Bastien Mallein
email: mallein@math.univ-paris13.fr

Quan Shi
email: quan.shi@amss.ac.cn

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

Keywords: 60G44, 60G51, 60J80, 60F20 Branching Lévy process, derivative martingale, spinal decomposition, Lévy process, perpetual integral

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet (σ 2 , a, Λ). We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a nontrivial limit in terms of (σ 2 , a, Λ). This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Lévy processes conditioned to stay positive.

Introduction

A branching random walk is a discrete-time particle system on the real line, which can be constructed as follows. It starts with an initial individual at position 0. This individual gives birth at time 1 to children, that are positioned according to a certain point process. Then each child gives birth at time 2 to offspring positioned around their parent according to an i.i.d. copy of that point process. The process continues, each generation of individuals giving birth independently to children positioned around their parent, according to a shifted copy of the same point process.

To describe this process, we introduce some notation. We denote by

P = x = (x 1 , x 2 , . . .) ∈ (-∞, ∞] N : x 1 ≤ x 2 ≤ • • • and lim n→∞ x n = ∞
the space of non-decreasing sequences x on (-∞, ∞] that converge to ∞. Equivalently, these sequences can be identified with Radon point measures µ on R with finite mass on (-∞, 0] through the identification

µ = n≥1 1 {xn<∞} δ xn ⇐⇒ x = inf {y ∈ R : µ((-∞, y]) ≥ n} , n ≥ 1 .
In particular, if µ = n j=1 x j , it is identified with the element x = (x 1 , . . . x n , ∞, ∞, • • • ) ∈ P. In other words, with this notation the point ∞ is taken as a cemetery state, and the Dirac mass δ ∞ is identified to 0. For y ∈ R, we denote by τ y x = (x n + y, n ≥ 1) the translation operator on P.

1

Observe that if there exists θ ≥ 0 such that c = E e -θx X 1 (dx) < ∞, then we have X k ∈ P for all k ∈ N a.s.; indeed, by the branching property we then have E e -θx X k (dx) = c k . Branching Lévy processes are the continuous-time counterparts of branching random walks. A branching Lévy process (with possibly infinite birth intensity) is a continuous-time particle system on the real line, in which particles move according to i.i.d. Lévy processes, and reproduce in a Poisson fashion, possibly on an everywhere dense countable set of times. Branching Lévy processes were introduced in [START_REF] Bertoin | Compensated fragmentation processes and limits of dilated fragmentations[END_REF], as an intermediate tool to study homogeneous growth-fragmentation processes. They can be constructed as increasing limits of Uchiyama-type continuous-time branching random walks, introduced in [START_REF] Uchiyama | Spatial growth of a branching process of particles living in R d[END_REF].

In [START_REF] Bertoin | Infinitely ramified point measures and branching Lévy processes[END_REF], it is proved that branching Lévy processes are the unique càdlàg (for the topology of vague convergence) P-valued processes Z that satisfy the following two properties.

Branching property: for all 0 ≤ s ≤ t, setting z = Z s ,

Z t = ∞ j=1 τ z j Z (j)
t-s in law, where Z (j) t-s are i.i.d. copies of Z t-s .

(

Exponential integrability: there exists θ > 0 such that for all t ≥ 0, E e -θx Z t (dx) < ∞.

(1.

3)

The class of branching Lévy processes generalizes several classical models, including the branching Brownian motion, continuous-type branching random walks [START_REF] Uchiyama | Spatial growth of a branching process of particles living in R d[END_REF], branching Lévy processes with finite birth intensity [START_REF] Kyprianou | A note on branching Lévy processes[END_REF], and, up to an exponential transform, homogeneous fragmentations [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] and compensated growth-fragmentations [START_REF] Bertoin | Compensated fragmentation processes and limits of dilated fragmentations[END_REF]. Observe that the branching property implies that for all t > 0, the process (Z nt , n ≥ 0) is a branching random walk. The law of a branching Lévy process is characterized by a triplet (σ 2 , a, Λ), with σ 2 ≥ 0, a ∈ R and Λ a sigma-finite measure on P with the following integrability conditions. We assume that

P 1 ∧ x 2 1 Λ(dx) < ∞, (1.4)
and that there exists θ ≥ 0 satisfying P e -θx 1 1 {x 1 <-1} + j≥2 e -θx j 1 {x j >-∞} Λ(dx) < ∞.

(1.5)

Informally, the branching Lévy process can be described as follows: each particle moves independently according to a Lévy process with diffusion coefficient σ, drift a, and Lévy jump measure the image of Λ by the application x → x 1 . Simultaneously, the particle gives birth to children in a Poissonian fashion, in such a way that at rate Λ(dx), a particle makes a jump of x 1 while simultaneously giving birth to offspring positioned at distances x 2 , x 3 , . . . from the pre-jump position of the parent. The set of individuals in a branching Lévy process can be indexed by using a generalization of the Ulam-Harris notation, introduced by Shi and Watson in [40, Section 2] for the study of growthfragmentation processes. This enumeration allows the description of the genealogy of particles in a branching Lévy process. For each t ≥ 0, we denote by N t the set of individuals alive at time t, and for all s ≤ t, by X s (u) the position at time s of a particle u ∈ N t or its ancestor, if the particle is not born yet at time s.

Condition (1.4) ensures that the trajectories of particles are well-defined Lévy processes. Condition (1.5) implies (1.3) (see [10, Theorem 1.2(ii)]), which ensures that for all t ≥ 0 the random measure u∈Nt δ Xt(u) almost surely belongs to P.

Remark 1.1. The definition we give here of a branching Lévy process differs from the ones in [START_REF] Bertoin | Infinitely ramified point measures and branching Lévy processes[END_REF][START_REF] Bertoin | Biggins' martingale convergence for branching Lévy processes[END_REF]. In those articles, the branching Lévy processes were constructed in such a way that they have finite mass on [0, ∞) a.s., instead of (-∞, 0]. We choose to change this convention in order to be consistent with the corresponding results for branching random walks in the boundary case, described in e.g. [START_REF] Biggins | Measure change in multitype branching[END_REF].

Under assumption (1.5), for all z ∈ C with R(z) = θ and t ≥ 0, we have by [START_REF] Bertoin | Infinitely ramified point measures and branching Lévy processes[END_REF]Theorem 1.2(ii)]

E u∈Nt e -zXt(u) = exp(tκ(z)), (1.6)
where

κ(z) := σ 2 z 2 2 -az + P j≥1 e -zx j -1 + zx 1 1 {|x 1 |<1} Λ(dx) (1.7)
is called the cumulant generating function of the branching Lévy process. Equation (1.6) and the branching property imply that W t (θ) := u∈Nt e -θXt-tκ(θ) , t ≥ 0 is a non-negative martingale. As such, its limit W ∞ := lim t→∞ W t exists a.s. and is finite. Whether the terminal value is degenerate (i.e. W ∞ (θ) = 0 a.s.) or not has fundamental importance in the study of branching processes, and has been investigated by a considerable amount of literature. Under the condition that κ (θ) :

= 1 i d dξ κ(θ + iξ)| ξ=0 exists and is finite, it is now well-known that θκ (θ) < κ(θ) (1.8)
is a necessary condition for the non-degeneracy of W ∞ (θ), and that, up to an additional integrability assumption on Λ, this condition is also sufficient. This result was first proved by Bigggins [START_REF] Biggins | Martingale convergence in the branching random walk[END_REF] in the context of branching random walks, by McKean [START_REF] Mckean | Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov[END_REF] (see also Neveu [START_REF] Neveu | Multiplicative martingales for spatial branching processes[END_REF]) for the branching Brownian motion (which is the only branching Lévy process with continuous trajectories).

Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] then gave an elementary proof of this statement, by introducing the spinal decomposition, a critical tool in the study of branching processes. Bertoin and Mallein [START_REF] Bertoin | Biggins' martingale convergence for branching Lévy processes[END_REF] then adapted the proof of Lyons to the settings of branching Lévy processes. A necessary and sufficient condition for the nondegeneracy of W ∞ regardless of the existence of κ (θ) was obtained by Alsmeyer and Iksanov [START_REF] Alsmeyer | A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks[END_REF] for branching random walks, and by Iksanov and Mallein [START_REF] Iksanov | A result on power moments of Lévy-type perpetuities and its application to the L p -convergence of Biggins' martingales in branching Lévy processes[END_REF] for branching Lévy processes, by using result on the finiteness of perpetual integrals.

In the boundary case, when (1.8) fails to hold, the role of the additive martingale is replaced by a different martingale, which is not uniformly integrable. More precisely, if there exists θ * > 0 such that

θ * κ (θ * ) = κ(θ * ), (1.9)
then the martingale W t (θ * ) converges to 0 almost surely, as stated above. However, the process

Z t := u∈Nt (θ * X t (u) + tκ(θ * ))e -θ * Xt(u)-tκ(θ * ) , t ≥ 0 (1.10)
is a signed martingale, often called the derivative martingale. The almost sure limit

Z ∞ = lim t→∞ Z t , if existing, is non-negative.
Assuming in addition that

κ (θ * ) := E u∈N 1 X 1 (u) + κ (θ * ) 2 e -θ * X 1 (u)-κ(θ * ) ∈ (0, ∞), (1.11)
Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] obtained sufficient integrability conditions for the non-degeneracy of Z ∞ for branching random walks. These conditions were shown to be necessary by Chen [START_REF] Chen | A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk[END_REF], who proved that if they do not hold, the derivative martingale converges almost surely to 0. For the branching Brownian motion, the optimal condition for the non-degeneracy of the limit of the derivative martingale was previously obtained by Yang and Ren [START_REF] Yang | Limit theorem for derivative martingale at criticality w.r.t. branching Brownian motion[END_REF]. It is worth noting that with a proper rescaling, the critical martingale W t will converge to Z ∞ (see e.g. [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] for a proof in the context of branching random walks and [START_REF] Bogus | Self-similar solutions of kinetictype equations: the boundary case[END_REF] for a continuous-time extension). When Z t converges to a non-degenerate limit Z ∞ , the random variable Z ∞ is related to the maximal displacement of the branching process. Lalley and Sellke [START_REF] Lalley | A conditional limit theorem for the frontier of a branching Brownian motion[END_REF] showed a deep connection between the derivative martingale and the asymptotic behaviour of the maximal displacement of particles for the branching Brownian motion. More precisely, they showed that for all y ∈ R,

lim t→∞ P max u∈Nt X t (u) ≤ √ 2t - 3 2 √ 2 log t + y = E exp -CZ ∞ e - √ 2y
, famously correcting an error in [START_REF] Mckean | Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov[END_REF], who used the critical additive instead of the derivative martingale (see the survey of Biggins and Kyprianou [START_REF] Biggins | Fixed points of the smoothing transform: the boundary case[END_REF] on that subject). The convergence in law of the maximum of a branching random walk was proved by Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF], which is again related to the derivative martingale. This result was then extrapolated to branching Lévy processes by Dadoun [START_REF] Dadoun | Asymptotics of self-similar growth-fragmentation processes[END_REF]. To sum up, obtaining necessary and sufficient conditions for the derivative martingale limit to be non-trivial is relevant to understand the asymptotic properties of extremal particles in branching processes. For branching Lévy processes with one-sided jumps, Shi and Watson [START_REF] Shi | Probability tilting of compensated fragmentations[END_REF] obtained sufficient conditions for the convergence of the derivative martingale, by adapting spinal decomposition arguments dating back to [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF][START_REF] Biggins | Measure change in multitype branching[END_REF]. The aim of this work is to obtain optimal integrability conditions for general branching Lévy processes, which is, however, a much harder question. Note that since (Z n , n ≥ 0) is a branching random walk, the optimal integrability conditions of Aïdékon and Chen allow us to immediately obtain a necessary and sufficient condition for the non-degeneracy of Z ∞ in terms of the law of Z 1 . However, this condition does not translate easily in terms of conditions on (σ 2 , a, Λ), as the connection between the two quantities is intricate.

