Explosion rates for continuous state branching processes in a L´evy environment

Here we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a L´evy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the inﬁnite mean case and provide necessary and suﬃcient conditions for the process to be conservative, i.e. that the process does not explode in ﬁnite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, ﬁrst found by Palau et al. [19] in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator. 2020 Mathematics


Introduction and main results
Let us consider (Ω (b) , F (b) , (F (b) t ) t≥0 , P (b) ) a filtered probability space satisfying the usual hypothesis.A continuous-state branching process (CSBP for short) Y = (Y t , t ≥ 0) defined on (Ω (b) , F (b) , (F (b) t ) t≥0 , P (b) ) is a [0, ∞]-valued Markov process with càdlàg paths whose laws satisfy the branching property, that is the law of Y started from x + y is the same as the law of the sum of two independent copies of Y but issued from x and y, respectively.More precisely, for all λ ≥ 0 and x, y ≥ 0 Moreover, the law of Y is completely characterised by the latter identity (see for instance Theorem 12.1 in [17]), i.e. for all y > 0 E (b) y e −λYt = e −yut (λ) , where t → u t (λ) is a differentiable function satisfying the following differential equation ∂ ∂t u t (λ) + ψ(u t (λ)) = 0 and u 0 (λ) = λ, where the function ψ is called the branching mechanism of the CSBP Y .It is well-known that the function ψ is either the negative of the Laplace exponent of a subordinator or the Laplace exponent of a spectrally negative Lévy process (see e.g.Theorem 12.1 in [17]).In this manuscript, we focus on the subordinator case, that is we assume where δ ≥ 0 and µ is a measure supported in (0, ∞) satisfying (0,∞) It is important to note that the function −ψ is also known in the literature as a Bernstein function, see for instance the monograph of Schilling et al. [23] for further details about this class of functions.
In this manuscript, we are interested in a particular extension of CSBPs by considering a random external force which affects the dynamics of the previous model.More precisely, we consider continuous-state branching processes in a random environment.Roughly speaking, a process in this class is a time-inhomogeneous Markov process taking values in [0, ∞] with 0 and ∞ as absorbing states.Furthermore, such processes satisfies a quenched branching property; that is conditionally on the environment, the process started from x + y is distributed as the sum of two independent copies of the same process but issued from x and y, respectively.
Actually the subclass of CSBPs in random environment that we are interested in pertains to instances where the external stochastic perturbation is driven by an independent Lévy process.This family of processes is known as continuous-state branching processes in Lévy environments (or CBLEs for short) and its construction have been given by He et al. [13] and by Palau and Pardo [19], independently, as the unique non-negative strong solution of a stochastic differential equation which will be specified below.
We now construct the demographic or branching terms of the model.Let N (b) (ds, dz, du) be a (F (b) t ) t≥0 -adapted Poisson random measure on R 3  + with intensity dsµ(dz)du, where µ is defined as above and satisfies the integral condition in (3).On the other hand, for the environmental term, we consider another filtered probability space (Ω (e) , F (e) , (F (e) t ) t≥0 , P (e) ) satisfying the usual hypotheses.Let us consider σ ≥ 0 and α real constants; and π a measure concentrated on R \ {0} such that Suppose that (B (e) t , t ≥ 0) is a (F (e) t ) t≥0 -adapted standard Brownian motion, N (e) (ds, dz) is a (F (e) t ) t≥0 -Poisson random measure on R + ×R with intensity dsπ(dz), and N (e) (ds, dz) its compensated version.We denote by S = (S t , t ≥ 0) a Lévy process, that is a process with stationary and independent increments and càdlàg paths, with the following Lévy-Itô decomposition S t = αt + σB (e z − 1)N (e) (ds, dz).
Note that S has no jumps smaller than or equals to -1 since the size of the jumps are given by the map z → e z − 1.
In our setting, the population size has no impact on the evolution of the environment and we are considering independent processes for the demography and the environment.More precisely, we work now on the space (Ω, F , (F t ) t≥0 , P) which is the direct product of the two probability spaces defined above, that is to say, Ω := Ω (e) × Ω (b) , F := F (e) ⊗ F (b) , F t := F (e) t ⊗ F (b) t for t ≥ 0, and P := P (e) ⊗ P (b) .Therefore, the continuous-state branching process Z = (Z t , t ≥ 0) in a Lévy environment S is defined on (Ω, F , (F t ) t≥0 , P) as the unique non-negative strong solution of the following SDE According to Theorem 1 in [19], the SDE (4) has pathwise uniqueness and strong solution up to explosion and by convention here it is identically equal to ∞ after the explosion time.
