t ≥ 0) a spectrally positive L´evy process with Laplace exponent −Ψ, killed at ∞ at an independent exponential random variable eλ with parameter λ := −Ψ(0) ≥ 0. Set (Rt , t ≥ 0) the process satisfying Z c t (9) Rt = z + Yt − Rs ds. 2 0 There is a unique process (Rt , t ≥ 0) satisfying (9), see Sato [Sat13, Chapter 3, Section 17 page 104]. By definition it is called an Ornstein-Uhlenbeck type process with L´evy process (Yt , 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   Z s c c Ψ(e− 2 u θ)du . (10) Ez (e−θRs ) = exp −θe− 2 s z + 0

In particular, by letting θ to 0, we see that the process (Rt , t ≥ 0) will never reach ∞ in finite time if it is not killed. In the unkilled case, it is shown in [Shi90] that if (Yt , t ≥ 0) is not a subordinator then the process (Rt , t ≥ 0) is irreducible in R. Namely, for any a ∈ R, if σa := inf{t ≥ 0, Rt ≤ a} then Pz (σa < ∞) > 0 for any z > 0. On the other hand, if −Ψ is the Laplace exponent of a subordinator with drift δ ≥ 0, then the process (Rt , t ≥ 0) is irreducible in ( 2δ c , ∞). Moreover, the process can be positive recurrent, null-recurrent or transient. 2in the sense that it does not depend on the L´ evy measure π

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[Shi90, Theorem 1.1] states that (Rt , t ≥ 0) is recurrent (in a pointwise sense) if E = ∞ and transient if E < ∞, where we recall ! Z θ Z 1 2 θ Ψ(u) E= exp du dx. c x u 0 x If λ = −Ψ(0) > 0, then one can easily see that E < ∞, so that explosion by jump can be seen as a particular case of transience. We work in the sequel with the process (Rt , t ≥ 0) stopped on first entry into (−∞, 0). Define R t∧σ ds and its right-inverse t 7→ Ct := inf{u ≥ 0; θu > t} ∈ [0, ∞]. The σ0 := inf{t ≥ 0, Rt < 0}, θt := 0 0 R s Lamperti time-change of the stopped process (Rt , t ≥ 0) is the process (Ztmin , t ≥ 0) defined by   RCt 0 ≤ t < θ∞ Ztmin = 0 t ≥ θ∞ and σ0 < ∞   ∞ t ≥ θ∞ and σ0 = ∞. A first consequence of this definition is that the process (Ztmin R σ0, tds≥ 0) hit its boundaries if and only if θ∞ < ∞. On the one hand, if σ0 < ∞ then ζ∞ = ∞ and ζ0 = θ∞ = 0 Rs . On the other hand, if σ0 = ∞ then ζ0 = ∞ R ∞ ds and ζ∞ = θ∞ = 0 R . Note that if λ > 0, then Rs = ∞ for any s ≥ eλ and the last integral is finite. s Lemma 1. The process (Ztmin , t ≥ 0) is a minimal logistic continuous-state branching process. Proof. Denote by L Y the generator of the (possibly killed) L´evy process (Yt , t ≥ 0) and L R the generator of (Rt , t ≥ 0), which acts on Cc2 ([0, ∞)) as follows c L R f (z) = L Y f (z) − f 0 (z). 2   Rt By Itˆ o’s formula (or by applying [SY84, Theorem 3.1]), one can see that the process f (Rt ) − 0 L R f (Rs )ds, t ≥ 0 Rt R C ds = t and then Ct = 0 Zsmin ds. is a local martingale. By definition of the time-change, for any t ∈ [0, θ∞ ), 0 t R s A (continuous) time-change of a local martingale remains a local martingale. Hence Z Ct Z t t ∈ [0, θ∞ ) 7→ f (RCt ) − L R f (Rs )ds = f (Ztmin ) − Zsmin L R f (Zsmin )ds 0

0

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 compact sets [0, t], so that it is a true martingale and (Ztmin , t ≥ 0) solves (MP).  Lemma 2. There exists a unique minimal logistic CSBP. Proof. We have seen above how to construct a solution to the martingale problem. OnlyRuniqueness has to be t justified. Consider any solution (Zt , t < ζ) to the martingale problem (MP). Set Ct := 0 Zs ds for t < ζ and Ct := Cζ for all t ≥ ζ. Let θt := inf{s ≥ 0 : Cs > t} and Rt := Zθt for any time t ∈ [0, Cζ ). By definition, RCt = Zt and thus Cζ = inf{t ≥ 0; Rt ∈ / (0, ∞)}. As in Lemma 1, but in the opposite direction, one sees that the process (Rt , t < Cζ ) solves the same martingale problem as an Ornstein-Uhlenbeck type process (with ˜ = Ψ + λ and c/2) stopped on first entry into (−∞, 0). The Ornstein-Uhlenbeck type process is parameters Ψ uniquely defined in law (see [Sat13, Chapter 3, section 17] where existence and uniqueness of solution R s pathwise du > t}, which entails that the to (9) are established). Moreover, one can readily check that Ct = inf{s ≥ 0, 0 R u law of (Zt , t ≥ 0) is uniquely determined by the law of (Rt , t ≥ 0).  We now gather some path properties of minimal logistic CSBPs obtained directly by time-change. Lemma 3. Assume that −Ψ is not the Laplace exponent of a subordinator. If E = ∞, then: P(Ztmin −→ 0) = 1, t→∞

If E < ∞, then R x 2Ψ(y) 1 −zx− θ cy dy dx xe R R ∞ 1 − x 2Ψ(y) dy e θ cy dx 0 x

R∞ Pz (Ztmin

−→ 0) =

t→∞

0

< 1.

