Bénédicte Haas
email: haas@ceremade.dauphine.fr

Asymptotic behavior of solutions to the fragmentation equation with shattering: an approach via self-similar Markov processes

Keywords: AMS subject classifications: 60J75, 60G18, 82C40 self-similar Markov processes, fragmentation equation, scaling limits, regular variation, quasi-stationary solutions

Asymptotic behavior of solutions to the fragmentation equation with shattering: an approach via self-similar

Markov processes

Introduction and main results

The fragmentation equation is commonly used to describe the evolution of a particles system where particles break up as time passes. In this paper, we are interested in models in which particles with mass xy, 0 < y < 1, are produced from the splitting of a particle with mass x at rate x α B(dy), where α ∈ R and B is a measure on ]0, 1[ such that 1 0 y(1 -y)B(dy) < ∞, and B (]0, 1[) > 0.

(

The corresponding weak form of the fragmentation equation is then

∂ t < µ t , f > = ∞ 0 x α 1 0 (f (yx) -f (x)y) B(dy) µ t (dx), (2)
where (µ t , t ≥ 0) denotes a family of measures on ]0, ∞[ and f any test function. The quantity µ t (dx) represents the average number per unit volume of particles having a mass in [x, x + dx[ at time t. The term between parentheses on the right-hand side of the equation models the loss of particles of mass x and the increase of particles of mass xy, 0 < y < 1, due to the fragmentation of particles of mass x. Note that the overall rate at which a particle with mass x splits is x α 1 0 yB(dy), which may be infinite. When B(dy) = b(y)dy and 1 0 yb(y)dy = 1, this equation is a weak form of the standard fragmentation equation with fragmentation kernel F (x, y) = x α-1 b(y/x), x > y, which describes the evolution of the density per unit volume of particles with mass x at time t, say n t (x), in the following way:

∂ t n t (x) = ∞ x
F (y, x)n t (y)dy -x 0 y x F (x, y)n t (x)dy.

(3)

This last equation was intensively studied both by physicists and mathematicians. Among the first papers on the topic, we may cite e.g. [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF][START_REF] Melzak | A scalar transport equation[END_REF]. We also refer to the book [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF] for a probabilistic approach of the microscopic mechanism of fragmentation and to the papers [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] and [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF] for discussions on the relations between the probabilistic model and the above equations. Let us add that the physical interpretation of the fragmentation equation imposes some constraints on the measure B. However other interpretations are possible and in the following we will be concerned with all measures B satisfying [START_REF] Banasiak | Shattering and non-uniqueness in fragmentation models-an analytic approach[END_REF].

In this paper, we focus on solutions to [START_REF] Bertoin | Lévy processes[END_REF] with finite and non-zero initial total mass. The fragmentation equation being linear, we suppose, without loss of generality, that ∞ 0 xµ 0 (dx) = 1. To be precise, we call a solution to [START_REF] Bertoin | Lévy processes[END_REF] starting from µ 0 , any family of measures (µ t , t ≥ 0) on ]0, ∞[ starting from µ 0 and such that • (µ t , t ≥ 0) satisfies [START_REF] Bertoin | Lévy processes[END_REF] for any test functions f ∈ C 1 c , the set of real-valued continuously differentiable functions on ]0, ∞[ with compact support

• the following natural "physical properties" are respected (id denotes the identity function)

m(t) :=< µ t , id > ≤ m(0) = 1, ∀t ≥ 0, (4)
and

µ 0 ([M, ∞[) = 0 for some M > 0 ⇒ µ t ([M, ∞[) = 0 ∀t ≥ 0. ( 5
)
Note the self-similarity of solutions: if (µ t , t ≥ 0) is a solution to [START_REF] Bertoin | Lévy processes[END_REF], so is (γ -1 µ tγ α • (γid) -1 ), for all γ > 0. Note also that if (µ t , t ≥ 0) is a solution to the equation with parameters (α, B), then for all c > 0, (µ ct , t ≥ 0) is a solution to the equation ( 2) with parameters (α, cB).