Further, the process Z satisfies the strong Markov property and the quenched branching property.For further details, we refer to He et al. [13] or Palau and Pardo [19] in more general settings.According to He et al. [13] and Palau and Pardo [19], the long-term behaviour of the process Z is deeply related to the behaviour and fluctuations of the Lévy process ξ = (ξ t , t ≥ 0), defined as follows z N (e) (ds, dz) where Note that, both processes (S t , t ≥ 0) and (ξ t , t ≥ 0) generate the same filtration.Actually, the process ξ is obtained from S by changing the drift term and the jump sizes as follows α → α and ∆S t → ln (∆S t + 1) , where ∆S t = S t − S t− , for t ≥ 0. Furthermore, we point out that the drift term α involves both branching and environment parameters.The relevance of this process stems from the fact that, in the finite mean case, the process (Z t e −ξt , t ≥ 0) is a quenched martingale which allows to study the long-term behaviour of the process Z (see Proposition 1.1 in [3]).Moreover, the law of the process (Z t e −ξt , t ≥ 0) can be characterised via a backward differential equation which is the analogue to (1) when the environment is fixed.In the infinite mean case, it is no longer true that (Z t e −ξt , t ≥ 0) is a quenched martingale, however, in this paper we show that the law of (Z t e −ξt , t ≥ 0) can be also characterised via a backward differential equation.
Recall that we say that Z is a conservative process if where P z denotes the law of Z starting at z ≥ 0. Thus when we think about the nonexplosion event, the first question that might arise is: under which conditions the process Z is conservative?When the environment is fixed, Grey in [12] provided a necessary and sufficient condition for the process to be conservative which depends on the integrability at 0 of the associated branching mechanism ψ.More precisely, a continuous-state branching process is conservative if and only if Observe that, a necessary condition is that ψ(0) = 0 and a sufficient condition is that ψ(0) = 0 and |ψ ′ (0+)| < ∞ (see for instance Theorem 12.3 in [17]).In contrast, in the particular case when the environment is driven by a Lévy process, Bansaye et al. [3] furnish a necessary condition under which the process Z is conservative.They proved that if the branching mechanism satisfies |ψ ′ (0+)| < ∞, then the associated CBLE is conservative (see Lemma 7 in [3]).It is worth noting that these results remain valid when the branching mechanism is given by the Laplace exponent of a spectrally negative Lévy process.Nevertheless, our focus here is on the case when the branching mechanism is given by ( 2) and satisfies ψ ′ (0+) = −∞.So our first aim is twofold, first we characterise the law of the process (Z t e −ξt , t ≥ 0) in the infinite mean case, which up to our knowledge is unknown, and then provide necessary and sufficient conditions for conservativeness when the branching mechanism is given as in (2) and satisfies ψ ′ (0+) = −∞.
Our second aim deals with the asymptotic behaviour of the non-explosion probability which, up to our knowledge, has only been studied for the case where the associated branching mechanism is given by the negative of the Laplace exponent of a stable subordinator (see Proposition 2.1 in [20]), that is where β ∈ (−1, 0) and C is a negative constant.According to [19], the non-explosion probability for a CBLE with branching mechanism as in (7) (stable CBLE for short) is given by where I 0,t (βξ) denotes the exponential functional of the Lévy process βξ, i.e.
Observe from (8), that the process Z explodes with positive probability.Hence our second aim is study the rates of the non-explosion probability of CBLEs in a more general setting rather than the stable case.According to Palau et al. [20] there are three different regimes for the asymptotic behaviour of the non-explosion probability that depends only on the mean of the underlying Lévy process ξ.They called these regimes: subcritical, critical and supercritical explosive depending on whether the mean of ξ is negative, zero or positive (see Proposition 2.1 in [20]).Up to our knowledge, this is the only known result in the literature about explosion rates for CBLEs.
In particular, we study the speed of the non-explosion probability of the process Z, when the branching mechanism ψ is given as in (2), in the critical and subcritical explosive regimes.In particular, we show that in the subcritical explosive regime, i.e. when the auxiliary Lévy process ξ drifts to −∞, and under an integrability condition, the limit of the non-explosion probability is positive.In the critical regime, i.e. when ξ oscillates, and in particular when ξ satisfies the so-called Spitzer's condition plus an integrability condition, the non-explosion probability decays as a regularly varying function at ∞.Our arguments use strongly fluctuation theory and the asymptotic behaviour of exponential functionals of Lévy processes.