Proof. By construction, {Ztmin −→ 0} = {σ0 < ∞} with σ0 := inf{t ≥ 0, Rt ≤ 0}. According to Patie [Pat05, t→∞

Proposition 3], for any z > a ≥ 0 and µ > 0 R∞ (11)

Ez [e

−µσa

]=

0 R∞ 0

Rx

xµ−1 e−zx−

xµ−1 e−ax−

θ

Rx θ

2Ψ(y) cy dy 2Ψ(y) cy dy

dx dx

.

LOGISTIC CSBPS: DUALITY AND REFLECTION AT INFINITY

9

In order to make the paper selfcontained, a simple proof of (11) is provided in the Appendix (see Lemma 30). One can easily check that if E = ∞, then R ∞ µ−1 −zx−R x 2Ψ(y) dy cy θ x e dx Ez [e−µσ0 ] = 0R −→ 1. R x 2Ψ(y) ∞ µ−1 − dy x e θ cy dx µ→0 0 Therefore for any z ∈ [0, ∞), Pz (σ0 < ∞) = 1. Now if E < ∞, then R ∞ −1 −zx−R x 2Ψ(y) dy cy θ x e dx Pz (σ0 < ∞) = lim Ez [e−µσ0 ] = 0R < 1. R x 2Ψ(y) ∞ −1 − dy µ→0 x e θ cy dx 0  Lemma 4. Assume that −Ψ is not the Laplace exponent of a subordinator. Set ζa := inf{t ≥ 0; Ztmin ≤ a}. For any z > a ≥ 0 and µ > 0, one has R ∞ µ−1 −zx−R x 2Ψ(y) dy R ζa min cy θ x e dx . Ez [e−µ 0 Zs ds ] = R0 R x 2Ψ(y) ∞ µ−1 −ax− dy cy θ x e dx 0 Rζ Proof. By the time-change σa = 0 a Zsmin ds a.s and the statement follows directly by (11).  The next lemma provides Theorem 1. Lemma 5 (Explosion). The minimal process explodes with positive probability if and only if E < ∞. Proof. On the event {σ0 < ∞}, (Ztmin , t ≥ 0) converges towards 0 almost-surely and thus does not explode. We then focus on the event {σ0 = ∞}. If λ > 0 then R ∞ 1explosion is trivial. Assume now λ = 0. By time-change, the process (Ztmin , t ≥ 0) explodes if and only if Rs ds < ∞. Assume first E = ∞. Let a > 0 and consider the successive excursions of (Rt , t ≥ 0) under a. Since the process is recurrent, there is an infinite number of such excursions. Let b0 := 0 and for any n ≥ 1, an := inf{t > bn−1 , Rt ≤ a}, bn := inf{t > an , Rt > a}. We see that Z ∞ X Z bn 1 X bn − an ds . ≥ ds ≥ Rs a 0 an Rs n≥1

n≥1

Since excursions are i.i.d and they have positive lengths, plainly Ez [e−

R∞ 0

ds Rs

1

, σ0 = ∞] ≤ Ez [e− a

P

n≥1 (bn −an )

] = 0.

Ztmin

Therefore θ∞ = ∞. Hence, < ∞ for all t > 0 and ∞ is inaccessible. Now consider the case E < ∞, the unstopped process (R , t ≥ 0) is transient, and the event {σ0 = ∞} has positive probability. One has to check t R∞ that the integral 0 R1s ds is finite almost-surely on the event {σ0 = ∞}. Recall the Laplace transform (11), one has   Z s

c

Ez (e−θRs ) = exp −θe− 2 s z +

c

Ψ(e− 2 u θ)du .

0 c

By the change of variables v = e− 2 u θ, we get Ez (e

−θRs

Rs 0

c

Ψ(e− 2 u θ)du = − 2c s

) = exp −θe

Z z+

Rθ

2

θe− c s

θ

c θe− 2 s

2Ψ(v) cv dv,

2Ψ(v) dv cv

therefore

!

c

and the same change of variables x = θe− 2 s provides, ! Z Z θ Z ∞ 2Ψ(v) 2 θ1 −θRs exp −xz + dv dx. Ez (e )ds = c 0 x cv x 0 Since E < ∞, the last integral is finite for any θ > 0. Let b > 0. By Tonelli, one has   Z bZ ∞ Z ∞ 1 − e−bRs Ez , σ0 = ∞ ds = Ez (e−θRs , σ0 = ∞)dsdθ Rs 0 0 0 Z bZ ∞ ≤ Ez (e−θRs )dsdθ 0

(12)