As a result, we instead refine the spinal decomposition method in [START_REF] Shi | Probability tilting of compensated fragmentations[END_REF] to prove the non-degeneracy of Z ∞ , for which we still have to overcome some substantial challenges. The problem relies on the analysis of a Lévy process ξ conditioned to stay positive, in particular, a necessary and sufficient condition for the finiteness of ∞ 0 f (ξ t )dt (a so-called perpetual integral), which is, to our knowledge, not available in the literature. We build in the forthcoming Proposition 3.3 a novel zero-one law for the finiteness of such perpetuities of Lévy processes conditioned to be positive, by using new techniques from [START_REF] Kolb | A characterization of the finiteness of perpetual integrals of Lévy processes[END_REF][START_REF] Baguley | General path integrals and stable SDEs[END_REF]. This is another main contribution of this work and might be of independent interest.

Main results

The aim of this article is to obtain necessary and sufficient conditions for the non-degeneracy of Z ∞ , the limit of the derivative martingale in a branching Lévy process. We first observe that up to a spacetime linear transform, the branching Lévy process can be assumed to be in the so-called boundary case, which will simplify computations and notation later on.

More precisely, assuming (1.9) and (1.11), set for all t ≥ 0, u ∈ N t and s ≤ t

Y s (u) = θ * X s (u) + sκ(θ * ),
with θ * the unique positive solution of the equation θκ (θ) -κ(θ) = 0. One immediately obtains that Y is a branching Lévy process with characteristic triplet (σ 2 θ * , a θ * , Λ θ * ) which can be expressed explicitly in terms of (σ 2 , a, Λ) and θ * . Moreover, Λ θ * satisfies (1.4) and (1.5) with θ = 1, and we have

log E u∈N 1 e -Y 1 (u) = log E u∈N 1 e -θ * X 1 (u) -κ(θ * ) = 0, E u∈N 1 Y 1 (u)e -Y 1 (u) = E u∈N 1 (θ * X 1 (u) + κ(θ * ))e -θ * X 1 (u)-κ(θ * ) = θ * κ (θ * ) -κ(θ * ) = 0, and E u∈N 1 Y 1 (u) 2 e -Y 1 (u) = E u∈N 1 (θ * X 1 (u) + κ(θ * )) 2 e -θ * X 1 (u)-κ(θ * ) = (θ * ) 2 κ (θ * ) ∈ (0, ∞).
Therefore, up to a linear transformation, we can assume without loss of generality that the branching Lévy process X satisfies

E u∈N 1 e -X 1 (u) = 1, E u∈N 1 X 1 (u)e -X 1 (u) = 0.
(2.1)

Then we say that X is in the boundary case, in the language of [START_REF] Biggins | Fixed points of the smoothing transform: the boundary case[END_REF]. The assumption (2.1) is classical when studying spatial branching processes around their critical parameter. A classification of branching random walks that can be assumed to be in the boundary case is given in the appendix of the arXiv version of [START_REF] Jaffuel | The critical barrier for the survival of branching random walk with absorption[END_REF], or in [START_REF] Bérard | Survival probability of the branching random walk killed below a linear boundary[END_REF]. Moreover, we also assume that

E u∈N 1 X 1 (u) 2 e -X 1 (u) ∈ (0, ∞), (2.2)
which is a second moment integrability condition, equivalent to (1.11).

Note that a branching Lévy process X with characteristic triplet (σ 2 , a, Λ) satisfies (2.1), i.e. is in the boundary case, if and only if (σ 2 , a, Λ) satisfies

           σ 2 2 = P j≥1 (1 + x j )e -x j -1 Λ(dx) a = σ 2 2 + P j≥1 e -x j -1 + x 1 1 {|x 1 |<1} Λ(dx), (2.3)
or equivalently κ(1) = κ (1) = 0, in terms of the cumulant generating function κ defined in (1.7).

Similarly, (2.2) can be rewritten as

P i≥1 x 2 i e -x i Λ(dx) < ∞ ⇐⇒ κ (1) ∈ (0, ∞). (2.4)
We now state our main result.

Theorem 2.1. Let (X t (u), u ∈ N t ) t≥0 be a branching Lévy process satisfying (2.1) and (2.2), and Z t = u∈Nt X t (u)e -Xt(u) for t ≥ 0. The derivative martingale (Z t ) t≥0 converges a.s. to a nonnegative non-trivial limit Z ∞ if and only if

P Y (x) log + (Y (x) -1) 2 + Y (x) log + ( Y (x) -1) Λ(dx) < ∞, (H)
where log + : x ∈ [0, ∞) → max(0, log(x)) and for any x ∈ P, we have set

Y (x) := ∞ i=1 e -x i and Y (x) := ∞ i=1 1 {x i ≥0} x i e -x i .
Remark 2.2. By the branching property of a branching Lévy process, if Z ∞ exists, then the event {Z ∞ =0} is an inherited property of the underlying Galton-Watson tree (see [START_REF] Shi | Branching random walks[END_REF]Discussion 5.4] for a related argument for the discrete-time branching random walk). As a result, either Z ∞ = 0 a.s. or Z ∞ > 0 a.s. on the survival set {N t = ∅, ∀t ≥ 0} of the branching Lévy process.

As observed above, the convergence of the derivative martingale in a generic branching Lévy process can be obtained from Theorem 2.1. We state here the result for a general branching Lévy process, without assuming the boundary condition.

Theorem 2.3. Let (X t (u), u ∈ N t ) t≥0 be a branching Lévy process satisfying (1.9) and (1.11), and

Z t = u∈Nt (θ * X t (u) + tκ(θ * ))e -θ * Xt(u)-tκ(θ * ) for t ≥ 0. The derivative martingale (Z t ) t≥0 con- verges a.s. to a non-negative non-trivial limit Z ∞ if and only if P Y (x) log + (Y (x) -1) 2 + Y (x) log + ( Y (x) -1) Λ(dx) < ∞, (H*)
where log + : x ∈ [0, ∞) → max(0, log(x)) and for any x ∈ P, we have set

Y (x) := ∞ i=1 e -θ * x i and Y (x) := ∞ i=1 1 {x i ≥0} θ * x i e -θ * x i .
Proof. Consider the branching Lévy process (X s (u) := θ * X s (u) + sκ(θ * ), t ≥ 0, u ∈ N t , s ≤ t). In particular, X satisfies the boundary condition and its Lévy measure Λ is the pushforward of Λ via the function x → θ * x. By Theorem 2.1, the derivative martingale of X , which is identical to that of X, converges a.s. to a non-negative non-trivial limit if and only if (H) holds for Λ . Rewriting this condition in terms of Λ leads to the result.

As we consider a general class of branching Lévy processes, Theorem 2.1 generalizes a few sufficient conditions previously obtained in the literature: [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] for fragmentations, [START_REF] Shi | Probability tilting of compensated fragmentations[END_REF] for growthfragmentations, and [START_REF] Cheek | The empirical mean position of a branching Lévy process[END_REF] for branching Lévy processes with finite birth rate. It also generalizes the necessary and sufficient condition obtained by [START_REF] Yang | Limit theorem for derivative martingale at criticality w.r.t. branching Brownian motion[END_REF] for branching Brownian motion.

The rest of the article is organised as follows. We introduce some well-known fact on Lévy processes conditioned to stay positive in the next section, and establish a necessary and sufficient condition for finiteness of associated perpetual integrals. In Section 4, we introduce the spinal decomposition of the branching Lévy process associated to the additive and the derivative martingales. In Section 5.2, we prove that under assumptions (2.1) and (2.2), (H) implies that the derivative martingale converges to a non-degenerate limit using the same classical argument as in [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF]. Finally, we prove the "necessary part" of Theorem 2.1 in Section 5.3.

The many-to-one lemma and Lévy processes conditioned to stay positive

We present in this section the many-to-one lemma, that links first moments of additive functionals of a branching Lévy process with the law of an associated Lévy process. Then, we introduce some estimates on Lévy processes, in particular defining the law of the Lévy process conditioned to stay positive. We end this section by establishing a novel necessary and sufficient condition for the finiteness of a perpetual integral of the Lévy process conditioned to stay positive.

The many-to-one lemma

One can observe that for all r ∈ R, the function Ψ : r ∈ R → κ(1 + ir) can be rewritten as

Ψ(r) = - σ 2 2 r 2 + i ar + R e irx -1 + rx1 {|x|<1} π(dx), (3.1)
where

a = a -σ 2 + P j≥1 x j e -x j 1 {|x j |<1} -x 1 1 {|x 1 |<1} Λ(dx), (3.2)
and π is the sigma-finite measure on R satisfying for all measurable non-negative functions f :

R f (x)π(dx) = P j≥1 f (x j )e -x j Λ(dx).
In other words, the function Ψ can be seen as the Lévy-Khinchine exponent of a Lévy process ξ with diffusion term σ 2 , drift a and jump measure π. This fact can be related to the celebrated many-to-one lemma in the context of branching random walks, which can be tracked back at least to the work of Kahane and Peyrière [START_REF] Peyrière | Turbulence et dimension de Hausdorff[END_REF][START_REF] Kahane | Sur certaines martingales de Benoit Mandelbrot[END_REF]: roughly speaking, the many-to-one lemma links additive moments of a branching random walk with random walk estimates.

Lemma 3.1 (Many-to-one lemma). Let (X t (u), u ∈ N t ) t≥0 be a branching Lévy process satisfying (2.1) and ξ a Lévy process with Lévy-Khinchine exponent Ψ. For any non-negative measurable function f and t ≥ 0, we have

E u∈Nt f (X s (u), s ≤ t) = E e ξt f (ξ s , s ≤ t) .
We refer to [10, Lemma 2.2] for a proof in branching Lévy processes settings, and to the proof of [9, Lemma 2.2] for the computation of Ψ. The many-to-one lemma can be thought of as a preliminary version of the spinal decomposition, that we describe in Section 4.2.