The supercritical explosive regime, i.e. when ξ drifts to ∞, remains unknown, except in the stable case.We may expect, similarly as in the stable case, that under some positive exponential moments of ξ, the non-explosion probability decays exponentially as time increases.But it seems that this case requires other techniques than the ones developed in this article.
As we said before, the long-term behaviour of the non-explosion probability of Z is deeply related to the behaviour and fluctuations of ξ.Therefore, a few knowledge on fluctuation theory of Lévy process are required in order to state our main results.

Preliminaries on Lévy processes
For simplicity, we denote by P (e) x the law of the process ξ starting from x ∈ R, and when x = 0, we use the notation P (e) for P x be the law of x + ξ under P (e) , that is the law of ξ under P (e) −x .In the sequel, we assume that ξ is not a compound Poisson process since it is possible that in this case the process visits the same maxima or minima at distinct times which can make our analysis more involved.
Let us introduce the running infimum and supremum of ξ, by ξ = (ξ t , t ≥ 0) and ξ = (ξ t , t ≥ 0), with It is well-known that the reflected process ξ − ξ (resp.ξ − ξ) is a Markov process with respect to the filtration (F (e) t ) t≥0 , see for instance Proposition VI.1 in [5].We denote by L = (L t , t ≥ 0) and L = ( L t , t ≥ 0) the local times of ξ − ξ and ξ − ξ at 0, respectively, in the sense of Chapter IV in [5].Next, define where L −1 and L −1 are the right continuous inverse of the local times L and L, respectively.The range of the inverse local times, L −1 (resp.L −1 ), corresponds to the set of times at which new maxima (resp.new minima) occur.Hence, the range of the process H (resp. H) corresponds to the set of new maxima (resp.new minima).The pairs (L −1 , H) and ( L −1 , H) are bivariate subordinators known as the ascending and descending ladder processes, respectively.The Laplace transform of (L −1 , H) is such that for θ, λ ≥ 0, where κ(•, •) denotes its bivariate Laplace exponent (resp.κ(•, •) for the descending ladder process).Similarly to the absorption rates studied in [3,9,8], the asymptotic analysis of the event of explosion and the role of the initial condition involve the renewal functions U and U, associated to the supremum and infimum respectively, which are defined, as follows [0,∞) The renewal functions U and U are finite, subadditive, continuous and increasing.Moreover, they are identically 0 on (−∞, 0], strictly positive on (0, ∞) and satisfy where C 1 , C 2 are finite constants (see for instance Lemma 6.4 and Section 8.2 in the monograph of Doney [10]).Moreover U(0) = 0 if 0 is regular upwards and U(0) = 1 otherwise, similarly U(0) = 0 if 0 is regular upwards and U (0) = 1 otherwise.Roughly speaking, the renewal function U(x) (resp.U(x)) "measures" the amount of time that the ascending (resp.descending) ladder height process spends on the interval [0, x] and in particular induces a measure on [0, ∞) which is known as the renewal measure.The latter implies A similar expression holds true for the Laplace transform of the measure U (dx) in terms of κ.For a more in-depth account of fluctuation theory, we refer the reader to the monographs of Bertoin [5], Doney [10] and Kyprianou [17].

Main results
Our first main result determines the law of the reweighted process (Z t e −ξt , t ≥ 0) via a backward differential equation in terms of ψ 0 (λ) := ψ(λ) + λδ and the process ξ.The function ψ 0 plays now the role of the branching mechanism for the CBLE Z since the demographic term δ can be added to the environment S without changing the structure of S and the definition of Z in (4).In fact, we observe that the resulting process (δt+S t , t ≥ 0) is still a Lévy process.To avoid any confusion we call ψ 0 as the pure branching mechanism of the CBLE Z.
We denote by P (z,x) the law of the process (Z, ξ) starting at (z, x).
Theorem 1.1.For every z > 0, x ∈ R, λ ≥ 0 and t ≥ s ≥ 0, we have where for any λ, t ≥ 0, the function ) is an a.s.solution of the backward differential equation and with terminal condition v t (t, λ, ξ) = λ.In particular, for every z, λ, t > 0 and x ∈ R, we have Our second main result furnishes a necessary and sufficient condition for a CBLE to be conservative, in our setting.The result is an extension of the original characterisation given by Grey [12] in the classical case for continuous-state branching process with constant environment.
Proposition 1.2.A continuous-state branching process in a Lévy environment with pure branching mechanism ψ 0 is conservative if and only if In what follows, we assume that the pure branching mechanism ψ 0 satisfies or in other words that Z may explode in finite time with positive probability.Under such assumption, we are interested in the asymptotic behaviour of the non-explosion probability in the following regimes: subcritical and critical explosive.