2 = c

Z

0 b

Z

dθ 0

0

θ

1 exp −xz + x

Z

θ

x

! 2Ψ(v) dv dx. cv

10

´ CLEMENT FOUCART

R θ R θ 2Ψ(v) The upper bound is finite since θ ∈ (0, b) 7→ 0 x1 e x cv dv dx is bounded. Thus  Z ∞ 1 − e−bRs ds, σ0 = ∞ < ∞. Ez Rs 0 R∞ −bRs We deduce then that on the event {σ0 = ∞}, 0 1−eRs ds < ∞ a.s. Since E < ∞ then on {σ0 = ∞}, R  −bRs ∞ ds ∼ R1s a.s. Therefore Pz 0 R < ∞|σ0 = ∞ = 1, and the process (Ztmin , t ≥ 0) Rs −→ ∞ a.s and 1−eRs s s→∞ s→∞ explodes.  Lemma 6. When E < ∞, explosion is almost-sure if and only if −Ψ is the Laplace exponent of a subordinator. Moreover, (Ztmin , t ≥ 0) is irreducible in ( 2δ c , ∞). Proof. We have seen in the proof of Lemma 5 that when E < ∞, the following two events coincide {ζ∞ = ∞} = {σ0 < ∞}. In the non-subordinator case, one has Pz (σ0 = ∞) < 1 since the unstopped process (Rt , 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 σ0 = ∞ a.s. Denote (Yt , t ≥ 0) the subordinator with Laplace exponent −Ψ. Since Yt ≥ z + δt for all t ≥ 0 Pz -a.s, a comparison argument in (9) entails that Rt ≥ rt for all t ≥ 0, Pz -a.s, with (rt , t ≥ 0) solution c − 2c t to drt = δdt − 2c rt dt with r0 = z. We deduce that Rt ≥ e− 2 t z + 2δ ) > 0 for all t ≥ 0, Pz -a.s. This c (1 − e entails Pz (σ0 = ∞) = 1 for any z > 0. Note that, by time-change, (Ztmin , t ≥ 0) is irreducible in ( 2δ  c , ∞). Lemma 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 (Ztmin , t ≥ 0) be a minimal logistic CSBP with λ > 0. By the time-change, (Rt , t ≥ 0) := (Zθt , t ≥ 0) is an Ornstein-Uhlenbeck type process killed at some exponential random variable eλ and stopped at its first entry in (−∞, 0). Since for any s < eλ , Rs < ∞, then Zt = RCt < ∞ for any t < θeλ . Therefore, the process cannotR explode before θeλ and on the event {σ0 = ∞}, explosion is made by a single jump which occurs at time e ds .  θ eλ = 0 λ R s Remark. We have seen that when the Ornstein-Uhlenbeck type process (Rt , 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 L´evy process (Yt , t ≥ 0) drifts ”slowly” towards ∞. 5. Infinity as an entrance boundary 5.1. Generalized Feller diffusions. Recall (Ut , t < τ ) with τ := inf{t ≥ 0; Ut ∈ / (0, ∞)} solution to the sde Z t Z tp cUs dBs − Ψ(Us )ds (13) Ut = x + 0

0

where (Bt , t ≥ 0) is a Brownian motion. The following observation is our starting point in the study of logistic continuous-state branching processes by duality. Lemma 8 (Generator duality). For all x ∈ [0, ∞[ and z ∈ [0, ∞[, let ex (z) = e−xz , then (14)

L ex (z) = A ez (x)

with

c 00 xf (x) − Ψ(x)f 0 (x). 2 Proof. One can readily check that for all x and z in ]0, ∞[, A f (x) =

c ∂ez (x) c ∂ 2 ez (x) L ex (z) = Ψ(x)zex (z) + xz 2 ex (z) = −Ψ(x) + x . 2 ∂x 2 ∂x2  Intuitively, integrating each side in (14) should provide a duality at the level of semi-groups of the form:  
min Ez e−xZt = Ex e−zUt . The study of the boundaries 0 and ∞ of (Ut , t ≥ 0) would then provide the nature of boundaries ∞ and 0 of (Ztmin , 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.

LOGISTIC CSBPS: DUALITY AND REFLECTION AT INFINITY

11

Lemma 9 (Boundaries). The boundaries 0 and ∞ of the diffusion with generator A are classified as follows:   Rθ Rθ 1) The boundary 0 is an exit if E = 0 x1 exp − 2c x Ψ(u) du dx = ∞, regular if E < ∞ and 0 ≤ 2λ u c < 1, and an entrance if 2λ c ≥ 1. R∞ 2) The boundary ∞ is inaccessible. It is an entrance boundary if R ∞ dx Ψ(x) = ∞.

dx Ψ(x)

< ∞ and a natural one if

When E < ∞, 0 is regular and there are several possibilities for extending the minimal diffusion after τ . In the next lemma, we denote by (Ut0 , t ≥ 0) the diffusion (13) with either 0 regular absorbing or exit. Lemma 10 (Exit law from (0, ∞)). Assume 0 ≤

2λ c

< 1.

1) Assume there exists z ≥ 0, such that Ψ(z) ≥ 0 (-Ψ is not the Laplace exponent of a subordinator), then for all x ≥ 0, Px ( lim Ut0 = 0) = 1. t→∞

2) Assume Ψ of the form Z

∞

(1 − e−vu )π(du)

Ψ(v) = −λ − δv − 0

with δ ≥ 0 and

R∞ 0

(1 ∧ u)π(du) < ∞. Recall the condition (A)

δ = 0 and π ¯ (0) + λ ≤ c/2

i) If (A) is satisfied then for all x ≥ 0, Px ( lim Ut0 = 0) = 1. t→∞

ii) If (A) is not satisfied then for all x ≥ 0, R∞ x

exp

Px ( lim Ut0 = 0) = 1 − Px ( lim Ut0 = ∞) = R +∞ t→∞ t→∞ 0

R

exp

y 2Ψ(z) cz dz θ

R y θ



dy  . 2Ψ(z) cz dz dy

5.2. Duality and entrance law. In this section, we assume E = ∞. Recall that it ensures the inaccessibility of ∞ for the process (Ztmin , t ≥ 0) and that 0 is an exit for the diffusion (Ut , t ≥ 0). Lemma 11 (Duality lemma). Assume E = ∞. For all z ∈ [0, ∞) and x ∈ (0, ∞), the following duality holds min