Lévy processes conditioned to stay positive

In this section, we denote by ξ a Lévy process with Lévy-Khinchine exponent Ψ given by (3.1). By (2.1), (2.2) and the many-to-one lemma, this Lévy process is centred with finite variance. In particular, it is oscillating, i.e. lim sup

t→∞ ξ t = lim sup t→∞ -ξ t = ∞ a.s..
We recall in this section the definition of the law of ξ conditioned to stay above a given level. We refer to [17, Section 2] for a self-contained construction of this law. For all x ∈ R, we denote by τ x := inf{t ≥ 0 : ξ t < x} the first passage time below level x of ξ, and P x the law of (x + ξ t , t ≥ 0), a Lévy process with Lévy-Khinchine exponent Ψ starting from x. To simplify notation, we also write τ := τ 0 and P := P 0 . We introduce the renewal function associated to this process, defined for all x ≥ 0 by

R(x) = E ∞ 0 1 {τ -x >t} dL t , (3.3)
where L is the local time at zero of the reflected process (ξ t -inf s≤t ξ s , t ≥ 0). We extend this definition by setting R(x) = 0 for all x < 0. Since ξ does not drift to -∞ (i.e. it is not the case that lim t→∞ ξ t = -∞ a.s.), we know from [17, Lemma 1] that the function R satisfies for all x ∈ R and t ≥ 0

R(x) = E R(ξ t + x)1 {τ -x >t} = E x (R(ξ t )1 {τ >t} ). (3.4)
Let us recall some additional properties of R. If 0 is a regular point for the Lévy process ξ (i.e. inf{t > 0 : ξ t = 0} = 0 a.s.), then R(0) = 0, otherwise we normalize L such that R(0) = 1. Moreover, the function R is finite, continuous, increasing, and x → R(x) -R(0) is sub-additive on [0, ∞). As ξ has zero mean and finite variance, there exist

0 < c 1 < c < c 2 < ∞ such that c 1 x ≤ R(x) ≤ c 2 (x + 1) and lim x→∞ R(x) x = c . (3.5)
See [START_REF] Bertoin | Lévy processes[END_REF]Theorem I.21] and [START_REF] Doney | Stability of the overshoot for Lévy processes[END_REF]Theorem 7]. By (3.4), the process R(ξt+x) R(x) 1 {τ -x >t} , t ≥ 0 is a non-negative P-martingale. Let (F t ) be the filtration associated to ξ. For all x > 0, we denote by P ↑

x the law defined for all t ≥ 0 by

P ↑ x Ft = R(ξ t ) R(x) 1 {τ >t} • P x | Ft . (3.6)
The probability P ↑ x is, in the sense of Doob's h-transform, the law of the Lévy process ξ started from x conditioned to stay positive.

Lévy processes conditioned to stay positive have been the subject of a large literature. We simply recall from [17, Proposition 1] that, for all x > 0, the process ξ is transient under P ↑ x . Moreover, [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF]Theorem 1] gives the following path-decomposition at the last passage time of the overall minimum. Lemma 3.2 ([17, Theorem 1]). We set ξ = inf s≥0 ξ s and m = sup{t ≥ 0 : min(ξ t , ξ t-) = ξ}. For all x > 0 and 0 ≤ y ≤ x, we have

P ↑ x (ξ ≥ y) = R(x -y) R(x) 1 {y≤x} .
Moreover, the process (ξ s+m -ξ, s ≥ 0) is independent of (ξ s , s ≤ m); the law of the former process, that we write P ↑ , does not depend on x.

Perpetual integrals of a Lévy process conditioned to stay positive

Given ξ a càdlàg process on R + and f a measurable function R + → R + a perpetual integral of ξ is a variable defined as ∞ 0 f (ξ s )ds. The main result of this section is an integral criterion on f for the finiteness of perpetual integrals of the Lévy process conditioned to stay positive. Proposition 3.3. Let ξ be a centred Lévy process with finite variance starting from x > 0. We denote by P ↑ x the law of this process conditioned to stay positive, as defined in (3.6). For every eventually non-increasing non-negative bounded function

f : [0, ∞) → [0, ∞), we have ∞ 0 f (ξ s )ds < ∞ P ↑ x -a.s. ⇐⇒ ∞ 0 yf (y)dy < ∞ ∞ 0 f (ξ s )ds = ∞ P ↑ x -a.s. ⇐⇒ ∞ 0 yf (y)dy = ∞. (3.7)
We mention that the assumptions on f could be relaxed by adding requirements on the law of ξ, such as requiring the law of ξ 1 to be absolutely continuous with respect to the Lebesgue measure. We also observe that

∞ 0 f (ξ s )1 {ξs≤C} ds < ∞, P ↑ x -a.s for all C > 0, (3.8)
by [17, Proposition 1] and boundedness of f , and that C 0 yf (y)dy < ∞ by boundedness of f . As a result, without loss of generality, we can assume the function f in Proposition 3.3 to be non-increasing.

If ξ is a Brownian motion under law P, then ξ under P ↑ x is a 3-dimensional Bessel process. In that case, a similar result is known [39, Exercise XI 2.5]. For a random walk conditioned to stay positive, the corresponding result is given by [20, Proposition 2.1]. For unconditioned Lévy processes, the perpetual integrals have been studied in [START_REF] Döring | Perpetual integrals for Lévy processes[END_REF][START_REF] Kolb | A characterization of the finiteness of perpetual integrals of Lévy processes[END_REF]. Recent development of Baguley, Döring and Kyprianou [START_REF] Baguley | General path integrals and stable SDEs[END_REF] gives a new characterization for transient strong Markov processes. However, to directly apply their criterion would require characterizing all so-called supportive sets of a Markov process X, i.e. all sets A such that P({X t , t ≥ 0} ⊆ A) > 0. It is unclear how to do this for conditioned Lévy processes. For this reason, Proposition 3.3 is not an immediate consequence of [START_REF] Baguley | General path integrals and stable SDEs[END_REF] but we will develop a proof based on an intermediate result developed there (Lemma 3.5 below).

We decompose the proof of Proposition 3.3 into three lemmas. In Lemma 3.4, we show that f (ξ s )ds is finite P ↑ x -a.s. if and only if its mean is finite, using Lemma 3.5. We then show in Lemma 3.6 the equivalence between the finiteness of the mean and the integral test of the proposition. Finally, we show that ∞ 0 f (ξ s )ds < ∞ has a 0-1 law, which yields the equivalence between the two statements in (3.7). Lemma 3.4. For any bounded non-increasing measurable function f ≥ 0 and x ≥ 0, we have

P ↑ x ∞ 0 f (ξ s )ds < ∞ = 1 ⇐⇒ ∃C > 0, ∞ 0 E ↑ x (f (ξ s )1 {ξs≥C} )ds < ∞.
The proof of Lemma 3.4 is based on the following observation that we take from [START_REF] Baguley | General path integrals and stable SDEs[END_REF]. For unconditioned Lévy processes, an analogous result is given in [START_REF] Kolb | A characterization of the finiteness of perpetual integrals of Lévy processes[END_REF]Lemma 4.5].

M N,p := y ∈ R + : P y ∞ 0 f (ζ s )ds < N > p . Suppose that x ∈ M N,p , then ∞ 0 E x (f (ζ s )1 {ζs∈MN,p} )ds < 2N/p 2 .
Proof of Lemma 3.4. We first observe that if there exists C > 0 such that

∞ 0 E ↑ x (f (ξ s )1 {ξs≥C} )ds = E ↑ x ∞ 0 f (ξ s )1 {ξs≥C} ds < ∞, then ∞ 0 f (ξ s )1 {ξs≥C} ds < ∞ P ↑
x -a.s.. Combining this and (3.8) completes the proof of the reverse part of the lemma.

We now turn to the direct part and assume that there exists x ≥ 0 such that ∞ 0 f (ξ s )ds < ∞ P ↑ x -a.s.. Then, given p ∈ (0, 1), we can choose N large enough such that

P ↑ x ∞ 0 f (ξ s )ds ≥ N < 1 -p,
i.e., with M N,p defined as in Lemma 3.5, we have x ∈ M N,p . it remains to prove that there exists C < ∞ such that [C, ∞) ⊂ M N,p . Then the desired statement follows from Lemma 3.5.

Set

T N = inf t : t 0 f (ξ s )ds ≥ N .
We first show that P ↑ y (T N < ∞) is comparable to inf 0≤z≤δy P ↑ y-z (T N < ∞) for large values of y. Setting ξ = inf s≥0 ξ s , we observe that, for all y, z ≥ 0,

P ↑ y+z (T N < ∞) ≤ P ↑ y+z (T N < ∞, ξ ≥ z) + P ↑ y+z (ξ < z) ≤ P ↑ y+z T N < ∞ ξ ≥ z + P ↑ y+z (ξ < z).
Moreover, the law of ξ under P ↑ y+z ( • | ξ ≥ z) is the same as the law of ξ + z under P ↑ y ( • | ξ ≥ 0) = P ↑ y . It follows that

P ↑ y+z T N < ∞ ξ ≥ z = P ↑ y ∞ 0 f (ξ s + z)ds < N ≤ P ↑ y (T N < ∞),
as f is non-increasing. Therefore, we have

P ↑ y (T N < ∞) ≥ P ↑ y+z (T N < ∞) -P ↑ y+z (ξ < z),
where

P ↑ y+z (ξ < z) = 1 - R(y)
R(y+z) by Lemma 3.2. Let ε ∈ (0, 1) and δ < ε/2. By (3.5), there exists C 1 (ε) > 0 such that for all y > C 1 (ε) and z ∈ [0, δy] we have

R(y) R(y+z) = P ↑ y+z (ξ ≥ z) > 1 -ε.
As a result, for all y ≥ C 1 (ε), we have

inf 0≤z≤δy P ↑ y-z (T N < ∞) ≥ P ↑ y (T N < ∞) -ε. (3.9)
We now set T y,δ = inf{s ≥ 0 :

ξ s ∈ [y(1 -δ), y]}.
As ξ is a Lévy process with finite variance under law P, it is well-known that the overshoot distribution of ξ is tight under law P ↑ x . Indeed, by [START_REF] Bertoin | Some applications of duality for Lévy processes in a halfline[END_REF]Lemma 3] this holds under P. Therefore, it also holds under P ↑ x due to the duality property [START_REF] Bertoin | Some applications of duality for Lévy processes in a halfline[END_REF]Corollary 2]. Therefore, we have

P ↑ x ( T y,δ < ∞) ≥ P ↑ x ξ inf{t>0:ξt>y(1-δ)} -y(1 -δ) ≤ δy -→ y→∞ 1.
Then we set C 2 (ε) such that P ↑ x ( T y,δ = ∞) < ε for all y ≥ C 2 (ε). Recall that x ∈ M N,p . For all y ≥ C(ε) := max(C 1 (ε), C 2 (ε)), using the strong Markov property and (3.9), we deduce that

P ↑ x ∞ 0 f (ξ s )ds ≥ N ≥ E ↑ x 1 T y,δ <∞ P ↑ ξ T y,δ ∞ 0 f (ξ s )ds ≥ N ≥ E ↑ x 1 T y,δ <∞ P ↑ y ∞ 0 f (ξ s )ds ≥ N -ε ≥ P ↑ y ∞ 0 f (ξ s )ds ≥ N (1 -ε) -ε.
By choosing ε ∈ (0, 1) small enough, we have

P ↑ y ∞ 0 f (ξ s )ds ≥ N < P ↑ x ∞ 0 f (ξ s )ds ≥ N + ε 1 -ε < 1 -p.
So we have [C(ε), ∞) ⊂ M N,p , completing the proof.