Many results on existence and uniqueness of solutions to [START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF] are available in the litterature. See e.g. [START_REF] Banasiak | Shattering and non-uniqueness in fragmentation models-an analytic approach[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Laurençot | On a class of continuous coagulation-fragmentation equations[END_REF] and the references therein. With the definition above, we have the following result on existence and uniqueness of solutions to [START_REF] Bertoin | Lévy processes[END_REF], which is a generalization of Theorem 1 of [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF] (see also [START_REF] Fournier | On small particles in coagulation-fragmentation equations[END_REF] for a similar approach). We recall that a subordinator is a non-decreasing Lévy process and that its distribution is characterized by two parameters: a non-negative drift coefficient and a so-called Lévy measure on ]0, ∞[ that governs the jumps of the process. See Section 2 for background on this topic. More precisely, this solution can be constructed via a subordinator ξ with zero drift and a Lévy measure given for any measurable function

g :]0, ∞[→ [0, ∞[ by ∞ 0 g(x)Π(dx) = 1 0 g(-ln(x))xB(dx), (6)
through the formula

∞ 0 f (x)xµ t (dx) := ∞ 0 E f x exp(-ξ ρ(x α t) ) xµ 0 (dx), for all measurable f :]0, ∞[→ [0, ∞[, (7)
where ρ is defined by ρ(t) := inf u ≥ 0 : u 0 exp(αξ r )dr > t .

(ii) This solution is unique as soon as µ 0 has a compact support.

When the family (µ t , t ≥ 0) is constructed via subordinators by [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF], some conditions on µ 0 and B for the existence of a density for µ t , t > 0, can be settled expliticitly. See e.g. [START_REF] Fournier | Existence of densities for jumping stochastic differential equations[END_REF]Proposition 3.10]. We also recall that there may be multiple solutions to the fragmentation equation when the assumption (4) is dropped. We refer to [START_REF] Banasiak | Shattering and non-uniqueness in fragmentation models-an analytic approach[END_REF] for some explicit examples.

The proof of Theorem 1.1, based on that of Theorem 1 in [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF], is postponed to the Appendix at the end of this paper.

The main purpose of this paper is to use the construction [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF] of solutions to the fragmentation equation to describe the long-time behavior of these solutions when α < 0. From another, but equivalent, point of view, our main results describe the large-time behavior of time-changed subordinators, as defined in Theorem 1.1, conditioned on non-extinction. These processes belong to the family of so-called self-similar Markov processes. We refer to Section 3 for a statement of our results in that context.

The study of large-time behavior of solutions to the fragmentation equation when α > 0 is investigated in details in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. We point out that some results of [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] can be re-demonstrated using a probabilistic approach: it mainly consists in combining the subordinator construction of solutions to the fragmentation equation with the description of large-time behavior of time-changed subordinators when α > 0 investigated in [START_REF] Bertoin | Entrance from 0+ for increasing semi-stable Markov processes[END_REF].

From now on, we consider α < 0. It is well-known that in such case, small particles split so quickly that they are reduced to a dust of zero-mass particles, so that the total mass of non-zero particles m(t) =< µ t , id > decreases as time passes. This phenomenon, sometimes called "shattering", was studied e.g. in [START_REF] Banasiak | Shattering and non-uniqueness in fragmentation models-an analytic approach[END_REF][START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF][START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF][START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF][START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF][START_REF] Wagner | Explosion phenomena in stochastic coagulation-fragmentation models[END_REF]. More precisely, one can check that the total mass m is strictly decreasing and strictly positive on [0, ∞[ and that m(t) → 0 as t → ∞. See the forthcoming Proposition 3.3 for a proof in our framework.

In order to describe the behavior of m(t) as t → ∞ more accurately, we introduce the following function defined for all t ≥ 0 by

φ(t) := 1 0 1 -x t xB(dx). (8)
It is not hard to check that the function t → t/φ(t) is continuous, strictly increasing on ]0, ∞[ and that its range is ](

1 0 | ln(x)|xB(dx)) -1 , ∞[. Note that the integral 1 0 | ln(x)|xB(dx) may be finite or infinite. Then introduce ϕ, the inverse of t → t/φ(t), (9)
which is well defined in a neighborhood of ∞. This function will play a key-role in the description of the long-time behavior of solutions to the fragmentation equation.