Ez [e−xZt ] = Ex [e−zUt ]. Proof. Recall (Rt , t ≥ 0) the Ornstein-Uhlenbeck type process. For any x > 0, by Itˆo’s formula, one sees that the process   Z t −xRt R e − L ex (Rs )ds, t ≥ 0 0

is a local martingale. By time-changing it, we obtain that   Z t Z −xZtmin min (Mt , t ≥ 0) := e − L ex (Zs )ds, t ≥ 0 , 0

is a local martingale. Since x > 0, then z 7→ L ex (z) is bounded and (MtZ , t ≥ 0) is a martingale. Consider now (Ut , t ≥ 0) a Ψ-generalized Feller diffusion independent of (Ztmin , t ≥ 0). By applying Itˆo’s formula, we have that for any z ≥ 0, and  > 0;   Z t∧τ U −zUt∧τ (Mt , t ≥ 0) := e − A ez (Us )ds, t ≥ 0 0

is a martingale, with τ := inf{t ≥ 0, Ut ≤ }. Recall the generator duality in Lemma 8, A ez (x) = L ex (z) and set g(z, x) := L ex (z). We apply Ethier-Kurtz’s duality result [EK86, Corollary 4.15 page 196] (with α = β = 0, τ = ∞, σ = τ ). Provided that their condition (4.50) holds, we obtain, for x ≥  Z t   min Ez [e−xZt ] − Ex [e−zUt∧τ ] = E 1t−s>τ g(Zsmin , U(t−s)∧τ ) ds 0 Z t−τ ∧t  =E L e (Zsmin )ds 0

From the last equality, it comes Ex [e

−zUt∧τ

] − Ez [e

−xZtmin

Z ] = −E 0

t−τ ∧t



L e (Zsmin )ds

min

= e−z − Ez [e−Zt−t∧τ ],

´ CLEMENT FOUCART

12

where we have obtained the last equality using the martingale (MtZ , t ≥ 0) conditionally given τ since τ is independent of (Ztmin , t ≥ 0). By letting  to 0, τ −→ τ0 a.s and the last equality provides →0

Ex [e

−zUt∧τ

] − Ez [e

−xZtmin

min ] −→ 1 − Pz (Zt−t∧τ < ∞). 0 →0

We know that under the condition E = ∞ the process (Ztmin , t ≥ 0) does not explode. Therefore the limit above min is 0 and Ez [e−xZt ] = Ex [e−zUt∧τ0 ] for all x > 0 and z ∈ [0, ∞). On the other hand, under the condition E = ∞, 0 is an exit of the diffusion and thus min Ez [e−xZt ] = Ex [e−zUt ]. It remains to verify the technical condition (4.50) in Ethier-Kurtz [EK86] page 192. Let T > 0 and  > 0
fixed. Since the random variable sups,t≤T exp −Us∧τ Ztmin is clearly bounded by 1, we only need to show the integrability of the random variable c sup |g(Ztmin , Us∧τ )|, where g(z, u) = Ψ(u)ze−uz + uz 2 e−uz . 2 s,t≤T For any mechanism Ψ, if u ≥  > 0, then |Ψ(u)| ≤ b u2 for some b > 0. Recall x ≥  > 0. For all s ≥ 0, Us∧τ ≥  a.s. under Px , therefore min min c |g(Ztmin , Us∧τ )| = |Ψ(Us∧τ )Ztmin e−Us∧τ Zt + Us∧τ (Ztmin )2 e−Ut∧τ Zt | 2 min min c 2 ≤ b Us∧τ Ztmin e−Us∧τ Zt + Us∧τ (Ztmin )2 e−Us∧τ Zt  2 c 2 min −Ztmin ≤ b Us∧τ Z e + t  Us∧τ b 2 c ≤ Us∧τ +    where in the second inequality we have used uz 2 e−uz ≤ u2 and in the third one, ze−z ≤ 1 . We now argue by 2 comparison in order to show that sups≤T Us∧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 a same Brownian motion (Bt , t ≥ 0), the process (Ut , t ≥ 0) strong solution to (13) with 0 exit and the process (Vt , t ≥ 0) strong solution to p dVt = cVt dBt + γ Vt dt, V0 = x. Both processes are adapted to the natural filtration of (Bt , 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 , Us ≤ Vs . Note that (Vt , t ≥ 0) is a supercritical Feller diffusion with branching mechanism Φ(u) = 2c u2 + γ u. It is easily checked that for any t ≥ 0, Vt has a second moment. Moreover, the process (Vs , s ≥ 0) is a submartingale and by Doob’s inequality   Ex sup Vs2 ≤ 4Ex [VT2 ] < ∞. s≤T

Since for any  > 0,

sup Us2 ≤ sup Vs2 then the proof is complete.

s≤T ∧τ



s≤T


Let Ptmin , t ≥ 0 be the semigroup of (Ztmin , t ≥ 0). Lemma 5 ensures that when E = ∞, ∞ is inaccessible. To see that ∞ is
an entrance boundary, we show in the following lemmas how to define a semigroup coinciding with Ptmin , t ≥ 0 over [0, ∞), with an entrance law from ∞. Lemma 12. For any t > 0, x 7→ Px (Ut = 0) is the Laplace transform of a certain probability measure ηt over [0, ∞). Moreover ηt → η0 := δ∞ weakly as t → 0. Proof. By taking limits as z → ∞ in the duality formula in Lemma 11, one has:  
min lim Ez e−xZt = lim Ex e−zUt = Px (Ut = 0) . z→∞

z→∞

Since 0 is an exit thanks to the assumption E = ∞, Px (Ut = 0) = Px (τ0 ≤ t) > 0. By L´evy ’s continuity theorem, x 7→ Px (τ0 ≤ t) is the Laplace transform of a certain finite measure ηt which is the weak limit of the law of Ztmin under Pz as z → ∞. Moreover, lim Px (τ0 ≤ t) = P0+ (τ0 ≤ t) = 1 (see e.g. [BS02, Section 10 in Chapter 2]) and ηt x→0

is a probability measure over [0, ∞). By continuity of the paths of (Ut , t ≥ 0), if x > 0, then lim Px (Ut = 0) = 0, t→0

and if x = 0 then lim Px (Ut = 0) = 1. This entails that ηt → δ∞ weakly as t → 0. t→0



From now on, we will work with the following definition of ex over [0, ∞]. For any x > 0, ex (z) = e−xz for all z ∈ [0, ∞] and e0 (z) = 1 for all z ∈ [0, ∞]. Note that e0+ (z) := x→0 lim ex (z) = 1{z0

LOGISTIC CSBPS: DUALITY AND REFLECTION AT INFINITY

13

min Lemma 13. R ∞For any function f ∈ Cb ([0, ∞]) and any t ≥ 0, set Pt f (z) := Pt f (z) for any z ∈ [0, ∞) and Pt f (∞) := 0 f (u)ηt (du). This defines a Feller semigroup (Pt , t ≥ 0) over [0, ∞]. Furthermore, if (Zt , t ≥ 0) is a c` adl` ag Markov process with semigroup (Pt , t ≥ 0), and T := inf{t > 0; Zt < ∞}, then P∞ (T = 0) = 1.