Lemma 3.6. Under the same assumptions as in Proposition 3.3, for any x ≥ 0, there exists

0 < c 1 < c 2 < ∞ such that c 1 ∞ x yf (y)dy ≤ ∞ 0 E ↑ x (f (ξ s ))ds ≤ c 2 ∞ 0 yf (y)dy.
Note that, by applying Lemma 3.6 with f (x) = f (x)1 {x≥C} and using the boundedness of f , we deduce that, under the same assumptions,

∃C > 0, ∞ 0 E ↑ x f (ξ s )1 {ξs≥C} ds < ∞ ⇐⇒ ∞ 0 yf (y)dy < ∞.
Proof. We first assume that ξ is not a compound Poisson process. It then follows from the change of measure (3.6) and [START_REF] Bertoin | Lévy processes[END_REF]Theorem VI.20] that

∞ 0 E ↑ x (f (ξ t )) dt = 1 R(x) ∞ 0 E x R(ξ t )f (ξ t )1 {t<τ } dt = C [0,∞) dR(r) [0,x] dR(z)R(x + r -z)f (x + r -z),
where C > 0 is a certain constant, τ = inf{s ≥ 0 : ξ s < 0} and R stands for the renewal function of -ξ. Additionally, since the unconditioned Lévy process ξ is centred with finite variance, by [START_REF] Doney | Stability of the overshoot for Lévy processes[END_REF]Theorem 7] and [6, Theorem I.21], the measures dR(• + z) and dR(• + z) converge vaguely toward multiples of the Lebesgue measures, as z → ∞. Using (3.5) as well, we deduce that there exist two constants

0 < c < C < ∞ such that c [0,x] dz [x-z,∞) yf (y)dy ≤ [0,∞) dR(r) [0,x] dR(z)R(x + r -z)f (x + r -z) ≤ C [0,x] dz [x-z,∞) yf (y)dy.
As the function f is bounded, this leads to the desired statement.

If ξ is a compound Poisson process, then the corresponding result on random walks leads to the conclusion. We recall that the span of the Lévy process ξ is defined as r := sup{s > 0 : P(ξ 1 ∈ sZ) = 0} ≥ 0, with the convention that sup ∅ = 0. Assuming that ξ is non-lattice (i.e. the span is r = 0), it is a consequence of [20, Equation (2.9)] and estimates on the renewal functions of random walks that can be found in [START_REF] Feller | An introduction to probability theory and its applications[END_REF]Chapter XII]. If ξ is lattice with span r > 0, a similar argument leads to

∞ 0 E ↑ x (f (ξ t )) dt < ∞ ⇐⇒ ∞ k=0 kf (kr + x) < ∞.
As f is eventually non-increasing, there exists

0 < c < C < ∞ such that c ∞ x yf (y)dy ≤ ∞ k=0 kf (kr + x) ≤ C ∞ 0 yf (y)dy,
completing the proof.

We now prove that the finiteness of a perpetual integral of a Lévy process conditioned to stay positive satisfies a zero-one law. Lemma 3.7. Under the assumptions of Proposition 3.3, for all x ≥ 0 we have

P ↑ x ∞ 0 f (ξ s )ds < ∞ ∈ {0, 1}.
Proof. Since ξ is transient under P ↑

x and f is bounded, we may assume without loss of generality that f is non-increasing on the entire half-line [0, ∞).

We introduce the function

ψ : x ∈ R + → P ↑ x ∞ 0 f (ξ s )ds < ∞ .
Note that ψ is measurable, non-negative and bounded, by standard properties of Markov processes. We first claim that (ψ(ξ t ), t ≥ 0) is a closed martingale. Indeed, for all x, t ≥ 0, as t 0 f (ξ s )ds < ∞ P ↑

x -a.s., we have

P ↑ x ∞ 0 f (ξ s )ds < ∞ F t = P ↑ x ∞ 0 f (ξ t+s )ds < ∞ F t = P ↑ ξt ∞ 0 f (ξ s )ds < ∞ = ψ(ξ t ), P ↑ x -a.s..
Therefore, (ψ(ξ t ), t ≥ 0) is a closed martingale. In particular, this yields

lim t→∞ ψ(ξ t ) = 1 { ∞ 0 f (ξs)ds<∞} , P ↑ x -a.s.. (3.10)
We next prove that ψ is non-increasing in x, using Lemma 3.2. With notation therein, we observe that, for all x > 0, m 0 f (ξ s )ds<∞ P ↑ x -a.s.. It follows that

ψ(x) = P ↑ x ∞ m f (ξ s )ds < ∞ = P ↑ x ∞ 0 f (ξ s+m -ξ + ξ)ds < ∞ = P ↑ ∞ 0 f (ξ s + v x )ds < ∞ ,
where v x is, under law P ↑ , an independent variable of ξ with the same law as the variable ξ under law P ↑

x . For all x < y, v x is stochastically dominated by v y . Hence, as f is non-increasing, ∞ 0 f (ξ s +v x )ds is stochastically larger than ∞ 0 f (ξ s + v y )ds. It follows that ψ is non-increasing and non-negative, and hence lim x→∞ ψ(x) = λ exists.

Since ξ t → ∞ P ↑ x -a.s. by transience, we deduce by (3.10) that

λ = lim t→∞ ψ(ξ t ) = 1 { ∞ 0 f (ξs)ds<∞} , P ↑ x -a.s..
This implies that λ ∈ {0, 1} and that ∞ 0 f (ξ s )ds < ∞ holds with probability 0 or 1, depending on the value of λ. This completes the proof.

Proof of Proposition 3.3. By Lemmas 3.4 and 3.6, we first observe that

∞ 0 f (ξ s )ds < ∞ P ↑ x -a.s. ⇐⇒ ∞ 0 yf (y)dy < ∞.
Then, as P ↑

x ∞ 0 f (ξ s )ds < ∞ ∈ {0, 1} by Lemma 3.7, we deduce by contraposition that

∞ 0 yf (y)dy = ∞ ⇐⇒ P ↑ x ∞ 0 f (ξ s )ds < ∞ < 1 ⇐⇒ P ↑ x ∞ 0 f (ξ s )ds < ∞ = 0,
completing the proof.

Truncated derivative martingales and the spinal decomposition

In this section, we use the renewal function of a Lévy process to introduce the truncated versions of a derivative martingale. We show that the non-degeneracy of the limit of the derivative martingale is equivalent to the uniform integrability of the truncated derivative martingales. We then give the spinal decomposition, describing the law of a branching Lévy process biased by the truncated martingales.

Truncated derivative martingales

Lemma 4.1. Let b > 0. We set for t ≥ 0:

Z b t = u∈Nt R(X u (t) + b)1 {infs≤t Xu(s)≥-b} e -Xu(t) . (4.1)
The process Z b := (Z b t , t ≥ 0) is a non-negative martingale, called the truncated derivative martingale, that converges a.s. to a limit Z b ∞ ≥ 0 as t → ∞.

Proof. The fact that Z b is a non-negative martingale is a straightforward consequence of the branching property of the branching Lévy process, the many-to-one lemma and equation (3.4).

The family (Z b , b ≥ 0) approaches the derivative martingale Z in the following sense.

Lemma 4.2. Under assumptions (2.1) and (2.2), Z t converges P-a.s. to a non-negative limit Z ∞ ≥ 0 as t → ∞. Moreover, with c > 0 the constant given in (3.5), there is the identity

c Z ∞ = lim b→∞ Z b ∞ , P-a.s..
Proof. By (3.5), the function R satisfies R(x) ∼ c x as x → ∞. Hence, for all ε > 0, there exists A ε > 0 such that x ∈ [R(x)/(c + ε), R(x)/(c -ε)] for all x ≥ A ε . In particular, for every t ≥ 0,

1 c + ε Z b t ≤ Z t + bW t ≤ 1 c -ε Z b t , on the event inf s≤t M s ≥ A ε -b ,
where W t = u∈Nt e -Xt(u) is the additive martingale and M s = inf u∈Ns X s (u). By [9, Theorem 1.1], under assumption (2.1) we have lim t→∞ W t = 0 a.s.. Since e -Mt ≤ W t → 0, it follows that inf t≥0 M t > -∞ a.s.. Therefore, letting t → ∞, we have

1 c + ε Z b ∞ ≤ lim inf t→∞ Z t ≤ lim sup t→∞ Z t ≤ 1 c -ε Z b ∞ on the event inf t≥0 M t ≥ A ε -b . Observe that b → Z b t is non-decreasing for all t ≥ 0, so b → Z b ∞ is a.s. non-decreasing. Therefore, lim b→∞ Z b ∞ exists a.s.. Then letting b → ∞, as lim b→∞ P(inf t≥0 M t ≥ A ε -b) = 1, we deduce that lim b→∞ 1 c + ε Z b ∞ ≤ lim inf t→∞ Z t ≤ lim sup t→∞ Z t ≤ 1 c -ε lim b→∞ Z b ∞ , P-a.s..
Finally, letting ε → 0 leads to the desired statement.

The previous lemma allows us to study the non-degeneracy of the limit of the derivative martingale Z via the uniform integrability of the truncated martingales Z b . Remark 4.4. In proving Theorem 2.1, we will also show that the following three facts are equivalent:

1. Z ∞ > 0 a.s. on the survival set of the branching Lévy process; 2. there exists b > 0 such that Z b is uniformly integrable; 3. for every b > 0, Z b is uniformly integrable.

Spinal decompositions of the branching Lévy process

The spinal decomposition consists in an alternative description of the law of a branching process biased by a non-negative martingale as a branching process with a special individual called the spine. This alternative description in turns allows us to study whether the martingale is uniformly integrable or not, thanks to the following classical argument.

Fact 4.5. Let (M, F) be a non-negative P-martingale with E(M 0 ) = 1 and set Q := M • P to be the law P biased by the martingale M , which means that, for each t ≥ 0, dQ dP Ft = M t a.s.. We have

(M t , t ≥ 0) is uniformly integrable ⇐⇒ lim inf t→∞ M t < ∞ Q-a.s.. M ∞ = 0 P-a.s. ⇐⇒ lim sup t→∞ M t = ∞ Q-a.s.. Fact 4.
5 is a consequence of [START_REF] Durrett | Probability: theory and examples[END_REF]Theorem 5.3.3], and the fact that (1/M t , t ≥ 0) is a non-negative Q-supermartingale, thus having a finite limit Q-a.s.. In view of Fact 4.5, we will study the spinal decomposition associated with a truncated martingale Z b and explore its asymptotic behaviour.

