Quasi-stationary distributions associated with explosive CSBP

We characterise all the quasi-stationary distributions and the Q-process associated with a continuous state branching process that explodes in finite time. We also provide a rescaling for the continuous state branching process conditioned on non-explosion when the branching mechanism is regularly varying at 0.


Introduction
Continuous-state branching processes (CSBP) are [0, ∞]-valued Markov processes that describe the evolution of the size of a continuous population. They have been introduced by Jirina [6] and Lamperti [10]. We recall some basic facts on CSBP and refer to Bingham [2], Grey [3], Kyprianou [7] and Le Gall [11] for details and proofs.
Observe that Ψ is also the Laplace exponent of a spectrally positive Lévy process, we refer to Theorem 1 in [10] for a pathwise correspondence between Lévy processes and CSBP.
By quasi-stationary distribution (QSD for short), we mean a probability measure µ on (0, ∞) such that P µ (Z t ∈ · | T > t) = µ(·) When µ is a QSD, it is a simple matter to check that under P µ the random variable T has an exponential distribution, the parameter of which is called the rate of decay of µ. The goal of the present paper is to investigate the QSD associated with a CSBP that explodes in finite time almost surely.

A brief review of the literature: the extinction case
Li [12] and Lambert [8] considered the extinction case T = T 0 < ∞ almost surely, so that Ψ (0+) ≥ 0 and (1.6) holds, and they studied the CSBP conditioned on nonextinction. We recall some of their results. When Ψ is subcritical, that is Ψ (0+) > 0, there exists a family (µ β ; 0 < β ≤ Ψ (0+)) of QSD where β is the rate of decay of µ β . These distributions are characterised by their Laplace transforms as follows Notice that Φ is well-defined thanks to (1.6). For any β > Ψ (0+) they proved that there is no QSD with rate of decay β, and that Equation (1.8) does not define the Laplace transform of a probability measure on (0, ∞). Additionally, the value β = Ψ (0+) yields the so-called Yaglom limit: When Ψ is critical, that is Ψ (0+) = 0, the preceding quantity converges to a trivial limit for all x > 0 and Equation (1.8) does not define the Laplace transform of a probability measure on (0, ∞). However, under the condition Ψ (0+) < ∞, they proved the following convergence (that extends a result originally due to Yaglom [15] for Galton-Watson processes) Finally in both critical and subcritical cases, for any given value t > 0 the process (Z r , r ∈ [0, t]) conditioned on s < T admits a limiting distribution as s → ∞, called the Q-process. The law of the Q-process is obtained as a h-transform of P as follows ∀x > 0, dQ x|Ft := Z t e Ψ (0)t x dP x|Ft
We start with an elementary remark: conditioning a CSBP on non-explosion does not affect the branching property. Consequently the law of Z t conditioned on T > t is infinitely divisible: if it admits a limit as t goes to ∞, the limit has to be infinitely divisible as well. Our result below shows that Ψ(+∞) plays a rôle analogue to Ψ (0+) in the extinction case. For any β > 0 there exists a unique quasi-stationary distribution µ β associated to the rate of decay β. This probability measure is infinitely divisible and is characterised by Additionally, the following dichotomy holds true: (i) Ψ(+∞) ∈ (−∞, 0). The limiting conditional distribution is given by The limiting conditional distribution is trivial: Let us make some comments. Firstly this theorem implies that λ → Φ(λ) is the Laplace exponent of a subordinator, and so, µ β is the distribution of a Φ-Lévy process taken at time β. Secondly there is a similarity with the extinction case: the limiting conditional distribution is trivial iff Ψ(+∞) = −∞ so that the dichotomy on the value Ψ(+∞) is the explosive counterpart of the dichotomy on the value Ψ (0+) in the extinction case. Also, note the similarity in the definition of the Laplace transforms (1.8) and (1.10). However, there are two major differences with the extinction case: firstly there is no restriction on the rates of decay. Secondly, even if the limiting conditional distribution is trivial when Ψ(+∞) = −∞, there exists a family of QSD.
The following theorem characterises the Q-process associated with an explosive CSBP. Let F t be the sigma-field generated by (Z r , r ∈ [0, t]), for any t ∈ [0, ∞).
The Q-process appears as the Ψ-CSBP from which one has removed all the jumps: only the deterministic part remains, see also the forthcoming Proposition 3.1. Notice that the Q-process cannot be defined through a h-transform of the CSBP: actually the distribution of the Q-process on D([0, t], [0, ∞)) is not even absolutely continuous with respect to that of the Ψ-CSBP, except when the Lévy measure ν is finite.
When Ψ(+∞) = −∞, Theorem 1.1 shows that the process conditioned on nonexplosion converges to a trivial limit. In the next theorem, under the assumption that the branching mechanism is regularly varying at 0 we propose a rescaling of the CSBP conditioned on non-explosion such that it converges to a non-trivial limit. Recall that we call slowly varying function at 0 any continuous map L : (0, ∞) → (0, ∞) such that for any a ∈ (0, ∞), L(au)/L(u) → 1 as u ↓ 0.  0+)) as t → ∞. Then the following convergence holds true: Observe that the limit displayed by this theorem is the Laplace transform of the QSD associated with Ψ(u) = −u 1−α .
The proof of Theorem 1.3 is inspired by calculations of Slack in [14] where it is shown that any critical Galton-Watson process with a regularly varying generating function can be properly rescaled so that, conditioned on non-extinction, it converges towards a non-trivial limit. For completeness we also adapt the result of Slack to critical CSBP conditioned on non-extinction.
We recover in particular the finite variance case (1.9) of Lambert and Li. Our result also covers the so-called stable branching mechanisms Ψ(u) = u 1+α with α ∈ (0, 1]. Organisation of the paper. We start with a study of continuous-time Galton-Watson processes (which are the discrete-state counterparts of CSBP): we provide a complete description of the QSD when this process explodes in finite time almost surely and compare the results with the continuous-state case. In the third section we prove Theorems 1.1, 1.2 and 1.3. Finally in the fourth section we prove Proposition 1.5.