First we focus on the subcritical explosive regime, i.e. when the Lévy process drifts to −∞.We recall that α and π are the drift term and the Lévy measure of ξ, respectively.We introduce the following real function We also introduce the function where μ(x) := µ(x, ∞).Further, let us denote by E 1 the exponential integral, i.e.
We can then formulate the following theorem.
Theorem 1.3 (Subcritical explosive regime).Suppose that (19) holds, ξ drifts to −∞, P (e) -a.s., and that there exists λ > 0 such that Then, for any z > 0, lim In particular, if Note that, for y > 0, the following inequality for the exponential integral holds Therefore, in the case E (e) ξ 1 ∈ (−∞, 0), a simpler condition than ( 23) is the following We also observe that since the mapping λ → Φ λ (u) is decreasing, we have that if (22) holds for some λ > 0 then it holds for any λ > λ.Moreover, note that log(1 + 1/yλ) = log(1/yλ) + log(yλ + 1) and since the problem is at zero only log(1/yλ) matters for the finiteness of the above integral.
Our next main result deals with the critical explosive regime.More precisely, we assume that ξ satisfies the so-called Spitzer's condition, i.e.
Bertoin and Doney in [6] showed that the later condition is equivalent to P (e) (ξ s ≥ 0) → ρ as s → ∞.Recall from (12) that κ(θ, λ) is the Laplace exponent of the ascending ladder process (L −1 , H).Now, according to Theorem IV.12 in [5], Spitzer's condition (A) is equivalent to θ → κ(θ, 0) being regularly varying at zero with index ρ ∈ (0, 1), that is In addition, the function θ → κ(θ, 0) may always be written in the form where ℓ is a slowly varying function at 0 + , i.e for all positive constant c, it holds According to Theorems VI.14 and VI.18 in [5] or Remark 3.5 in [16], Spitzer's condition determines the asymptotic behaviour of the probability that the Lévy process ξ remains negative.In other words, under Spitzer's condition (A) and for x < 0, we have where we recall that U(•) is the renewal function for the ascending ladder-height, defined in (13).
In order to control the effect of the environment on the event of non-explosion we need other assumptions.The following integrability condition is needed to guarantee the non-explosion of the process in unfavourable environments.Let us assume In particular, from ( 14), we observe that the previous condition is satisfied if On the other hand, we shall assume that the pure branching process of the CBLE Z is lower bounded by a stable branching mechanism whose associated CBLE explodes with positive probability.More precisely, we assume that there exists β ∈ (−1, 0) and C < 0 such that ψ 0 (λ) ≥ Cλ 1+β for all λ ≥ 0. (C) The above condition is necessary to deal with the functional v t (s, λ, ξ) and to obtain an upper bound for the speed of non-explosion when the sample paths of the Lévy process have a high running supremum (see Proposition 5.4 below for details).Roughly speaking, our aim is to show, under the above conditions, that the probability of non-explosion varies regularly at ∞ with index ρ.Theorem 1.4 (Critical-explosion regime).Suppose that (19) holds and that conditions, (A), (B) and (C) are also fulfilled.Then, for any z > 0, there exists The previous result provides evidence that the asymptotic behaviour of the nonexplosion probability is deeply related to the fluctuations of the Lévy environment ξ.
The remainder of this paper is devoted to the proofs of the main results.In Section 2, we present the proofs of Theorem 1.1 and Proposition 1.2.In Section 3, the proof of Theorem 1.3 is given.In Section 4, we introduce continuous-state branching processes in a conditioned Lévy environment.This conditioned version is required to study the long-term behaviour of the non-explosion probability in the critical regime.Section 5 is devoted to the long-term behaviour results for the critical regime.

Proofs of Theorem 1.1 and Proposition 1.2
We first deal with the proof of Theorem 1.1 which relies on the extension of the classical Carathéodory's theorem for ordinary differential equations, that we state here for completeness.For its proof the reader is referred to Theorems 1.1, 2.1 and 2.3 in Person [22].Assume that the function f : iii) there exists a Lebesgue-integrable function m on the interval I such that Then there exists an absolutely continuous function u(x) such that On the other hand, by integration by parts, we note that the function |ψ 0 | = −ψ 0 can also be rewritten as follows where we recall that μ(x) = µ(x, ∞).The previous expression will be useful for what follows.