To this end, we begin by introducing the spinal decomposition associated to the critical additive martingale (W t = u∈Nt e -Xt(u) , t ≥ 0). Let P be the law of the branching Lévy process and F its natural filtration. Using the P-martingale (W, F), we define the measure P by

P := W • P. (4.2)
This change of measure was considered in [START_REF] Bertoin | Biggins' martingale convergence for branching Lévy processes[END_REF][START_REF] Iksanov | A result on power moments of Lévy-type perpetuities and its application to the L p -convergence of Biggins' martingales in branching Lévy processes[END_REF] to study the asymptotic behaviour of additive martingales, and is the counterpart of results of Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] for branching random walks.

To obtain an alternative representation of P, we construct a new branching process, with a distinguished individual called the spine. Specifically, define a sigma-finite measure Λ on P × N by Λ(dx, dk) := j≥1 e -x j Λ(dx)δ j (dk).

(4.3)

Let β be a Brownian motion and N (dt, dx, dk) an independent Poisson random measure on R + ×P×N with intensity dt ⊗ Λ(dx, dk). Define a Lévy process ξ by the following compensated Poisson integral

ξ t = σβ t + at + [0,t]×P×N x k 1 {|x k |<1} N c (dt, dx, dk) + [0,t]×P×N x k 1 {|x k |≥1} N (dt, dx, dk), (4.4)
with a the quantity defined in (3.2). The process ξ is well-defined and finite, thanks to equations (1.4) and (1.5) (see [START_REF] Bertoin | Biggins' martingale convergence for branching Lévy processes[END_REF] for more details on this construction). Moreover, ξ is a Lévy process with Lévy-Khinchine exponent Ψ defined in (3.1).

The branching Lévy process with spine is then constructed as follows. The spine particle follows the trajectory of ξ, while making offspring according to the point process N . More precisely, for all atoms (t, x, k) of N , the spine particle jumps at time t from position ξ t-to ξ t = ξ t-+ x k , while for all j = k, it creates a new particle at position ξ t-+ x j . Each newborn particle then starts from its current birth time and location a new independent branching Lévy process with law P. The set of particles alive at time t is again denoted by N t . For u ∈ N t , let (X s (u), s ≤ t) be the trajectory of this particle. The label of the spine particle at time t is written as w t ∈ N t . The law of the branching Lévy process with spine (X, N , w) thus defined is denoted by P. The spinal decomposition is the following result.

Lemma 4.6. The law of (X, N ) is the same under laws P and P. Moreover, one has

P(w t = u | F t ) = e -Xt(u) /W t , ∀t ≥ 0. (4.5)
We refer to [START_REF] Shi | Probability tilting of compensated fragmentations[END_REF]Theorem 5.2] for the proof of the spinal decomposition for branching Lévy processes, and to [9, Lemma 2.3] for a simple argument based on the spinal decomposition of branching random walks which could be adapted to our settings. The spinal decomposition was introduced by Lyons, Pemantle and Peres in [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF] for Galton-Watson processes. The result was then refined by Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] to study additive martingales in a branching random walk, and was further extended to general martingales based on additive functionals of a branching random walk in [START_REF] Biggins | Measure change in multitype branching[END_REF].

We now discuss a similar extension in the settings of branching Lévy processes. Consider for every b > 0 the law Q b defined by

Q b = Z b R(b) • P.
Thanks to Lemma 4.6, one straightforwardly obtains a spinal decomposition result for the law

Q b . Lemma 4.7. Let b > 0, we define a measure Q b by setting for all t ≥ 0 d Q b d P Ft = R(X t (w t ) + b) R(b) 1 {infs≤tXs(wt)>-b} . (4.6)
Then the law of (X, N ) is the same under laws Q b and Q b .

Proof. Let V be an F t -measurable random variable. We observe that

E Q b (V ) = E V R(X t (w t ) + b) R(b) 1 {infs≤t Xs(wt)>-b} = E V W t u∈Nt e -Xt(u) R(X t (u) + b) R(b) 1 {infs≤t Xs(u)>-b} ,
by conditioning on (X, N ) and using Lemma 4.6. Then, as {W t = 0} ⊂ {Z b t = 0}, one has

E Q b (V ) = E Z b t R(b)W t V = E Q b (V ).
In light of Lemma 4.7 and the spinal construction of P, we can still describe Q b as the law of the particle system in which the spine particle follows the trajectory ξ, while making offspring according to the point process N . However, we stress that under law Q b , ξ is a Lévy process with characteristic exponent Ψ conditioned to stay above level -b, in the sense of Section 3.2; in other words,

ξ under Q b has the same distribution as (ξ -b) under P ↑ b . (4.7)
In the rest of the article, we apply the results of Section 3.2 to the process ξ + b under law Q b , which is a Lévy process conditioned to stay positive. Under Q b , N is also no longer a Poisson random measure with intensity dt ⊗ Λ(dx, dk). We can nevertheless compute its compensator in the following sense. Lemma 4.8. For any non-negative predictable function W on R + × P × N, we have

E Q b R + ×P×N W (t, x, k)d N (dt, dx, dk) = E Q b R + ×P×N W (t, x, k)1 b+ ξ t-+x k >0 R(b + ξ t-+ x k ) R(b + ξ t-) dt ⊗ Λ(dx, dk) . Proof. Write Z t := R( ξt+b) R(b) 1 inf s≤t ξs>-b
, which is the density process for the change of measure in Lemma 4.7. It is well-known the predictable compensator (see e.g. [27, Section II.1]) of the Poisson random measure of N under P is given by its intensity dt ⊗ Λ(dx, dk). Since we have that, for any non-negative predictable function

W on R + × P × N, E R + ×P×N Z t W (t, x, k) N (dt, dx, dk) = E R + ×P×N Z t-1 b+ ξ t-+x k >0 R(b + ξ t-+ x k ) R(b + ξ t-) W (t, x, k) N (dt, dx, dk) ,
and that

(t, x, k) → 1 b+ ξ t-+x k >0 R(b+ ξ t-+x k ) R(b+ ξ t-)
is predictable, the result follows from Girsanov's theorem for random measures (see e.g. [27, Theorem III.3.17(b)]).

Proof of the main result

Using Fact 4.5, Lemma 4.7 and Corollary 4.3, we now prove Theorem 2.1. We begin by giving alternative expressions of the condition (H) that will help us use Proposition 3.3. We then show that (H) implies the non-degeneracy of the limit of the derivative martingale and finally that the derivative martingale converges to 0 if (H) does not hold.

Equivalent integral tests for (H)

We give here some equivalent formulas for (H). Recall that, for all

x ∈ P, Y (x) = k≥1 e -x k and Y (x) = k≥1 x k 1 {x k ≥0} e -x k . Introduce the quantity Y (x) = k≥1 (1 + x k 1 {x k ≥0} )e -x k = Y (x) + Y (x).
We first rewrite (H) in terms of the variable Y (x).

Lemma 5.1. Under (2.1) and (2.2), (H) is equivalent to

P Y (x) log + (Y (x) -1) 2 Λ(dx) < ∞. Proof. Let B = {x ∈ P : Y (x)≥2}. Since {x ∈ P : Y (x)≥2 or Y (x)≥2} ⊂ B, we have 0 = B c Y (x) log + (Y (x) -1) 2 Λ(dx) = B c Y (x) log + ( Y (x) -1)Λ(dx).
So it suffices to prove that

B Y (x) log + (Y (x) -1) 2 Λ(dx) < ∞ ⇐⇒ B Y (x) log + (Y (x) -1) 2 + Y (x) log + ( Y (x) -1)Λ(dx) < ∞.
Additionally, we have

Λ(B) = Λ x ∈ P : Y (x)≥2 < ∞. (5.1)
Indeed, as sup x∈R xe -x 1 {x≥0} ≤ e -1 , we deduce that Y (x) ≤ Y 2 (x) + 1 + e -1 for all x ∈ P, where we have set

Y 2 (x) := k≥2 (1 + x k 1 {x k ≥0} )e -x k . So we have {Y (x)≥2} ⊂ Y 2 (x) ≥ 1 -e -1 .
We also note that

P Y 2 (x)Λ(dx) ≤ P k≥2 (2 + x 2 k )e -x k Λ(dx) < ∞, (5.2)
where the finiteness of the last integral follows from (2.4). As a result, we have

P Y (x)1 {Y (x)≥2} Λ(dx) ≤ P (Y 2 (x) + 1 + e -1 )1 {Y 2 (x)≥1-e -1 } Λ(dx) ≤ C P Y 2 (x)Λ(dx) < ∞. (5.3)
This implies (5.1). Therefore, we may assume, without loss of generality, that Λ is a probability distribution on P. But in that case, the equivalence is a direct consequence of [2, Lemma B.1] (for the reverse part) and [START_REF] Mallein | Genealogy of the extremal process of the branching random walk[END_REF]Lemma A.1] (for the direct part).

Using Lemma 5.1, we also write (H) as the following integral test, which will be used later on to apply Proposition 3.3. Lemma 5.2. For all r ≥ 0, we set P (r) = x ∈ P : Y (x) ≤ e r/3 + 1 . We have

(H) =⇒ ∞ 0 P (r) c rY (x) + Y (x) Λ(dx)dr < ∞.
Proof. We observe that, by the Fubini-Tonelli theorem

∞ 0 P (r) c rY (x) + Y (x)Λ(dx)dr = P 3 log + (Y (x)-1) 0 rY (x) + Y (x)drΛ(dx) = 9 2 P Y (x) log + (Y (x) -1) 2 Λ(dx) + 3 P Y (x) log + (Y (x) -1)Λ(dx).
By Lemma 5.1, the finiteness of the first integral is equivalent to (H). It remains to prove that the second term is finite under (H). By the same arguments as in the proof of Lemma 5.1, we may assume that Λ is a probability measure. Then the finiteness of the second term follows from [2, Lemma B.1].

The sufficient part

Let X be a branching Lévy process satisfying (2.1) and (2.2) and Z b its truncated derivative martingale defined in (4.1). In this section, we assume that (H) holds and prove that Z b is uniformly integrable for all b > 0, which by Corollary 4.3 is enough to deduce the sufficient part of Theorem 2.1. Lemma 5.3. Under assumption (H), Z b is uniformly integrable for all b > 0.

Proof. By Fact 4.5, to prove that

(Z b t , t ≥ 0) is uniformly integrable, it is enough to prove that Q b (lim inf t→∞ Z b t < ∞) = 1
. By Lemma 4.7, we can equally prove that Q b (lim inf t→∞ Z b t <∞)=1. We write G = σ( ξ, N ) for the sigma-algebra generated by the spine ξ and the birth times and places of its children. By the conditional Fatou lemma, we have

E Q b lim inf t→∞ Z b t G ≤ lim inf t→∞ E Q b Z b t G , Q b -a.s..
To conclude the proof, it is therefore enough to show that

lim inf t→∞ E Q b Z b t G < ∞ Q b -a.s.. (5.4)
By the spinal decomposition of Q b and Lemma 4.1, we have the following Q b -a.s. identity:

E Q b Z b t G = R(b + ξ t )e -ξt + [0,t]×P×N i =k R(b + ξ s-+ x i )e -( ξ s-+x i ) N (ds, dx, dk).
As ξ under Q b is a centred Lévy process conditioned to stay above -b, it is transient, i.e. lim t→∞ ξ t = ∞ Q b -a.s.. Therefore, letting t → ∞, Q b -a.s. we have:

lim inf t→∞ E Q b Z b t G = [0,∞)×P×N i =k
R(b+ ξ s-+x i )e -( ξ s-+x i ) N (ds, dx, dk).