Most of our main results rely on the following hypothesis on the measure B:

the function u :]0, 1[→ 1-u 0 xB(dx) varies regularly at 0 with an index -β ∈] -1, 0], (H)
which, in particular, ensures that φ and ϕ are regularly varying functions at ∞, with respective indices β and 1/(1 -β). See Section 2.2 for details and background on regular variation.

Last, we mention that the large-time behavior of solutions to the fragmentation equation will depend strongly on the structure of the initial measure µ 0 , mainly on the manner it distributes weight near ∞. The statements of our results are therefore split into two parts, according as to wether the initial measure has compact support (Subsection 1.1) or not (Subsection 1.2). Subsection 1.3 deals with the quasi-stationary solutions.

Initial measure µ 0 with compact support

In this subsection, we adopt the following hypotheses and notations:

-α < 0 -the measure µ 0 has a compact support, i.e. µ 0 ([M, ∞[) = 0 for some M > 0 -(µ t , t ≥ 0) denotes the unique solution to the fragmentation equation (2) starting from µ 0 .

The supremum of the support of µ 0 is the real s such that µ 0 (]s, ∞[) = 0 and µ 0 (]s -ε, s]) > 0 for all ε < s.

Thanks to the self-similarity of solutions, we can, and will, always suppose that this supremum is equal to 1. In such frame, we have the following results.

Proposition 1.2. For all λ < φ(∞) := lim x→∞ φ(x), there exists a constant

C λ < ∞ such that m(t) ≤ C λ exp(-λt), ∀t ≥ 0.
More precisely, under the hypothesis (H),

-ln(m(t)) ∼ t→∞ (1 -β) |α| ϕ(|α|t).
In particular, t → -ln(m(t)) is regularly varying at ∞ with index 1/(1 -β).

Together with the following theorem, this gives a complete description of the large-time behavior of (µ t , t ≥ 0). Here, two positive functions g and h are said asymptotically equivalent if g(x)/h(x) → 1 as x → ∞.

Theorem 1.3. Suppose (H) and 0 | ln(x)|xB(dx) < ∞. Then for all continuous bounded test functions

f :]0, ∞[→ R, 1 m(t) ∞ 0 f ϕ(|α|t) |α|t 1/|α| x xµ t (dx) → t→∞ ∞ 0 f (x)xµ ∞ (dx),
where xµ ∞ (dx) is a probability distribution on ]0, ∞[ that is characterized by its moments

∞ 0 x |α|n xµ ∞ (dx) = φ(|α|)φ(2|α|)...φ(n|α|), n ≥ 1. ( 10
)
The function t → ϕ(|α|t)/(|α|t) can be replaced by any asymptotically equivalent function.

It is interesting to compare this result with that obtained by Escobedo et al. [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] when the parameter α is positive. As already mentioned, part of their result can be rediscovered and completed by using Bertoin and Caballero's paper [START_REF] Bertoin | Entrance from 0+ for increasing semi-stable Markov processes[END_REF]. With our notations and under the assumptions 0 | ln(x)|xB(dx) < ∞ and α > 0, the asymptotic behavior of the solution (µ t , t ≥ 0) to the fragmentation equation (α, B) starting from µ 0 = δ 1 can be described as follows:

∞ 0 f t 1/α x xµ t (dx) → t→∞ ∞ 0 f (x)xη ∞ (dx),
for all continuous bounded functions f :]0, ∞[→ R. The measure xη ∞ (dx) is a probability measure on ]0, ∞[. Interestingly, the measure B is then only involved in the description of the limit measure η ∞ , not on the "shape" of the speed of decrease of masses to 0.