Proof. The subalgebra generated by the maps {ex (·), x ≥ 0} is separating [0, ∞] (recall that e0 (z) = 1 for any z ∈ [0, ∞]) and therefore by the Stone-Weierstrass theorem, is dense in Cb ([0, ∞]) for the supremum norm. By duality, for any x ≥ 0, Pt ex (z) = Ex [e−zUt ] when z ∈ [0, ∞). The map z 7→ Pt ex (z) is therefore continuous on [0, ∞[. The continuity at z = ∞ holds since by definition Pt ex (∞) = lim Pt ex (z). By density, z 7→ Pt f (z) is z→∞

continuous on [0, ∞] with any f ∈ Cb ([0, ∞]). Hence Pt Cb ([0, ∞]) ⊂ Cb ([0, ∞]). We show now that (Pt , t ≥ 0) is a semigroup. Since it coincides with the semigroup (Ptmin , t ≥ 0) on [0, ∞[, then for any s, t ≥ 0, any function f ∈ Cb ([0, ∞]) and any z ∈ [0, ∞) Pt+s f (z) = Pt Ps f (z). For z = ∞, we have Z Pt+s f (∞) = lim Pt+s f (z) = lim Pt Ps f (z) = Ps f (y) ηt (dy) . z→∞

z→∞

The last equality above holds since Ps f ∈ Cb ([0, ∞]). This provides Pt+s f (∞) = Pt Ps f (∞). It remains to justify the continuity of (Pt , t ≥ 0) at 0. That is to say Pt f (z) −→ f (z) for any z ∈ [0, ∞] and any f ∈ Cb ([0, ∞]). Since t→0

(Ut , t ≥ 0)
is a diffusion (in particular with continuous paths and with infinite life-time) then t 7→ Pt ex (z) = Ex e−zUt is continuous, in particular continuous at 0. Hence, t 7→ Pt f (z) is continuous at zero for any z ∈ [0, ∞). For z = ∞, since ηt −→ δ∞ weakly, then Pt f (∞) −→ f (∞). The Feller property of (Pt , t ≥ 0) ensures the t→0

t→0

existence of a Markov process (Zt , t ≥ 0) with semigroup (Pt , t ≥ 0) and c`adl`ag paths. We show now that ∞ is instantaneous. Since for every t > 0, ηt is a probability over R+ , then P∞ (T < t) = P∞ (Zt < ∞) = ηt (R+ ) = 1. Letting t to 0 provides P∞ (T = 0) = 1.  We give in the next lemma, an alternative proof for the property of entrance at ∞, based on arguments that are not involving duality. Lemma 14. Define ζa := inf{t ≥ 0; Ztmin ≤ a} for any a ≥ 0. For any large enough positive a, one has 4 . sup Ez (ζa ) ≤ ca z≥a

Proof. Recall G the generator of a CSBP with mechanism Ψ. Let h(z) = z1 . One has  Z ∞  γ 1 1 1 σ2 z − + 1{h≤1} h 2 π(dh) G h(z) = 2 + + z z z+h z z 0  Z Z 1  ∞ 2 γ 1 σ h 1 π(dh) + − π(dh). = 2 + − h z z z+h z z+h 1 0 By Lebesgue’s theorem, G h(z) −→ 0. Since L h(z) = G h(z) + 2c , then there exists a > 0 such that for all z ≥ a, z→∞

L h(z) ≥ 4c . Since by assumption, E = ∞, the process (Ztmin , t ≥ 0) does not explode and there exists a localizing sequence of stopping times (Tm , m ≥ 1) such that Tm −→ ∞ almost-surely and (Mt∧Tm , t ≥ 0) is a bounded m→∞ martingale, where Z t Mt = h(Ztmin ) − L h(Zsmin )ds. 0

By the optional stopping theorem, Ez [Mζa ∧Tm ] = h(z) and we obtain, letting m to ∞ "Z # ζa 1 1 c min min Ez [h(Zζa )] − h(z) = − = Ez L h(Zs )ds ≥ Ez (ζa ). a z 4 0
4 1 1 We conclude that Ez (ζa ) ≤ c a − z for any z ≥ a. The entrance property can be deduced by following the proof of Kallenberg [Kal02, Theorem 20.13].  5.3. Longterm behavior and stationarity. We now show Corollary 1, Corollary 2 and Theorem 5 in the case E = ∞. R ∞ du Lemma 15 (Corollary 1: accessibility of 0). Let ζ0 := inf{t > 0; Zt = 0}. If Ψ(u) < ∞ then for any z ≥ 0, R ∞ du Pz (ζ0 < ∞) > 0. If Ψ(u) = ∞ then for any z > 0, Pz (ζ0 = ∞) = 1. Proof. For all z ∈ R+ , Pz (ζ0 ≤ t) = lim Ez [e−xZt ] = lim Ex [e−zUt ] = E∞ [e−zUt ]. x→∞

Therefore Pz (ζ0 ≤ t) > 0Rif and only if E∞ [e ∞ du (Ut , t ≥ 0) if and only if Ψ(u) < ∞.