The discrete case
A discrete-state branching process (Z t , t ≥ 0) is a continuous-time Markov process taking values in Z + ∪ {+∞} that verifies the branching property (we refer to Chapter V of Harris [4] for the proofs of the following facts). It can be seen as a Galton-Watson process with offspring distribution ξ where each individual has an independent exponential lifetime with parameter c > 0. Let us denote by φ(λ) = ∞ k=0 λ k ξ(k), ∀λ ∈ [0, 1] the generating function of the Galton-Watson process. We denote by P n the law on the space D([0, ∞), Z + ∪ {+∞}) of Z starting from n ∈ Z + ∪ {+∞}, and E n the related expectation operator. The semigroup of the DSBP is characterised via the Laplace transform (see Let τ be the lifetime of Z, that is, the infimum of the extinction time τ 0 and the explosion time τ ∞ . Taking the limits r ↓ 0 and r ↑ 1 in (2.1) one gets In this section, we assume that there is explosion in finite time almost surely. Results of Chapters V.9 and V.10 of [4] then entail that the smallest solution of the equation φ(x) = x equals 0 (and so ξ(0) = 0) and that 1− , r ∈ (0, 1] From the Markov property, we deduce that τ has an exponential distribution under P µ , the parameter of which is called the rate of decay of µ.  (1)). There is a unique quasi-stationary distribution µ β associated with the rate of decay β if and only if β is of the form nβ 0 , with n ∈ N. It is characterised by its Laplace For any initial condition n ∈ N we have Let us make some comments. First there exists only a countable family of QSD. This is due to the restrictive condition that our process takes values in Z + ∪ {∞}. Also, observe the similarity with Theorem 1.1: indeed a DSBP can be seen as a particular CSBP starting from an integer and whose branching mechanism is the Laplace exponent of a compound Poisson process with integer-valued jumps. In particular ν({k}) = c ξ(k + 1) for all integer k ≥ 1. Hence the quantity c(1 − ξ(1)) in the DSBP case corresponds to ν(0, ∞) in the CSBP case. Finally we mention that the Q-process associated with an explosive DSBP is the constant process, that is, the DSBP with the trivial generating function F (t, r) = r. This fact can be proved using calculations similar to those in the proof below or it can be deduced from Theorem 1.2 and the remarks above.
Proof. We start with the proof of the uniqueness of the QSD for a given rate of decay β > 0. Let µ be a QSD and let β > 0 be its rate of decay. Then we have for all t ≥ 0 Since F (Φ(r), 1−) = r we get ∀r ∈ (0, 1], e −βΦ(r) = k µ({k})r k which ensures the uniqueness of the QSD for a given rate of decay. We now prove that whenever β = nβ 0 with n ∈ N, the last expression is indeed the Laplace transform of a probability measure on N.