(5.5)

We divide the above integral into two parts, depending on the relative values of x and ξ t-at the atom (t, x, k) of N . More precisely, recalling that

Y (x) = j≥1
1 + x j 1 {x j >0} e -x j and P (r) = x ∈ P : Y (x) ≤ e r/3 + 1 , we set

A (1) := [0,∞)×P×N 1 x∈P ( ξ s-+b) i =k R(b + ξ s-+ x i )e -( ξ s-+x i ) N (ds, dx, dk), A (2)
:= [0,∞)×P×N 1 x ∈P ( ξ s-+b) i =k R(b + ξ s-+ x i )e -( ξ s-+x i ) N (ds, dx, dk),
so that (5.5) can be rewritten as

lim inf t→∞ E Q b Z b t G = A (1) +A (2) Q b -a.s.
. We now prove that A (1) and A (2) are both Q b -a.s. finite.

An direct application of Lemma 4.8 leads to

E Q b A (1) = [0,∞) E Q b e -ξs h (1) ( ξ s +b) ds,
where we set h (1) (y) = 1 R(y) P (y) k≥1 j =k R(y + x k )R(y + x j )e -x k -x j Λ(dx).

By (3.5) and the subadditivity of x → x1 {x>0} , for all x ∈ R and y ≥ 0 we have R(x + y) ≤ c 2 (1 + y)(1 + x1 {x>0} ). Therefore,

h (1) (y) ≤ C (1 + y) 2 R(y) P (y) k≥1 j =k (1 + x k 1 {x k >0} )(1 + x j 1 {x j >0} )e -x k -x j Λ(dx) ≤ C (1 + y) 2 y P (y) 1 + x 1 1 {x 1 >0} e -x 1 Y 2 (x) + Y 2 (x)Y (x) Λ(dx),
where Y 2 (x) = k≥2 1 + x k 1 {x k >0} e -x k . As a result, we have

h (1) (y) ≤ C (1 + y) 2 y P (y) (Y (x) + C)Y 2 (x)Λ(dx) ≤ C (1 + y) 2
y (e y/3 + 1)

P Y 2 (x)Λ(dx).
Since P Y 2 (x)Λ(dx) < ∞ by (5.2), we have h (1) (y) ≤ Cy -1 (1 + y) 2 (e y/3 + 1) and hence y → ye -y h (1) (y) is integrable on [0, ∞). Combining this fact, Lemma 3.6 and (4.7), we conclude that

E Q b (A (1) ) ≤ C ∞ 0 E Q b e -ξs h (1) ( ξ s + b) ds ≤ Ce b ∞ 0 E ↑ b e -ξs h (1) (ξ s ) ds < ∞,
which implies that A (1) < ∞ Q b -a.s.. We now turn to A (2) . Observe that A (2) is the integral of a random point measure, whose total mass is given by

M (2) = [0,∞)×P×N 1 x ∈P ( ξ s-+b)
N (ds, dx, dk).

As for all z ≥ 0, i =k R(z + x i )e -(z+x i ) is finite for Λ-almost all x, the finiteness of A (2) is a consequence of the finiteness of M (2) . Using first Lemma 4.8 and (3.5) and then (4.7), we have

E Q b M (2) = ∞ 0 E Q b P ξs+b c k≥1 R(b + ξ s + x k ) R(b + ξ s ) e -x k Λ(dx) ds ≤C ∞ 0 E Q b 1 R(b+ ξ s ) P ξs+b c (b+ ξ s )Y (x) + Y (x) Λ(dx) ds ≤C ∞ 0 E ↑ b 1 ξ s P (ξs) c ξ s Y (x) + Y (x) Λ(dx) ds.
By Lemma 5.2, under assumption (H) we have

∞ 0 P (r) c rY (x) + Y (x) Λ(dx)dr < ∞.
Since the function r → P (r) c 1 r rY (x) + Y (x) Λ(dx) is bounded by (5.3) and clearly decreasing, Lemma 3.6 shows that M (2) and hence A (2) are Q b -a.s. finite.

We conclude that A (1) + A (2) < ∞ Q b -a.s., from which we deduce by (5.5) that (5.4) holds, completing the proof.

Proof of the sufficient part of Theorem 2.1. We assume that (H) holds. Using Lemma 5.3, (Z b t ) is uniformly integrable for all b > 0. Therefore, by Corollary 4.3, we obtain that Z ∞ is non-degenerate, which completes the proof.

The necessary part

In this section, we prove that if (H) does not hold, then Z ∞ = 0 P-a.s.. Taking Corollary 4.3 into account, it suffices to prove that Z b ∞ = 0 P-a.s. for all b > 0. By Fact 4.5 and Lemma 4.7, the problem boils down to proving the following lemma. Its proof relies crucially on a conditional version of the Borel-Cantelli lemma, obtained by [START_REF] Chen | A short note on the conditional Borel-Cantelli lemma[END_REF], for a sum of non-negative random variables. A simplified version of this result can be stated as follows.

Lemma 5.5 ([19]). Let (V n , n ≥ 1) be a sequence of non-negative random variables defined on a probability space (Ω, H, P) and

(H n , n ≥ 0) a filtration. Let U n = E(V n | H n-1 ), n ≥ 1. 1. n≥1 V n < ∞ a.s. on { n≥1 U n < ∞}. 2. If (V n ) is bounded and (H n )-adapted, then n≥1 U n < ∞ a.s. on n≥1 V n <∞ .
Proof of Lemma 5.4. Fix b > 0. We prove this lemma by contraposition, showing that

Q b lim sup t→∞ Z b t < ∞ > 0 ⇒ (H).
Recall from Lemmas 4.6 and 4.7 the construction of Q b from P via a change of probabilities and the spinal decomposition for both. With notation therein, for each atom (t, x, k) of N , by (3.5) we have

Z b t ≥ j≥1 R(b + ξ t-+ x j )e -( ξ t-+x j ) ≥ c 1 e -ξ t- j≥1 b + ξ t-+ x j + e -x j .
The assumption Q b (lim sup t→∞ Z b t < ∞) > 0 implies that there exist T, A > 0 such that Q b ∀(t, x, k) atom of N : t≤T or j≥1 b+ ξ t-+x j + e -x j ≤Ae ξ t-> 0.

(5.6)

Write F for the natural filtration of the branching Lévy process with spine. Let

G := (t, x, k) atoms of N : j≥1 b + ξ t-+ x j + e -x j > Ae ξ t-+b . Since j≥1 b + y + x j + e -x j ≤ max(b + y, 1)Y (x),
Q b (#G < ∞) > 0. (5.7)
We now prove that condition (5.7) implies (H). To use Lemma 5.5, we need a time-discretization argument. Specifically, enumerate the atoms of N such that Y (x) > 2 in increasing order of time by {(τ n , x n , k n ), n ≥ 1}. Then (τ n ) are F-stopping times that do not accumulate Q b -a.s.. Set τ 0 = 0 and

B n = (t,x,k) atom of N 1 {t∈[τn,τ n+1 )} 1 j≥1 b+ ξ t-+x j + e -x j >Ae ξ t-+b , n ≥ 0.
Then we have #G = n≥1 B n . Note that each B n is F τn -measurable and takes values in {0, 1}.

As the family (B n ) is bounded, it follows from the second part of Lemma 5.5 that

Q b n≥1 B n < ∞ = Q b n≥1 E Q b B n F τ n-1 < ∞ . (5.8)
We use the Markov property and Lemma 4.8 to obtain

E Q b B n F τ n-1 = E Q b τn τ n-1 h( ξ r )dr F τ n-1 , Q b -a.s., (5.9)
where the function h : R → R is defined by

h(y) := P 1 j≥1 (b+y+x j ) + e -x j >Ae y+b k≥1 e -x k R(b + y + x k ) R(b + y) Λ(dx).
We deduce by (5.7), (5.8) and (5.9) that

Q b n≥1 E Q b τn τ n-1 h( ξ r )dr F τ n-1 < ∞ > 0.
Then, using the first part of Lemma 5.5, which does not require the boundedness, we deduce that (5.11) in view of (4.7). To directly apply Proposition 3.3 to the perpetual integral, we would need f to be eventually non-increasing, which is not necessarily the case. Instead, we use various non-increasing lower bounds of f to show that the right-hand side of (5.11) successively implies the three following integral tests:

Q b ∞ 0 h( ξ r )dr < ∞ > 0. ( 5
Q b ∞ 0 f ( ξ r + b)dr < ∞ > 0, which is equivalent to P ↑ b ∞ 0 f (ξ r )dr < ∞ > 0,
P Y (x) log + ( Y (x) -1)Λ(dx) < ∞, (5.12)
P Y (x) log + (Y (x) -1)Λ(dx) < ∞, (5.13)
P Y (x) log + (Y (x) -1) 2 Λ(dx) < ∞. (5.14)
Note that (5.12) and (5.14) immediately imply (H), which will complete the proof. Equation (5.13) is needed as a first step to prove (5.14). We first prove (5.12). For y ≥ 0, we have (y + x j ) + ≥ (x j ) + . Therefore for all y ≥ 0, f (y) ≥ f 1 (y) := P 1 {y>0} 1 j≥1 (x j ) + e -x j >Ae y k≥1 e -x k (x k ) + y + 1 Λ(dx).

As f 1 is non-increasing and (5. where the finiteness of the last two terms are from (5.1) and (5.3). We now turn to the proof of (5.13). To this end, for each y ≥ 0, we divide Y (x) into The last integral is finite by (5.3), so (5.18) follows. Similarly, we deduce (5.19) by using Lemma 3.6 and the observation that |x j |e -x j 1 {x j <-1} Λ(dx), which is finite by (2.4). We now turn to (5.17). For all y ≥ 2, using the fact that (x j + y) + ≥ y1 {x j >-y/2} /2 ≥ 1 {x j >-y/2} , we obtain f (y) ≥ 1 (y + 1) P 1 {Y y + (x)>Ae y } Y y + (x)Λ(dx) = f 3 (y). Now (5.17) follows from (5.11). This completes the proof of (5.13). Finally, we turn to the proof of (5.14), by following similar steps as the proof of (5.13). Using Fubini's theorem, (5.3) and Proposition 3.3, we observe that (5.14) is a consequence of which is finite as we have already proved (5.13), on account of (5.3). Similarly, we have x 2 j e -x j 1 {x j <-1} Λ(dx), which is finite by (2.4). Then we can deduce (5.21) and (5.22), by using Fibini's theorem and Lemma 3.6, in the same way as we prove (5.18). We thus only have to prove (5.20). For all y ≥ 2, since (y + x j ) + ≥ 1 {x j ≥-y/2} (y + 1)/3, we have

f (y) ≥ 1 3 P 1 {Y y + (x)>Ae y } Y y + (x)Λ(dx).
Then (5.20) follows from (5.11), which completes the proof of (5.14).