We come back to the case α < 0. Note that when

1-u 0 xB(dx) ∼ u -β as u → 0 for some β ∈ [0, 1[, we have φ(t) ∼ Γ(1 -β)t β and therefore (ϕ(|α|t)/|α|t) 1/|α| ∼ C α,β t β/((1-β)|α|) as t → ∞, where C α,β = (|α| β Γ(1 -β)) 1/((1-β)|α|) . When moreover 0 | ln(x)|xB(dx) < ∞, the Theorem 1.3 then reads 1 m(t) ∞ 0 f C α,β t β/((1-β)|α|) x xµ t (dx) → t→∞ ∞ 0 f (x)xµ ∞ (dx) for all continuous bounded test functions f :]0, ∞[→ R.
The existence and uniqueness of a measure µ ∞ on ]0, ∞[ satisfying [START_REF] Chow | Moment conditions for the existence and nonexistence of optimal stopping rules for S n /n 1[END_REF] actually hold without any assumption of regular variation on the measure B or on its behavior near 0. See the discussion near formula [START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF] in Section 3 for details. Some properties of the measure µ ∞ (tail behavior near 0, near ∞) are given in Section 5. In Subsection 1.3, we discuss its links with the quasi-stationary solutions to the fragmentation equation.

The proof of Theorem 1.3 consists in describing the behavior of the mass of a random typical non-dust particle, which is defined as follows: at each time t, choose a particle at random among the particles with a strictly positive mass, with a probability proportional to its mass. I.e. if M (t) denotes the mass of this random particle, the distribution of M (t) is given by

M (t) d ∼ xµ t (dx) m(t) .
Otherwise said, in terms of the subordinator ξ related to the equation by [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF], M (t) is distributed as M (0) exp(-ξ ρ(M(0) α t) ) conditioned to be strictly positive, with M (0) independent of ξ. In terms of M , the statement of Theorem 1.3 rephrases as follows

ϕ(|α|t) |α|t 1/|α| M (t) d → M ∞ ,
where M ∞ is a random variable with distribution xµ ∞ (dx). Note the special case

1 0 xB(dx) < ∞, where ϕ(t)/t → 1 0 xB(dx) < ∞.
Then we have that M (t) converges in distribution to a non-trivial limit. In the other cases satisfying the assumptions of Theorem 1.3, ϕ(t)/t → ∞ and therefore M (t)

P → 0.
Using this random approach, we can also specify the behavior of masses that decrease at different speeds to 0, as follows.

Proposition 1.4. Assume (H) and κ :=

1 0 | ln(x)|xB(dx) < ∞. (i) Suppose moreover that the support of B is not included in a set of the form {a n , n ∈ N} for some a ∈]0, 1[. Then for all measurable functions g : [0, ∞[→]0, ∞[ converging to 0 at ∞, g(t) α m(t) g(t)(ϕ(|α|t)/|α|t) 1/α 0 xµ t (dx) → t→∞ 1 |α|κ ; (ii) For all measurable functions g : [0, ∞[→]0, ∞[ converging to ∞ at ∞, -if g |α| (t)t/ϕ(t) converges to ∞ at ∞, ∞ g(t)(ϕ(|α|t)/|α|t) 1/α xµ t (dx) = 0 for all t sufficiently large -if g |α| (t)t/ϕ(t) converges to 0 at ∞ and 0 < β < 1 : lim sup t→∞ 1 φ -1 (g(t) |α| ) ln ∞ g(t)(ϕ(|α|t)/|α|t) 1/α xµ t (dx) m(t) ≤ - β |α| ,
where φ -1 denotes the inverse of φ.

Note that the first assertion of (ii) is obvious since g(t)(ϕ(|α|t)/|α|t) 1/α → ∞ (which means that for t sufficiently large, it is larger than 1, the supremum of the support of µ t ).

We finish this section with the following result on the remaining mass at time t of particles of mass 1 when µ 0 ({1}) > 0. The measure µ ∞ is that introduced in Theorem 1.3.

Proposition 1.5. Suppose µ 0 ({1}) > 0 and set φ(∞) := 1 0 xB(dx) ∈]0, ∞]. Then for all t ≥ 0, µ t ({1}) = exp (-tφ(∞)) µ 0 ({1}). When moreover (H) is satisfied and 0 | ln(x)|xB(dx) < ∞ and φ(∞) < ∞, µ t ({1}) m(t) → t→∞ φ(∞) 1/|α| µ ∞ {φ(∞) 1/|α| }
and this limit is non-zero if and only if

1 B(dx)
1-x < ∞. This means that under the assumptions of Proposition 1.5, for large times, the remaining total mass of 1-mass particles is proportional to the total mass of non-zero particles when 1 (1 -x) -1 B(dx) < ∞, whereas it is negligible compared to the total mass of non-zero particles when = 1 (1 -x) -1 B(dx) = ∞. We point out that the convergence of Proposition 1.5 is not necessarily true when µ 0 ({1}) = 0 (since then µ t ({1}) = 0 for all t ≥ 0, whereas the term in the limit may be strictly positive).