Quasi-stationary distributions and Q-process in the explosive case
Consider a branching mechanism Ψ of the form (1.3). It is well-known and can be easily checked from (1.1) that for any t ≥ 0 the law of Z t under P x is infinitely divisible. Consequently u(t, ·) is the Laplace exponent of a (possibly killed) subordinator (see Chapter 5.1 [7]). Thanks to the Lévy-Khintchine formula, there exist a t , d t ≥ 0 and a Borel measure w t on (0, Note that a t = u(t, 0+) is positive iff the CSBP has a positive probability to explode in finite time. In the genealogical interpretation, the measure w t gives the distribution of the clusters of individuals alive at time t who share a same ancestor at time 0, while the coefficient d t corresponds to the individuals at time t who do not share their ancestor at time 0 with other individuals. For further use, we write the integral version of (1.2): 2) The following result shows that the drift d t is left unchanged when replacing Ψ by Ψ Q of Theorem 1.2: this means that the Q-process is obtained by removing all the clusters in the population. Proof. Corollary p.1049 in [13] entails that d t = 0 for all t > 0 whenever σ > 0 or (0,1) hν(dh) = ∞. We now assume the converse, namely that Ψ fulfils (1.4) so that Ψ(u)/u → D as u → ∞. A direct computation shows that d t = lim λ→∞ u(t, λ)/λ. Then for any t ≥ 0, λ > 0 If q ∈ (0, ∞), then for all λ > q and all 0 ≤ s ≤ t we have q < u(t, λ) ≤ u(s, λ) ≤ λ thanks to (1.2) and by (3.2) we deduce that u(t, λ) ↑ ∞ as λ → ∞. If q = ∞, then for all λ > 0 and all 0 ≤ s ≤ t we have λ ≤ u(s, λ) ≤ u(t, λ) thanks to (1.2) and obviously u(t, λ) ↑ ∞ as λ → ∞. Since Ψ(u)/u ↑ D as u → ∞ the dominated convergence theorem applied to (3.3) yields that log(u(t, λ)/λ) → −Dt as λ → ∞.
Until the end of the section, we assume that Ψ verifies (1.7) and that q = ∞. Consequently under P x , Z explodes in finite time almost surely and a t = u(t, 0+) > 0 for all t > 0. An elementary calculation entails We introduce for all λ ≥ 0, Φ(λ) := 0 λ du/Ψ(u). This non-negative, increasing function admits a continuous inverse, namely the function t → a t . Also, thanks to Equation (3.2) we deduce the identities

Proof of Theorem 1.1
First we compute the necessary form of the QSD. Fix β > 0 and suppose that µ β is a QSD with rate of decay β. We get for all t ≥ 0 Consequently there is at most one QSD corresponding to the rate of decay β. Now suppose that the preceding formula defines a probability distribution on (0, ∞) then the following calculation ensures that it is quasi-stationary: We now assume Ψ(+∞) ∈ (−∞, 0) and we prove that λ → e −βΦ(λ) is indeed the Laplace transform of a probability measure µ β on (0, ∞).
From (3.2) and the definition of Φ we get that Using again (3.2) and the fact that Ψ is non-positive, we get that a t → ∞ and u(t, λ) → ∞ as t → ∞. Since Ψ(u) → −ν(0, ∞) as u → ∞, one deduces that and therefore Since Φ(λ) → 0 as λ ↓ 0, we deduce that λ → e −Φ(λ) x ν(0,∞) = e −βΦ(λ) is the Laplace transform of a probability measure on [0, ∞). Moreover, it does not charge 0 since Φ(λ) → ∞ as λ → ∞. We now suppose Ψ(+∞) = −∞. An easy adaptation of the preceding arguments ensures that for any x, Hence the limiting distribution is trivial: it is a Dirac mass at infinity. However, let us prove that λ → e −βΦ(λ) is indeed the Laplace transform of a probability measure µ β on (0, ∞). For every > 0, define the branching mechanism Observe that for any u ≥ 0, Ψ (u) ↓ Ψ(u) as ↓ 0. Thus by monotone convergence we deduce that The first part of the proof applies to Ψ , and therefore the l.h.s. of the preceding equation is the Laplace exponent taken at λ of an infinitely divisible distribution on (0, ∞). Since the r.h.s. vanishes at 0 and goes to ∞ at ∞, it is the Laplace exponent of an infinitely divisible distribution on (0, ∞).