−zUt

x→∞

] > 0. By Lemma 9, ∞ is an entrance boundary of the diffusion 

´ CLEMENT FOUCART

14

Lemma 16 (Corollary 2: stationarity). If the assumption (A) (with λ = 0) is not satisfied then for all x ≥ 0 R  R∞ y exp θ 2Ψ(z) dz dy
cz x R  . Ez e−xZt −→ L(x) := R ∞ y 2Ψ(z) t→∞ exp dz dy cz 0 θ Moreover L is the Laplace transform of a probability measure carried over ( 2δ c , ∞), where δ = − lim

u→∞

Ψ(u) u .

If the

assumption (A) is satisfied or −Ψ is not the Laplace exponent of a subordinator then for all x ≥ 0,
Ez e−xZt −→ 1. t→∞

Proof. According to Lemma 10, (Ut , t ≥ 0) exits (0, ∞) either by 0 or by ∞. Thus, for any z ∈ (0, ∞] and x ≥ 0, Ez (e−xZt ) = Ex (e−zUt 1{ lim Ut =0} + e−zUt 1{ lim Ut =∞} ) −→ Px ( lim Ut = 0). t→∞

t→∞

t→∞

t→∞

A direct application of Lemma 10 provides the two first convergences. The support of the stationary measure is  ( 2δ c , ∞) by Lemma 6. Lemma 17 (Theorem 5). Assume Ψ(z) ≥ 0 for some z > 0 then R ∞ du 1) If −→ 0 a.s. Ψ(u) = ∞, then Zt > 0 for any t ≥ 0 a.s. and Zt t→∞ R ∞ du 2) If Ψ(u) < ∞, then (Zt , t ≥ 0) get absorbed at 0 in finite time almost-surely. R ∞ dz Proof. Note first that Lemma 3 ensures that Zt −→ 0 a.s. If Ψ(z) = ∞, we have seen in Lemma 15 that t→∞ R ∞ dz −zUt 0 is inaccessible. Assume now ) −→ 1, thus Ψ(z) < ∞, then by Lemma 10-1) Pz (ζ0 ≤ t) = E∞ (e t→∞

Pz (ζ0 < ∞) = 1.

 6. Infinity as regular reflecting or exit boundary

In this subsection, we assume that E is finite. Hence, 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. Recall Ψk the branching mechanism associated to the triplet (σ, γ, πk ) with πk (du) = π|]0,k[ (du) + (¯ π (k) + λ)δk , that is Z
σ2 2 π (k) + λ). Ψk (x) = x − γx + e−xu − 1 + xu1u∈]0,1[ π(du) + (e−xk − 1)(¯ 2 ]0,k[ Note that for all x > 0, Ψk (x) −→ Ψ(x). k→∞

(k)

Lemma 18. For any k ≥ 0, denote (Ut , t ≥ 0) the unique solution to q (k) (k) (k) dUt = cUt dBt − Ψk (Ut )dt. There exists a probability space on which, with probability 1: (k)

Ut

(k+1)

≤ Ut

, for all k ≥ 1 and t ≥ 0. (k)

If 0 ≤ 2λ c < 1, then as k goes to ∞, the sequence of processes (Ut , t ≥ 0) converges pointwise almost-surely towards a process (Ut0 , t ≥ 0) solution to (13) absorbed at 0. If 2λ c ≥ 1, then as k goes to ∞, the sequence of (k) processes (Ut , t ≥ 0) converges almost-surely towards (Ut , t ≥ 0) unique solution to (13). Proof. One can plainly check that for any k ≥ 0 and x ≥ 0, Ψk (x) ≥ Ψk+1 (x). Since |Ψ0k (0+)| < ∞ for any k ≥ 0, the branching mechanisms Ψk are locally Lipschitz on [0, ∞) and therefore by applying the comparison theorem for SDEs ([RY99, Theorem IX.3.7]), one has on some probability space   (k+1) (k) Px Ut ≥ Ut for all t ≥ 0 and all k ≥ 0 = 1. (∞)

The existence of the limiting process (Ut , t ≥ 0) (in a pointwise sense) is ensured by monotonicity. Set (k) (∞) (k) τ k := inf{t ≥ 0, Ut ∈ / (0, ∞)} and τ ∞ := inf{t ≥ 0, Ut ∈ / (0, ∞)}. Let A (k) be the generator of (Ut , t ≥ 0).

LOGISTIC CSBPS: DUALITY AND REFLECTION AT INFINITY

15

The diffusive part in A (k) and A are the same, therefore for any g ∈ Cc2 ((0, ∞)): ||A (k) g − A g||∞ = sup |(Ψ(x) − Ψk (x))g 0 (x)| x∈[0,∞[

! Z −xu −xk 0 (e − 1)π(du) + (1 − e )(¯ π (k) + λ) g (x) = sup −λ + x∈[0,∞[ ]k,∞] ≤ 2 sup |e−xk π ¯ (k)g 0 (x)|+λ sup |e−xk g 0 (x)|. x∈[0,∞[

x∈[0,∞[

For all k ≥ 1 π ¯ (k) ≤ π ¯ (1) and for any x > 0, lim e

−xk

k→∞

= 0, since g 0 has a compact support in (0, ∞), then (k)

||A (k) g − A g||∞ −→ 0. Hence, for large enough k ≥ 1, ||A (k) g||∞ ≤ 1 + ||A g||∞ and A (k) g(Us ) −→ k→∞

(∞)