Proof of Theorem 1.1.The first part of the proof follows from similar arguments as those used in [4] and [19] in the case of finite mean (i.e., when |ψ ′ 0 (0+)| < ∞) whenever there is an a.s.solution of the backward differential equation (16).We present its proof for the sake of completeness.
We first deduce an explicit expression for the reweighted process Z t e ξt , for t ≥ 0. In order to do so, we consider the function f (x, y) = xe −y and apply Itô's formula (see e.g.Theorem II.5.1 in [14]).Then observing that f and applying directly Theorem II.5.1 in [14], we see By replacing all the factors in the above identity and recalling the definition of α below identity (5), we obtain Next, we fix λ ≥ 0 and t ≥ s ≥ 0. We assume for now that (v t (s, λ, ξ), s ∈ [0, t]) is solution of the backward differential equation (16).For simplicity on exposition, we denote by H t (s) = exp{−Z s e −ξs v t (s, λ, ξ)} and we apply again Itô's formula to the processes (Z s e −ξs , s ≤ t) and (v t (s, λ, ξ), s ≤ t) with the function f (x, y) = e −xy , that is Now using (16), we get H t (r)Z r e −ξr e ξr ψ 0 v t (r, λ, ξ)e −ξr dr H t (r)Z r e −ze −ξr vt(r,λ,ξ) − 1 µ(dz)dr where N (b) (ds, dz, dr) denotes the compensated version of N (b) (ds, dz, dr).By taking conditional expectations in both sides we get as expected.
In order to show the existence of equation ( 16), we will appeal to the extended version of Carathéodory's existence Theorem 2.1.Fix ω ∈ Ω (e) and t, λ ≥ 0. Denote by f : In the following we omit the notation ω for the sake of brevity.First, we observe that the mapping s → f (s, θ) is measurable for each fixed θ ∈ R. Indeed, the latter follows from the fact that ξ = (ξ s , s ≥ 0) possesses càdlàg paths and is (F (e) t ) t≥0 -adapted.More precisely, the application (s, ω) t -measurable implying that the mapping s → f (s, θ) is B([0, t])-measurable for each fixed θ ∈ R and ω ∈ Ω (e) .Furthermore, we have that the function θ → f (s, θ) is continuous for each fixed s ∈ [0, t] since ψ 0 is continuous.Therefore, according to Theorem 2.1, the proof is completed once we have shown that there exists an integrable function m on [0, t], such that, for any Note that, for any (s, θ) On the other hand, since |ψ 0 | is a Bernstein function, it is well known that there exists c, d > 0, such that |ψ 0 (θ)| ≤ c + dθ for any θ ≥ 0 (see for instance, Corollary 3.8 in [23]).It turns out that, Note that m is an integrable function on [0, t] since the Lévy process ξ has càdlàg paths.Finally, thanks to Theorem 2.1, there exists an a.s.solution of ( 16).
The functional v t (s, λ, ξ) has useful monotonicity properties as it is stated in the following lemma.In the forthcoming sections, we will make use of these properties.Lemma 2.2.For any λ ≥ 0 and t ≥ 0, the mapping s → v t (s, λ, ξ) is decreasing on [0, t].For any s ∈ [0, t], the mapping λ → v t (s, λ, ξ) is increasing on [0, ∞).
We conclude this section with the proof of Proposition 1.2.Before we do so, let us make an important remark about the non-explosion and extinction probabilities.It is easy to deduce, by letting λ ↓ 0 in (17) and with the help of the Monotone Convergence Theorem, that the non-explosion probability is given by With this in hand, we may now observe that the process Z is conservative if and only if On the other hand we also observe that, by letting λ ↑ ∞ in (17) and using again the Monotone Convergence Theorem, the probability of extinction is such that Moreover, from Lemma 2.2 and since v t (t, λe −x , ξ − x) = λe −x , we have that which combined with the previous identity clearly implies that Proof of Proposition 1.2.Fix t > 0 and recall from ( 10) that (ξ t , t ≥ 0) and (ξ t , t ≥ 0) denote the running infimum and supremum of the process ξ, respectively.First, we assume that the pure branching mechanism ψ 0 satisfies (18), that is to say, From Theorem 1.1, we see that the backward differential equation ( 16) can be rewritten as follows Now, we recall that |ψ 0 | is an increasing and non-negative function.Then appealing to the definition of the running infimum and supremum of ξ, we observe that the following inequality holds e −ξ t λ e −ξ t vt(0,λ,ξ) where in the last equality we have used the change of variables z = e −ξ t v t (s, λ, ξ).Next, letting λ ↓ 0 in the previous inequality, we get e ξ t e ξ t e −ξ t lim λ↓0 vt(0,λ,ξ) Thus, taking into account our assumption, we are forced to conclude that lim In other words, the process is conservative.