1.2 Initial measure µ 0 with unbounded support

We still suppose that α < 0 and we denote by (µ t , t ≥ 0) the solution to the fragmentation equation ( 2) starting from µ 0 and constructed via a subordinator by formula [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. The asymptotic behavior of the mass m(t) is then strongly modified by the presence of large masses and depends on the behavior as t → ∞ both of φ(t) and µ 0 ([t, ∞[). We investigate two particular cases: exponential and power decreases of µ 0 ([t, ∞[) as t → ∞.

Theorem 1.6. Assume (H) and that µ 0 possesses a density, say u 0 , in a neighborhood of ∞ such that

ln (u 0 (x)) ∼ ∞ -Cx γ
for some γ > 0.

(i) Then, -ln (m(t)) ∼ ∞ C α,β,γ C (1+(1-β)γ/|α|) -1 h(t)
where h is the inverse, well-defined in the neighborhood of ∞, of t → t 1+|α|/γ /φ(t) and

C α,β,γ = 1 + |α| -1 γ(1 -β) |α| 1/(1-β) γ γ(1-β) γ(1-β)+|α| .
In particular, -ln(m(t)) varies regularly at ∞ with index 1/(1 -β + |α|/γ). (ii) Suppose moreover that 0 | ln(x)|xB(dx) < ∞, which ensures that the function ln(m) is differentiable on ]0, ∞[. Then, if the derivative (ln(m)) ′ is regularly varying at ∞, one has, for all continuous bounded test functions

f :]0, ∞[→ R, 1 m(t) ∞ 0 f h(t) C α,β,γ,C t 1/|α| x xµ t (dx) → t→∞ ∞ 0 f (x)xµ ∞ (dx),
where µ ∞ (dx) is the measure introduced in Theorem 1.3 and

C α,β,γ,C = C α,β,γ C (1+(1-β)γ/|α|) -1 1 -β + |α|/γ .
Assuming that the derivative (ln(m)) ′ is regularly varying at ∞ may seem demanding. Actually, this assumption is also needed to get Theorem 1.3, but we are able to show it is always satisfied under the hypotheses of this theorem (see Lemma 3.9). Unfortunately, it seems difficult to adapt this proof to the case when the measure µ 0 has an unbounded support. However, according to a classical result on regular variation (the Monotone Density Theorem), (ln(m)) ′ varies regularly at ∞ as soon as ln(m) varies regularly at ∞ and (ln(m)) ′ is monotone near ∞, which can be checked in some particular cases.

There is also the following result on the decrease of the mass m when the density u 0 of µ 0 has a power decrease near ∞. Proposition 1.7. Assume that µ 0 possesses a density u 0 in a neighborhood of ∞ such that

u 0 (x) ∼ ∞ Cx -γ for some γ > 2. Then, m(t) ∼ ∞ C ′ t γ-2 α , with C ′ = |α| -1 C ∞ 0 m(u)u 2-γ α -1 du < ∞,
where m denotes the total mass of the solution to fragmentation equation with the same parameters α, B as that considered here, and with initial distribution δ 1 , the Dirac mass at 1.

Quasi-stationary solutions

A quasi-stationary solution to the fragmentation equation ( 2) is a solution (µ t , t ≥ 0) such that µ t = m(t)µ 0 , ∀t ≥ 0, with m(t) =< µ t , id >. These quasi-stationary solutions are closely related to the measure µ ∞ introduced in the statement of Theorem 1.3. We have already mentioned that the existence and uniqueness of such a measure µ ∞ satisfying [START_REF] Chow | Moment conditions for the existence and nonexistence of optimal stopping rules for S n /n 1[END_REF] hold without any assumption of regular variation on the measure B or on its behavior near 0. The interesting fact is that, whatever the conditions on B, this measure and its self-similar counterparts µ (λ) ∞ := λ -1 µ ∞ • (λid) -1 , λ > 0, are the only initial measures leading to quasi-stationary solutions to the fragmentation equation (2).