A g(Us

k→∞

) a.s. for any s ≥ 0. One can deduce that for any function g ∈ Cc2 ((0, ∞)), the process Z t (∞) ∞ A g(Us(∞) )ds t ∈ [0, τ ) 7→ g(Ut ) − 0 (∞)

is a martingale (see for instance [EK86, Lemma 5.1 page 196]). The process (Ut , t < τ ∞ ) solves the same martingale problem as (Ut , t < τ ). The latter problem being well-posed (see for instance [Dur96, Section 6.1, (∞) Theorem 1.6]), (Ut , t < τ ∞ ) and (Ut , t < τ ) have the same law. It remains only to identify the behavior of (∞) the process (Ut , t ≥ 0) after its first explosion time τ ∞ . Recall that by Lemma 9, ∞ is inaccessible, so that (∞) (∞) ∞ τ ∞ = inf{t ≥ 0, Ut = 0} a.s. If 2λ = ∞ a.s. and (Ut , t ≥ 0) has the same c ≥ 1 then 0 is an entrance, τ ∞ law as (Ut , t ≥ 0). If 0 ≤ 2λ < ∞ with positive probability. Let t ≥ 0. On c < 1, then 0 is regular or exit and τ (k) (∞) the event {τ ∞ < ∞}, by pointwise almost-sure convergence, Ut+τ ∞ = lim Ut+τ ∞ . By monotonicity τ ∞ ≥ τ k k→∞

(k)

(k)

(k)

(∞)

a.s. for all k ≥ 1, moreover 0 is an exit for (Ut , t ≥ 0) therefore Ut+τ ∞ = Ut+τ k = 0 and Ut+τ ∞ = 0 for any (∞)

t ≥ 0. We conclude that (Ut

, t ≥ 0) is the diffusion whose generator is A with 0 absorbing.

 (k)

The next lemmas will provide proofs of Theorem 3 and Theorem 4. For any k ≥ 1, denote by (Zt , t ≥ 0) (k) the logistic CSBP defined on [0, ∞] with mechanisms Ψk . According to Lemma 13, the processes (Zt , t ≥ 0) are Feller. In the sequel we work with their c`adl`ag versions. (k)

Lemma 19. Assume E < ∞ and 0 ≤ 2λ adl` ag c < 1, the sequence ((Zt )t≥0 , k ≥ 1) converges weakly towards a c` Feller process (Zt , t ≥ 0) valued in [0, ∞] such that for all z ∈ [0, ∞], all t ≥ 0, and all x ∈ [0, ∞[ 0

Ez [e−xZt ] = Ex [e−zUt ] where (Ut0 , t ≥ 0) is the Ψ-generalized Feller diffusion (13) with 0 regular absorbing. (k)

(k)

(k)

Proof. Denote by (Pt , t ≥ 0) the semi-group of (Zt , t ≥ 0) and (pt (z, ·), z ∈ [0, ∞], t ≥ 0) its transition (k) kernel. Let t ≥ 0 and z ∈ [0, ∞] be fixed. For any k ≥ 1, by Lemma 11, one has for all x ≥ 0, Pt ex (z) := (k) (k) (k) (k) Ez (e−xZt ) = Ex (e−zUt ) where (Ut , t ≥ 0) is defined in Lemma 18. Since, for any t ≥ 0, Ut converges (k) (∞) 0 (∞) almost-surely towards Ut as k goes to infinity, then lim Ez (e−xZt ) = Ex (e−zUt ) = Ex (e−zUt ). Therefore k→∞

(k)

pt (z, ·) converges weakly as k goes to ∞ towards some probability pt (z, ·) over [0, ∞] satisfying Z 0 Pt ex (z) := e−xy pt (z, dy) = Ex [e−zUt ] for any z ∈ [0, ∞), [0,∞] Z Pt ex (∞) := e−xy pt (∞, dy) = Px (Ut0 = 0). [[0,∞]

Since z ∈ [0, ∞] 7→ Pt ex (z) is continuous, then by density, Pt f is continuous for any function f ∈ Cb ([0, ∞]). We stress that at this stage, we do not know that (Pt , t ≥ 0) forms a semigroup. We establish now that for any (k) (k) x ≥ 0, (Pt ex , k ≥ 1) converges uniformly towards Pt ex . For any x ≥ 0, define ϕk := ||Pt ex − Pt ex ||∞ . Recall (k) (∞) that Ut ≤ Ut for any t ≥ 0, Px -almost-surely. Therefore, for any z ∈ [0, ∞] and t ≥ 0, i h i h (∞) (k) (∞) (k) Ex e−zUt − e−zUt = Ex (e−zUt − e−zUt )1{U (k) c/2. Therefore Z y    Z ∞ Z ∞ 2Ψ(z) 2 s(∞) = exp dz dy ≤ Kθ,y0 + exp cθ (x) − (λ + π ¯ (x)e−θx ) ln y dy cz c θ θ y0 Z ∞ −θx 2 = Kθ,y0 + Kθ,x y − c (λ+¯π(x)e ) dy < ∞ θ

where Kθ,y0 and Kθ,x are some constants. • Assume that π ¯ (0) = +∞ and define πb (dx) by π ¯b (x) = π ¯ (x∨b). For all b > 0 and all x ≥ 0, π ¯b (x) ≤ π ¯ (x). The measure πb is finite and one can choose b such that π ¯b (0) > c. One has for y large enough: Z ∞ −θx Z ∞ −θx e − e−yx e − e−yx π ¯ (x)dx ≥ π ¯b (x)dx x x 0 0 Applying the same argument as above with λ = 0 and πb instead of π, one obtains s(∞) < ∞.