On the other hand, we assume that the process is conservative or equivalently that (33) holds.We will proceed by contradiction, thus we suppose that Similar to the above arguments, we deduce that e −ξ t λ e −ξ t vt(0,λ,ξ) Taking ǫ > 0 sufficiently small, we see dz − e ξ t e ξ t ǫ e −ξ t vt(0,λ,ξ) Hence, by taking λ ↓ 0 in the above inequality and using (33), we have t ≤ 0, which is a contradiction.Therefore, we deduce that the pure branching mechanism ψ 0 satisfies (18).
3 Subcritical explosive regime: proof of Theorem 1.3 The proof of Theorem 1.3 follows similar ideas as those used in the proof of Proposition 3 in Palau and Pardo [19].
Proof of Theorem 1.3.Let z > 0 and x ∈ R. We begin by observing, from ( 4) or (29), that P (z,x) (Z t < ∞) does not depend of the initial value x of the Lévy process ξ.More precisely, from (4) we can observe that Z t depends only on the initial condition Z 0 = z.Thus, again from (29), we see Hence, as soon as we can establish that lim t→∞ v t (0, 0, ξ − x) < ∞, P (e) x − a.s., our proof is completed.From Lemma 2.2, we see that the mapping s → v t (s, λe −x , ξ − x) is decreasing and since v t (t, λe −x , ξ − x) = λe −x , we have v t (s, λe −x , ξ − x) ≥ λe −x for all s ∈ [0, t].It follows that, for s ∈ [0, t] and λ > 0, Therefore by integrating, Moreover, from Lemma 2.2, we have that the mapping λe Using the definition of Φ λ in (20), we get We recall that in this regime the process −ξ drifts to ∞, P x -a.s.Thus, in order to prove that the integral in ( 35) is finite, let us introduce ς = sup{t ≥ 0 : −ξ t ≤ 0} and observe that x -a.s., it follows that the first integral in the right-hand side above is finite x -a.s.For the second integral, we may appeal to Theorem 1 in Erickson and Maller [11] or the main result in Kolb and Savov [15].It is important to note that the main result in [15] extends the result in [11] but both results coincide in this particular case.Despite that the aforementioned Theorem is written in terms of Lévy processes drifting to +∞, it is not so difficult to see that it can be rewritten, using duality, in terms of Lévy processes drifting to −∞ which is how we are using it here.More precisely, since Φ λ is a finite positive, non-constant and non-increasing function on [0, ∞) and condition (22) holds, Theorem 1 in [11] guarantees that x − a.s.
Furthermore, if E (e) ξ 1 ∈ (−∞, 0), then lim x→∞ A ξ (x) is finite.In particular, it follows by integration by parts, that the integral condition ( 22) is equivalent to Moreover, we have Now by the definition of the exponential integral given in ( 21), we deduce that condition ( 22) is equivalent to which concludes the proof.

CSBPs in a conditioned Lévy environment
Throughout this section, we shall suppose that the Lévy process ξ satisfies Spitzer's condition (A) which in particular implies that the process oscillates.
As mentioned earlier, the asymptotic behaviour of the non-explosion probability is related to fluctuations of the Lévy process ξ, specially to its running supremum.For our purpose, we need to introduce the definition of a CSBP conditioned to stay negative, which roughly speaking means that the running supremum of the auxiliary Lévy process ξ is negative.Let us therefore spend some time in this section gathering together some of the facts of this conditioned version.
Similarly to the definition of Lévy processes conditioned to stay positive and following a similar strategy as in the discrete framework in Afanasyev et al. [1], we would like to introduce a continuous-state branching process in a Lévy environment conditioned to stay negative as a Doob-h transform.The aforementioned process was first investigated by Bansaye et al. [3] with the aim to study the survival event in a critical Lévy environment.Lemma 4.1 (Bansaye et.al. [3]).Let z, x > 0. The process { U (ξ t )1 {ξ t >0} , t ≥ 0} is a martingale with respect to (F t ) t≥0 and under P (z,x) .