In sum, assuming that Q b lim sup t→∞ Z b t < ∞ > 0, we deduce (5.12), (5.13) and (5.14). Therefore, (H) holds.

Proof of the necessary part of Theorem 2.1. Assume that (H) does not hold. By Lemma 5.4, we have By Fact 4.5, this implies that lim t→∞ Z b t = 0, P-a.s.. Using Corollary 4.3, we conclude that Z ∞ = 0 P-a.s., completing the proof.

Lemma 3 . 5 (

 35 [START_REF] Baguley | General path integrals and stable SDEs[END_REF] Proposition 2.7]). Let ζ be a transient strong Markov process on R + and denote by P x the law of ζ starting from ζ 0 = x. For any non-negative measurable function f and N, p > 0, let

Corollary 4 . 3 .

 43 If there exists b > 0 such that Z b is uniformly integrable, then Z ∞ is non-degenerate. On the other hand, if Z b ∞ = 0 a.s. for all b > 0, then Z ∞ = 0 a.s.. Proof. Assume first there exists b > 0 such that Z b is uniformly integrable, then P(Z b ∞ > 0) > 0. As b → Z b ∞ is non-decreasing, we deduce by Lemma 4.2 that P(Z ∞ > 0) > 0. If Z b ∞ = 0 a.s. for every b > 0, then Z ∞ = 0 a.s. by Lemma 4.2.

Lemma 5 . 4 .

 54 If (H) does not hold, then for every b > 0, lim sup t→∞ Z b t = ∞, Q b -a.s..

. 10 )P 1 {y>0} 1 j≥1

 1011 Define a function f : R → R by f (y) := (y+x j ) + e -x j >Ae y k≥1 e -x k (y + x k ) + y + 1 Λ(dx).By (3.5), there exists c > 0 such that h( • ) ≥ cf ( • + b). Therefore, (5.10) leads to:

1 j≥1P 1 Y

 11 [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] implies P ↑ b ∞ 0 f 1 (ξ r )dr < ∞ > 0, using Proposition 3.3 and Fubini's theorem, we deduce that∞ > ∞ 0 (y + 1)f 1 (y)dy = ∞ 0 dy P (x j ) + e -x j >Ae y k≥1 e -x k (x k ) + Λ(dx) = P Y (x) log + ( Y (x)/A)Λ(dx).This yields (5.12), since we haveP Y (x) log + ( Y (x)-1)Λ(dx) -P Y (x) log + ( Y (x)/A)Λ(dx) ≤ A log A • Λ A ≥ Y (x) ≥ 2 + log A (x)>A Y (x)Λ(dx) < ∞,(5.15)

1 2 dyPY↑ b ∞ 2 f 2 ↑ b ∞ 2 f 3 2 f 4 2 f 5 2 f 4 2 1 ξ r + 1 1 1 ≤ C log 2 PY

 222232425242112 {x j ≥-y/2} e -x j and Y y -(x) := j≥1 1 {x j <-y/2} e -x j .Notice that for all x ∈ P, L ∈ R and y ∈ R + , we haveY (x)1 {Y (x)≥L} = Y y + (x) + Y y -(x) 1 {Y (x)≥L} = Y y + (x)1 {Y y + (x)≥L} +Y y + (x)1 {Y (x)≥L≥Y y + (x)} +Y y -(x)1 {Y (x)≥L} . Additionally, since Y (x) ≥ Y y + (x) ≥ Y (x)/2 whenever Y y + (x) ≥ Y y -(x), we have Y y + (x)1 {Y (x)≥L≥Y y + (x)} ≤ Y y + (x)1 {Y (x)≥L≥Y y + (x)≥Y y -(x)} + Y y + (x)1 {Y y + (x)<Y y -(x)} ≤ Y (x)1 {Y (x)≥L≥Y (x)/2} + Y y -(x),As a result, we haveY (x)1 {Y (x)≥L} ≤ Y y + (x)1 {Y y + (x)≥L} + Y (x)1 {Y (x)≥L≥Y (x)/2} + 2Y y -(x).(5.16)By similar arguments as in(5.15), to show that (5.13) holds, it is enough to prove thatP Y (x)(log(Y (x)/A) -2) + Λ(dx) = ∞ (x)1 {Y (x)≥Ae y } Λ(dx) < ∞,where the equality is due to Fubini's theorem. By Proposition 3.3, this is equivalent toP (ξ r )dr < ∞ > 0, where f 2 (y) := 1 y + 1 P Y (x)1 {Y (x)≥Ae y } Λ(dx).In view of (5.16), it suffices to prove thatP (ξ r )dr < ∞ > 0, where f 3 (y) := 1 y + 1 P Y y + (x)1 {Y y + (x)≥Ae y } Λ(dx), (ξ r )dr < ∞, where f 4 (y) := 1 y + 1 P Y (x)1 {Y (x)≥Ae y ≥Y (x)/2} Λ(dx), (5.18) E ↑ b ∞ (ξ r )dr < ∞, where f 5 (y) := 1 y + 1 P Y y -(x)Λ(dx). (5.19) Using Fubini's theorem and Lemma 3.6, we have E ↑ b ∞ (ξ r )dr = P Y (x)Λ(dx) E ↑ b ∞ {Y (x)≥Ae ξr ≥Y (x)/2} dr {Y (x)≥Ae y ≥Y (x)/2} dy (x)1 {Y (x)≥A} Λ(dx).

1 2 drP 1 2 f 6 2 f 7 2 f 8 Y

 121262728 {Y (x)>Ae y } Y (x)Λ(dx) < ∞ ⇐⇒ P ↑ b ∞ {Y (x)>Ae ξr } Y (x)Λ(dx) < ∞ > 0,which, using again(5.16), is a consequence of the three conditionsP ↑ b ∞ (ξ r )dr < ∞ > 0,where f 6 (y) := P Y y + (x)1 {Y y + (x)≥Ae y } Λ(dx), (ξ r )dr < ∞, where f 7 (y) := P Y (x)1 {Y (x)≥Ae y ≥Y (x)/2} Λ(dx), (ξ r )dr < ∞, where f 8 (y) := (x)1 {Y (x)≥Ae y ≥Y (x)/2} Λ(dx) ≤ C P log + (Y (x)/A -2)Y (x)Λ(dx),

  = ∞, Q b -a.s., for all b > 0.

  we may assume that A is large enough such that G ⊂ {(t, x, k) atoms of N : Y (x) > 2}. By (5.3), we have Λ(Y (x) > 2) < ∞, where Λ is defined by(4.3). So the first coordinates of the atoms in G cannot accumulate in finite time P-a.s.. As Q b

	| Ft	is
	absolutely continuous with respect to P | Ft for any fixed time t > 0, they cannot accumulate in finite
	time Q b -a.s. either. Therefore, (5.6) implies that

Acknowledgments

The authors would like to thank Samuel Baguley, Loïc Chaumont, Leif Döring and Andreas Kyprianou for valuable inputs in the proof of Proposition 3.3. We also thank Haojie Hou and an anonymous referee for their helpful comments.

Funding

Part of this work was supported by a PEPS JCJC grant. BM is partially supported by the Projet ANR-16-CE93-0003 (ANR MALIN). QS was partially supported by the SNSF Grant P2ZHP2_171955.


Bastien Mallein
email: mallein@math.univ-paris13.fr

Pavel Tesemnikov
email: tesemnikov.p@gmail.com

We consider an inhomogeneous version of the Barak-Erdős graph, i.e. a directed Erdős-Rényi random graph on {1, . . . , n} with no loop. Given f a Riemann-integrable non-negative function on [0, 1] 2 and γ > 0, we define G(n, f, γ) as the random graph with vertex set {1, . . . , n} such that for each i < j the directed edge (i, j) is present with probability p

, independently of any other edge. We denote by Ln the length of the shortest path between vertices 1 and n, and take interest in the asymptotic behaviour of Ln as n → ∞.

On the length of the shortest path in a sparse Barak-Erdős graph

Introduction

The Barak-Erdős graph, introduced by Barak and Erdős in [START_REF] Amnon | On the maximal number of strongly independent vertices in a random acyclic directed graph[END_REF], is a random directed graph with no loop constructed as follows. Given n ∈ N and p ∈ (0, 1), the Barak-Erdős graph G(n, p) is a graph with vertex set {1, . . . , n} such that for each i < j, the edge (i, j) from vertex i to vertex j is present with probability p, independently of any other directed edge. This graph is a directed acyclic version of the well-known Erdős-Rényi graph, introduced by Erdős and Rényi in [START_REF] Erdős | On the evolution of random graphs[END_REF]. It can be used to model community food webs in ecology [START_REF] Newman | A stochastic theory of community food webs iv: Theory of food chain lengths in large webs[END_REF], or the task graph for parallel processing in computer sciences [START_REF] Gelenbe | An approximation of the processing time for a random graph model of parallel computation[END_REF].

In particular, the length (number of edges) of the longest (directed) path, denoted M n , has been the subject of multiple studies, as M n + 1 is the number of steps needed to complete the task graph assuming maximal parallelization. Newman [START_REF] Charles | Chain lengths in certain random directed graphs[END_REF] proved that Mn n converges in law to a deterministic function p → C(p). Increasingly precise bounds were obtained on this function and its generalizations by [START_REF] Foss | Extended renovation theory and limit theorems for stochastic ordered graphs[END_REF][START_REF] Chernysh | Coupling any number of balls in the infinite-bin model[END_REF][START_REF] Foss | Limiting properties of random graph models with vertex and edge weights[END_REF][START_REF] Mallein | Two-sided infinite-bin models and analyticity for Barak-Erdős graphs[END_REF][START_REF] Mallein | Barak-Erdős graphs and the infinite-bin model[END_REF][START_REF] Foss | Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation[END_REF].

In the present article, we take interest in the length L n of the shortest path between vertices 1 and n in this graph, which has been much less studied. It is worth noting that for fixed value of p, one has

P(L n = 1) = p and lim n→∞ P(L n = 2) = 1 -p, as with probability 1 -(1 -p 2 ) n-2
, there is a vertex j ∈ {2, . . . n -1} connected to both 1 and n, hence L n is equal to 1 or 2 with high probability. In particular, the length of the shortest path in dense graphs remains tight. This fact is mentioned in [START_REF] Tesemnikov | On the asymptotics for the minimal distance between extreme vertices in a generalised Barak-Erdős graph[END_REF], which takes interest in the asymptotic behaviour of L (γ) n , the length of the shortest path between vertices 1 and n in a graph with connexion constant p n = n -γ , i.e. in the limit of sparse graphs. There, it is shown that for all k ≥ 2,

lim n→∞ P(L (γ) n ≤ k) = 0 if 1 - 1 k < γ.
We extend this result in the present article by obtaining the convergence in distribution of

L (γ)
n for all γ ∈ (0, 1). We consider here the asymptotic behaviour of the length of the shortest path between 1 and n in a space-inhomogeneous version of the Barak-Erdős graph, defined as follows. Let f be a Riemann-integrable positive function on [0, 1] 2 and γ ∈ (0, 1). For each n ∈ N and i < j, we set p

(n) i,j = f (i/n,j/n) n γ
. The spaceinhomogeneous sparse Barak-Erdős graph G(n, f, γ) is defined as a graph with vertex set {1, . . . , n} such that for each i < j, the directed edge (i, j) is present with probability p (n) i,j . We remark that for γ = 0, the sequence G(n, f, 0) is a sequence of graphs converging to a graphon (a scaling limit of random graphs introduced by Lovász and Szegedy [START_REF] Lovász | Limits of dense graph sequences[END_REF]). More generally, putting a mass n γ to each edge of the graph G(n, f, γ), the sequence of weighted graphs converges to the graphon f . We refer to [START_REF] Lovász | Large networks and graph limits[END_REF] for definitions and results linked to graphons, and to [START_REF] Glasscock | What is . . . a graphon? Notices[END_REF] for a short self-contained introduction.