LOGISTIC CSBPS: DUALITY AND REFLECTION AT INFINITY

23

Assume now that condition (A) hold. Namely, δ = 0, π ¯ (0) < +∞ and π ¯ (0) + λ ≤ c/2. By using (22), there exists y0 such that for y ≥ y0 , Z y Z 2Ψ(z) ¯ (z) 2  y  2 x −θz π 2 2 − + e dz ≤ kθ (x) + 1 + ln dz ≤ kθ (x) + 1 − ln(θ) + (λ + π ¯ (0)) ln(y) cz c θ c z c c θ x/y Since π ¯ (0) + λ ≤ c/2, then Z

∞

y −2(¯π(0)+λ)/c dy = ∞

s(∞) ≥ Kθ,y0 + Kθ,x θ

We show now (22): Z ∞

−θz

e

−e

0

Z

x/y

e−θz

= 0

Z = |0

x/y

e−θz

−yz


π ¯ (z) dz − z

Z

x

π ¯ (z) dz z

e−θz

x/y


π ¯ (z) − e−yz dz − z

Z

x

e−yz

x/y

π ¯ (z) dz + z

Z

∞

e−θz − e−yz

x


π ¯ (z) dz z

Z xy Z ∞
π
π ¯ (z) π ¯ (u/y) ¯ (z) − e−yz dz − e−u du + e−θz − e−yz dz . z u z x x {z } | {z } | {z }

I1 (x,y)

I2 (x,y)

By changing variable, one has I1 (x, y) =

Rx 0

u

e−θ y − e−u



I3 (x,y)

π ¯ (u/y) du. u

Since y 7→ π ¯ (u/y) is non-decreasing and

converges to π ¯ (0) as y goes to ∞, one can apply the monotone convergence theorem, this provides Z x du < ∞. I1 (x, y) −→ π ¯ (0) (1 − e−u ) y→+∞ u 0 The monotone convergence theorem readily applies to I2 (x, y): Z ∞ −u e I2 (x, y) −→ π ¯ (0) du < ∞. y→+∞ u x
R∞ R ∞ e−θz π¯ (z) By Lebesgue’s theorem, I3 (x, y) = x e−θz − e−yz π¯ (z) dz < ∞. Finally, z dz −→ x z y→∞

Z kθ (x) = π ¯ (0) 0

x

1 − e−u du + u

Z

∞

x

 Z ∞ e−u π ¯ (z) du + dz < ∞. e−θz u z x 

Recall (Rt , t ≥ 0) the Ornstein-Uhlenbeck type process with parameters Ψ and c/2 as introduced in Section 4. We establish the formula (11) of the Laplace transform of σa := inf{t ≥ 0, Rt ≤ a}, needed in Lemma 3 and Lemma 4. Lemma 30. Assume that Ψ(z) ≥ 0 for some z ≥ 0. For any a ≥ 0 and µ > 0 R ∞ µ−1 −zx−R x 2Ψ(y) dy cy θ x e dx Ez [e−µσa ] = R0 . R x 2Ψ(y) ∞ µ−1 −ax− dy cy θ x e dx 0 Rx

2Ψ(y)

Proof. For any µ > 0, define gµ (x) = xµ−1 e− θ cy dy . The function gµ solves the following equation  c c Ψ(x) − µ + gµ (x) + xgµ0 (x) = 0 for all x ≥ 0. 2 2 Rθ R∞ Rθ We check now that 0 gµ (x)dx < ∞. Note that either 0 Ψ(u) du = −∞ or 0 Ψ(u) du ∈ (−∞, ∞). In both u Rb R b µ−1u cases, for any b > 0 there is a constant C > 0 such that 0 gµ (x)dx ≤ C 0 x dx < ∞ since µ > 0. By assumption Ψ is not the Laplace exponent of a subordinator, so there exists a ∈ (0, ∞) such that for all u ≥ a, Ψ(u) ≥ Ψ(a) > 2µ c . Then, for some other constant C, Z ∞ Z ∞ Z ∞ R 2Ψ(u) R x 2Ψ(a) µ−1 − ax cu du gµ (x)dx = C x e dx ≤ C xµ−1 e− a cu du dx a a Za∞ 2Ψ(a) ≤C x−( c −µ)−1 dx < ∞. a

24

´ CLEMENT FOUCART

R∞ Set fµ (z) = 0 e−xz gµ (x)dx for any z ≥ 0. This is a C02 decreasing function. For any z ∈ (0, ∞) and x ∈ (0, ∞), LR ex (z) = Ψ(x)ex (z) + 2c xzex (z). We now verify that fµ is an eigenfunction of the generator LR : Z ∞ Z ∞
c R R L fµ (z) − µfµ (z) = gµ (x) L ex (z) − µex (z) dx = gµ (x)(Ψ(x) + xz − µ)ex (z)dx 2 Z ∞ 0 Z0 ∞ c (Ψ(x) − µ)e−xz gµ (x)dx + = ze−xz xgµ (x)dx 2 0 0   Z ∞ Z ∞ x=∞ c  −xz = (Ψ(x) − µ)e−xz gµ (x)dx + −e xgµ (x) x=0 + e−xz (gµ (x) + xgµ0 (x))dx 2 0 Z0 ∞   c c 0 −xz (Ψ(x) − µ + )gµ (x) + xgµ (x) e dx = 0. = 2 2 0 By Itˆ o’s formula (see e.g. [Pat05, Lemma 7] for a similar calculation), the process (e−µt fµ (Rt ), t ≥ 0) is local martingale. Since (Rt , t ≥ 0) has no negative jumps, and the function fµ is decreasing, one has for any t ≤ σa , Rt ≥ a and fµ (Rt ) ≤ fµ (a) Pz -a.s, for all z ≥ a. Therefore, (e−µ(t∧σa ) fµ (Rt∧σa ), t ≥ 0) is a bounded martingale, and by the optional stopping theorem, one get Ez [e−µσa ] =

fµ (z) . fµ (a) 

Acknowledgements: The author would like to thank Ger´onimo Uribe-Bravo for a precious help in the proof of Lemma 11 and thanks Adrian Gonzales-Casanova for many helpful discussions. This work is partially supported by the French National Research Agency (ANR): ANR GRAAL (ANR-14-CE25-0014) and by LABEX MME-DII (ANR11-LBX-0023-01). References [Ber96]

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