With this in hand, they introduce the law of a continuous-state branching process in a Lévy environment ξ conditioned to stay positive as follows, for Λ ∈ F t , z, x > 0, where U is the renewal function defined in (13).It is natural therefore to cast an eye on similar issues for the study of non-explosion events in a Lévy environment.In contrast, we introduce here the process Z in a Lévy environment ξ conditioned to stay negative.Recall that ξ is the dual process of ξ.Appealing to duality and Lemma 4.1, we can see that the process {U(−ξ t )1 {ξ t <0} , t ≥ 0} is a martingale with respect to (F t ) t≥0 and under P (z,x) with z > 0 and x < 0. Then we introduce the law of the continuous-state branching process in a Lévy environment ξ conditioned to stay negative, as follows: for Λ ∈ F t for z > 0 and x < 0, Intuitively speaking, P ↑ (z,x) and P ↓ (z,x) correspond to the law of (Z, ξ) conditioning the random environment ξ to not enter (−∞, 0) and (0, ∞), respectively.
The following convergence result is crucial for Theorem 1.4.The proof can be obtained directly using duality and Lemma 3.2 in [3].Lemma 4.2.Fix z > 0, x < 0 and assume that Spitzer's condition (A) holds.Let R s be a bounded real-valued F s -measurable random variable.Then More generally, let (R t , t ≥ 0) be a uniformly bounded real-valued process adapted to the filtration (F t , t ≥ 0), which converges P ↓ (z,x) -a.s., to some random variable R ∞ .Then Recall from Theorem 1.1 that the quenched law of the process (Z t e −ξt , t ≥ 0) is completely characterised by the functional v t (s, λ, ξ).In the case of conditioned environment we have a similar result.We formalise this in the following lemma whose proof essentially mimics the steps of Proposition 3.3 in [3] and identity (31).Lemma 4.3.For each z > 0, x < 0 and λ ≥ 0, we have In particular, x exp {−zv t (0, 0, ξ − x)} , and The following lemma states that, with respect to P ↓ (z,x) , the population has positive probability to be finite forever.In other words, Z has a positive probability to be finite when the running supremum of the Lévy environment is negative.The statement holds under the moment condition (B) of the Lévy measure µ.Lemma 4.4.Assume that the Lévy process ξ satisfies Spitzer's condition (A) and condition (B).Then, for z > 0 and x < 0, we have Note that such behaviour is similar to the behaviour in the subcritical explosive regime (i.e. when the environment drifts to −∞) given in Theorem 1.3.
Proof.Let z > 0 and x < 0. From Lemma 4.3, we already know the formula, Then, similarly as in Theorem 1.3, in order to deduce our result we will show that lim We recall from the proof of Theorem 1.3 that for all λ > 0 lim Now, if the right-hand side of the above inequality is finite P (e),↓ x − a.s.then (38) holds.The result is thus proved once we show that First, with the help of Fubini's Theorem and the definition of the measure P (e),↓ x in terms of P (e) x (the law of the dual of ξ) we obtain Now, applying Theorem VI.20 in Bertoin [5] to the dual process ξ = −ξ and the function f (y) = U(y)Φ λ (y), y ≥ 0, we deduce that, there exists a constant k > 0 such that For the sake of simplicity we take k = 1 (we may choose a normalisation of the local time in order to have k = 1).Observe that, for any z ∈ [0, −x] and y ≥ 0, we have y − x − z ≤ y − x.Further, since U(•) and Φ λ (•) are increasing functions, we deduce that U(y Next recall that we may rewrite the function Φ λ as follows, where in the second inequality and in the last identity we have used, respectively, inequality (14) and identity (15), but in terms of U and after taking the first derivative.Now, we consider which is clearly finite from condition (B).Hence putting all pieces together, we obtain This concludes the proof.
It is important to note that in Baguley et al. [2] there is a necessary and sufficient condition (integral test) for the finiteness of path integrals for standard Markov processes, see Theorem 2.3 and comments below, in terms of their potential measures.The latter is in line with our necessary condition for the finiteness of Throughout this section, we shall suppose that the Lévy process ξ satisfies Spitzer's condition (A).
The strategy of our proof follows similar arguments as in [3], where the extinction event has been considered, that is we split the event {Z t < ∞} into two events by considering the behaviour of the running supremum of the environment.More precisely, we split the nonexplosion event into either unfavourable environments, i.e. when the running supremum is negative, or favourable environments, i.e. when the running supremum is positive.