The main result of the article is the following estimate on the asymptotic behaviour of L n for γ = 1 -1 k . Theorem 1.1. Let f be a Riemann-integrable non-negative function on [0, 1] 2 and k ∈ N. We fix γ = 1 -1 k and we set

c k (f ) = 0<u1<...<u k-1 <1 k-1 j=0 f (u j , u j+1 )du 1 • • • du k-1 ∈ [0, ∞],
with u 0 = 0 and u k = 1. Writing L n for the length of the shortest path between vertices 1 and n in G(n, f, γ), we have

lim n→∞ P(L n = k + 1) = 1 -P(L n = k) = exp (-c k (f )) .
By coupling arguments, Theorem 1.1 can be extended to describe the convergence in distribution of L n as n → ∞ for any space-inhomogeneous Barak-Erdős graph.

Corollary 1.2. Let f be a Riemann-integrable positive function on [0, 1] 2 and γ > 0. For k ≥ 2, we have

lim n→∞ P(L n ≤ k) =      0 if k < 1 1-γ , 1 -e -c k (f ) if k = 1 1-γ , 1 otherwise.
Observe that for γ = 1, the Barak-Erdős graph becomes unconnected, so that L n = ∞ with positive probability. In the present article, we do not treat the case np n → ∞ with n 1-ε p n → 0 for all ε > 0. However, a phase transition should be observed for the asymptotic behaviour of L n when p n ≈ log n n , as the graph becomes disconnected.

Some examples and applications

A class of inhomogeneous Barak-Erdős graphs previously studied are strongly inhomogeneous graphs. In this class of graphs, the probability of presence of the edge (i, j) is given by θ (j-i) α n β , with θ > 0, β > 0 and α ∈ (-1, β). This model can be constructed as an inhomogeneous Barak-Erdős graph, setting f (x, y) = θ(y -x) α and γ = β -α. Applying Corollary 1.2, we observe that for

any k ≥ 2, if 1 -1 k-1 < β -α < 1 -1 k , we have L n → k in probability. Similarly, if β -α = 1 -1 k , we set c k = θ k S k k j=1 t α j dt 1 • • • dt k-1 = θ k Γ(1 + α) k Γ(k(1 + α)) ,
where

S k = {(t 1 , . . . t k ) ∈ [0, 1] k : t 1 +• • •+t k = 1}. We conclude by Theorem 1.1 that L n converges in distribution to e -c k δ k + (1 -e -c k )δ k+1 as n → ∞.
Remark that using the coupling given in Proposition 2.3, for a similar model with α ≤ -1, we can obtain

lim n→∞ L n = k in probability if β -α = 1 - 1 k -1 , 1 - 1 k .
This result is an extension of Tesemnikov's [START_REF] Tesemnikov | On the asymptotics for the minimal distance between extreme vertices in a generalised Barak-Erdős graph[END_REF] estimates on the length of the shortest path in the inhomogeneous Barak-Erdős graph, setting β = 0. Outside of the boundary cases, the convergence in probability of L n to k ∈ N can be obtained through first-and second-moment methods, with using Lemma 2.1. We handle the boundary cases by using the Chen-Stein method, showing in Lemma 2.2 that the law of the number of paths of length k is close to a Poisson distribution for n large enough.

Another example of interest in the case of Barak-Erdős graphs with exponential density of connexion. Setting λ, µ ∈ R and γ > 0, we consider a Barak-Erdős graph with p

(n) i,j = e λ(j-i)/n+µ n γ
. In this case, we have

c k (f ) = 0<u1<...<u k-1 <1 k-1 j=0 e λ(uj+1-uj )+µ du 1 • • • du k-1 = e λ+kµ (k -1)! ,
and Theorem 1.1 and Corollary 1.2 apply.

Proof of the main result

For each k ≤ n, we denote by

Γ k (n) = {ρ ∈ N k+1 : ρ 0 = 1 < ρ 1 < • • • < ρ k-1 < ρ k = n}
the set of increasing paths of length k from 1 to n. As a first step towards estimating the length of the shortest path in a space-inhomogeneous Barak-Erdős graph, we compute the mean number of paths of length k.

Lemma 2.1. Let γ > 0 and f a Riemann-integrable non-negative function. For n ∈ N, we write Z n (k) for the number of paths of length k between 1 and n in G(n, f, γ), we have

E(Z n (k)) ∼ n (k-1)-kγ c k (f ) as n → ∞.
Proof. By linearity of the expectation, we have

E(Z n (k)) = ρ∈Γ k (n) k-1 j=0 p (n) ρj ,ρj+1 = n (k-1)-kγ 1 n k-1 ρ∈Γ k (n) k-1 j=0 f ρj n , ρj+1 n
.

Then lim n→∞

1 n k-1 ρ∈Γ k (n) k-1 j=0 f ρj n , ρj+1 n = c k (f )
, by Riemann integration, which completes the proof.

In particular, we remark that under the assumptions of Theorem 1.1, the mean number of paths of length k in G(n, f, γ) converges to c k (f ). Using this observation, we now prove that the number of paths of length k converges to a Poisson-distributed random variable.

Lemma 2.2. With the notation and assumptions of Theorem 1.1, we have

lim n→∞ Z n (k) = P (c k (f )) in distribution.
Proof. We use the Chen-Stein method [START_REF] Louis | Poisson Approximation for Dependent Trials[END_REF][START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF] to prove the convergence in distribution of Z n (k). More precisely, we show that for all j ∈ N we have

lim n→∞ jP(Z n (k) = j) -E(Z n (k))P(Z n (k) = j -1) = 0. (2.1)
Together with a tightness argument (due to the fact that E(Z n (k)) converges), it proves that Z n (k) converges in distribution to a Poisson variable with parameter

lim n→∞ E(Z n (k)) = c k (f ). Let j ∈ N, we rewrite jP(Z n (k) = j) = E   ρ∈Γ k (n) 1 {ρ open} 1 {Zn(k)=j}   , ( 2.2)
where ρ is said to be open if all edges (ρ i , ρ i+1 ) are present in the graph. Moreover for all ρ ∈ Γ k (n), we have

|P(Z n (k) = j|ρ open) -P(Z n (k) = j -1)|
≤ P there exists a path of length k between 1 and n distinct from ρ that shares an edge with ρ .

Indeed, to construct a graph with same law as G(n, f, γ) conditionally on ρ being open, it is enough to add to the graph G(n, f, γ) the edges (ρ i , ρ i+1 ) for all 1 ≤ i ≤ k if these are not already present. If opening these edges creates new paths, then these paths would have to share at least one edge with ρ. We remark that if there exists a path of length k between 1 and n distinct from ρ andthat shares an edge with ρ, then there exists 0 ≤ i 1 < i 2 ≤ k and 2 ≤ < k such that there exists a path of length between ρ i1 and ρ i2 that does not intersect ρ. Writing Y i1,i2, the number of such paths, with the same method as in Lemma 2.1, we compute

E (Y i1,i2, ) = ρi 1 <ρ 1 <•••<ρ -1 <ρi 2 -1 q=0 p (n) ρ q ,ρ q+1 ≤ n -γ ρi 1 <ρ 1 <•••<ρ -1 <ρi 2 -1 q=0 f ρ q n , ρ q+1 n → 0 as n → ∞,
using that there are at most n -1 paths of length between ρ i1 and ρ i2 . Therefore, by union bound, we deduce that

lim n→∞ P(Z n (k) = j|ρ open) -P(Z n (k) = j -1) = 0,
which implies, by (2.2), that

ρ∈Γ k (n) P(Z n (k) = j and ρ open) -P(Z n (k) = j -1) E(Z n (k)) = o(E(Z n (k))).
As E(Z n (k)) is bounded, we obtain (2.1).

We remark that sup n∈N E(Z n (k)) < ∞, hence (Z n (k)) is tight. Consider any subsequence (n j ) so that Z nj (k) converges in distribution as j → ∞. Writing Y a random variable with this distribution, we have for all j ∈ N:

jP(Y = j) = c k (f )P(Y = j -1), using that E(Z nj (k)) → c k (f ). Hence P(Y = j) = c k (f ) j j! P(Y = 0), with P(Y > n) → 0 as n → ∞. We conclude that Y is a P(c k (f )) random variable.
As any converging subsequence of (Z n (k)) is converging to P(c k (f )) in law, we conclude that Z n (k) converges to P(c k (f )) in law as n → ∞.

Before turning to the corollary, we introduce the following coupling estimate, which loosely states that a more connected graph will have a shorter shortest path between 1 and n.

Proposition 2.3. Let G n , G n be two inhomogeneous Barak-Erdős graphs such that an edge between i and j is present with probability

p (n) i,j and p (n) i,j respectively. If p (n) i,j ≤ p (n)
i,j for any i and j, then there exists a coupling between G n and G n such that L n ≥ L n .

Proof. We assume G n to be constructed on some probability space. Take any existing edge (i, j) of G n and do the following procedure: chosen edge is kept in graph with probability p (n) i,j /p (n) i,j and deleted with remained probability. This procedure creates a random graph distributed exactly as G n and is a subgraph of G n . Therefore, as no new edge was added, the length of the shortest path cannot have decreased.

Proof of Corollary 1 . 2 .

 12 We assume first that k < 1 1-γ . Then, by Lemma 2n (j)) = 0, therefore P(L n ≤ k) → 0 by Markov inequality.The case k = 1 1-γ is covered by Theorem 1.1. Finally, if k >1 1-γ , then for all A > 0, the Barak-Erdős graph G(n, f, γ) can be coupled with G(n, Af, k-1 k ) for n large enough, using Proposition 2.3. Therefore lim infn→∞ P(L n ≤ k) ≥ 1 -e -A k c k (f ) , using Theorem 1.1 and that c k (Af ) = A k c k (f ). As f is positive, c k (f ) ispositive, and letting A → ∞ we conclude that P(L n ≤ k) → 1.

Acknowledgements. The research of both authors was supported by joint grant (19-51-15001) of the Russian Foundation for Basic Research and from the French CNRS' PRC grant.