Proof of Theorem 1.2
Fix t ≥ 0. Since we are dealing with non-decreasing processes and since the asserted limiting process is continuous, the convergence of the finite-dimensional marginals suffices to prove the theorem (see for instance Th VI.3.37 in [5]). By Proposition 3.1, we know that u Q (t, λ) = λe −Dt is the function related to Ψ Q via (1.2). Hence we only need to prove that for all n ≥ 1, all n-uplets 0 ≤ t 1 ≤ . . . ≤ t n ≤ t and all coefficients λ 1 , . . . , λ n > 0 we have Thanks to an easy recursion, we get To prove (3.5), we proceed via a recurrence on n. We check the case n = 1. Recall that u(t, λ)/λ → d t as λ → ∞. Then the concavity of λ → u(t, λ) (that can be directly checked from (3.1)) implies that ∂ λ u(t, λ) → d t as λ → ∞. Writing u(t + s, 0+) = u t 1 , u(t + s − t 1 , 0+) , the preceding arguments and the fact that Suppose now that the result holds at rank n − 1 ≥ 1, that is, (3.5) holds true for all (n−1)-uplets of times and coefficients. In particular u t 2 − t 1 , λ 2 + . . . + u t n − t n−1 , λ n + u(t + s − t n , 0+) . . . − u(t + s − t 1 , 0+) Therefore the argument of the case n = 1 applies and shows that u t 1 , λ 1 + u t 2 − t 1 , λ 2 + . . . + u t n − t n−1 , λ n + u(t + s − t n , 0+) . . . − u(t + s, 0+) which is the desired result since d r+r = d r d r for all r, r ≥ 0 by Proposition 3.1.

Proof of Theorem 1.3
Recall the notation a t = u(t, 0+) and that a t → ∞ as t → ∞. Since u → Ψ(u)/u is strictly increasing from −∞ to D, there exists a positive function f such that We rely on two lemmas, whose proofs are postponed to the end of the subsection.
From the latter lemma, we deduce where we use (3.6) at the second line. The theorem is proved.
Proof of Lemma 3.2. Recall the definition of Φ. An integration by parts yields that for all u ∈ [0, ∞) Recall from Theorem 2 in [9] that vΨ (v)/Ψ(v) → 1−α as v ↓ 0. Therefore an elementary calculation ends the proof.
Proof of Lemma 3.3. For all t ∈ [0, ∞), a t ≤ u(t, λ f (t) −1 ). We write Suppose that t → u(t, λ f (t) −1 ) − a t is bounded for large times. The fact that Ψ (v)/Ψ(v) goes to 0 as v → ∞ together with (3.7) then entail which in turn proves the lemma. We are left with the proof of the boundedness of t → u(t, λ f (t) −1 ) − a t for large times. Fix k ∈ (−D, ∞). Since Ψ (v) ↑ D as v → ∞, for t large enough we get from (3.7) that Ψ(v) ≥ Ψ(a t )−k(v −a t ) for all v ∈ [ a t , u(t, λ f (t) −1 )].
A simple calculation then yields Using log(1 + v) ≥ v/2 for v small and since Φ λf (t) −1 → 0, we get for t large enough From (3.6), we deduce that t → u(t, λ f (t) −1 ) − a t is bounded for large times.