Before we prove Theorem 1.4, we introduce several useful results.Lemmas 4.2 and 4.4 allow us to establish the following result which describes the limit of the non-explosion probability when the associated environment is conditioned to be negative.Proposition 5.1.Suppose that conditions (A) and (B) are satisfied.Then for every z > 0 and x < 0, there exists 0 < c(z, x) < ∞ such that Proof.We begin by defining the decreasing sequence of events A t = {Z t < ∞} for t ≥ 0, and also the event A ∞ = {∀t ≥ 0, Z t < ∞}.Now, observe that Let (R t := 1 At , t ≥ 0) be a uniformly bounded process adapted to the filtration (F t , t ≥ 0).Note that, the process (R t , t ≥ 0) converges P ↓ (z,x) -a.s. to a random variable R ∞ = 1 A∞ .Then, by appealing to Lemma 4.2, we have Therefore, by using the asymptotic behaviour of the probability that the Lévy process ξ remains negative given in (25), we get where c(z, x) := E ↓ (z,x) R ∞ / √ π.Furthermore, from Lemma 4.4, we have which completes the first claim.
Proof.Let x < 0 and s ≤ t.We begin by noting Now, recall that under Spitzer's condition (A), the function κ(•, 0) is regularly varying at 0 or more precisely, from (25), we have where ℓ is the slowly varying function at ∞ defined in (24) as ℓ(t) = ℓ(1/t).Hence, for t and s large enough, we have where C 1 is a positive constant.On the other hand, according to Potter's Theorem in [7] we deduce that, for any A > 1 and η > 0 there exists Therefore, for t ≥ s ≥ t 1 where C 2 is a positive constant.Now plugging the later inequality back into (41), we get, as it was claimed, where C 3 is a constant bigger than 1.
Recall that I 0,t (βξ) denotes the exponential functional of the Lévy process βξ defined in (9).Our next result will be useful to control the probability of non-explosion under the event that {ξ t > 0}.

Further, lim
where ϑ a is a positive measure on (0, ∞).
For our purposes, we use f (x) = x a exp(−y(Cβ) −1/β x −1/β ) which is bounded and continuous.Thus by the latter identity we deduce that, for any a ∈ (0, 1), which is clearly continuous.The latter implies that there exists The following result makes precise the statement that only paths of the Lévy process with a very low running supremum give a substantial contribution to the speed of nonexplosion.Now, letting λ ↓ 0 and taking into account that β ∈ (−1, 0) and C < 0, we deduce the following inequality for all t ≥ 0, where I 0,t (β(ξ − x)) is the exponential functional of the Lévy process β(ξ − x).Hence, the quenched non-explosion probability given in (29) may be bounded in terms of this functional.That is to say, for all t > 0, Therefore conditioning on the environment, we obtain that for any y > x, Let w = x−y and t 0 > 0. Now, we split the event {τ − w ≤ t−ǫ} for 3t 0 < t and 0 < ǫ < 1, as follows By the monotonicity of the mapping t → I 0,t (−βξ), we have that, under the event {0 < τ − w ≤ (t − t 0 )/2}, the following inequalities hold Similarly, under the event {(t − t 0 )/2 < τ − w ≤ t − ǫ}, we obtain Next, appealing to the strong Markov property of ξ, we deduce Thus from Lemma 5.3, for t sufficiently large, we have where C β (ze −w ) are defined as in (42).Using the same arguments as above and Lemmas 5.2 and 5.3, we obtain the following sequence of inequalities for t sufficiently large, Hence plugging this back into (45) (similarly as in the proof of Lemma 4.4 in [18]), we get lim sup For every z > 0 and x < 0, denote by c(z, x) the constant defined in Proposition 5.1.Our next result provides some useful properties of the mapping x → c(z, x)U(−x).Proof.Note that the strictly positivity follows from the facts that the renewal function U(−x) is a strictly positive function on (−∞, 0) and by Lemma 5.1.Since P x (ξ t < 0) ≤ P y (ξ < 0) for x ≥ y, we observe that the left-hand side on (39) is decreasing in x < 0, so does the map x → c(z, x)U(−x).Now, in order to see that the function is bounded from above, we first observe that similarly as in (45), we have where z → C β (z) is the continuous function defined in Lemma 5.3.With this in hand, and considering that the mapping x → c(z, x)U(−x), on (−∞, 0), is decreasing and strictly positive, we conclude that for every z > 0 the limit B(z) exists and it is finite and strictly positive.
With Propositions 5.1 and 5.4 in hand, we may now proceed to the proof of Theorem 1.4.The proof follows the same arguments as those used in Theorem 1.2 in [3], we provide its proof for the sake of completeness.
Next, using that the function κ(•, 0) is regularly varying at 0, and then appealing to Potter's Theorem in Bingham et al. [7], we see that, for any A > 1 and η > 0, We observe that y ′ is a sequence which may depend on ς and z.Further, this sequence y ′ goes to −∞ as ς goes to 0. Thus, for any sequence y ς (z), we have