Theorem 1.8. For all λ > 0, let (µ ∞ and constructed via a subordinator by [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. Then for all t ≥ 0,

µ (λ) ∞,t = exp(-λ α t)µ (λ) ∞ = m(t)µ (λ) ∞ .
Reciprocally, if (µ t , t ≥ 0) is a quasi-stationary solution to the fragmentation equation, then there exists a λ > 0 such that (µ t , t ≥ 0) = (µ

(λ) ∞,t , t ≥ 0).
Organization of the paper. We start in Section 2 with some background on subordinators and regular variation. Section 3 is the core of this paper: our main results on large-time behavior of self-similar Markov processes conditioned on non-extinction are stated and proved there. Together with Theorem 1.1, these results imply Theorems 1.3, 1.6 and 1.8, as well as Propositions 1.2 and 1.7. Section 4 is devoted to the proof of Proposition 1.4. Some properties of the limit measure µ ∞ are given in Section 5, and used to prove Proposition 1.5. Last, some specific examples are discussed in Section 6 and the proof of Theorem 1.1 is detailed in the Appendix.

Background on subordinators and regular variation 2.1 Subordinators

A subordinator is a non-decreasing Lévy process, i.e. a non-decreasing càdlàg process with stationary and independent increments. We recall here the main properties we need in this paper and refer to the Chapter 3 of [START_REF] Bertoin | Lévy processes[END_REF] for a more complete introduction to the subject.

The distribution of a subordinator (ξ t , t ≥ 0) starting from ξ 0 = 0 is characterized by its so-called Laplace exponent

φ : [0, ∞[→ [0, ∞[ through the identity E [exp(-λξ t )] = exp(-tφ(λ)), ∀λ, t ≥ 0.
According to the Lévy-Khintchine formula [2, Theorem 1, Chapter 1], there exists a real d ≥ 0 and a measure

Π on ]0, ∞[, ∞ 0 (1 ∧ x)Π(dx) < ∞, such that φ(λ) = dλ + ∞ 0 (1 -exp(-λx))Π(dx), ∀λ ≥ 0.
The measure Π governs the jumps of the subordinator: the jumps process of ξ is a Poisson point process with intensity Π.

We will need the strong Markov property of subordinators ([2, Proposition 6, Chapter 1]): given a subordinator ξ and a stopping time T with respect to the filtration (F t , t ≥ 0) generated by ξ, then, conditionally on {T < ∞}, the process (ξ t+T -ξ T , t ≥ 0) is independent of F T and is distributed as ξ. Last, we recall that the semi-group of a subordinator possesses the Feller property ([2, Proposition 5, Chapter 1]).

From now on, all subordinators considered in this paper start from 0 and have a drift d = 0. Their distribution is therefore completely determined by their Lévy measure Π. Note that when Π is related to a measure B on ]0, 1[ through the formula (6), the above expression of φ coincides with that given by formula [START_REF] Caballero | [END_REF], i.e.

φ(λ) = ∞ 0 (1 -exp(-λx))Π(dx) = 1 0 (1 -x λ )xB(dx), ∀λ ≥ 0.

Regular variation

A function f :]0, ∞[→]0, ∞[ is said to vary regularly at ∞ (resp. 0) with index γ ∈ R if for all a > 0 f (ax) f (x) → a γ as x → ∞ (resp. 0).
We refer to [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF] for background on the topic. In particular, we have already implicitely used that the inverse, when it exists, of a function regularly varying at ∞ with index γ > 0 is also regularly varying at ∞, with index 1/γ (see Section 1.5.7 of [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]).

Note that when the Lévy measure Π is related to the fragmentation measure B by the formula (6), our main assumption (H) reads "u ∈]0, ∞[→ ∞ u Π(dx) varies regularly at 0 with index -β". It is classical that this is equivalent to the fact that the function φ varies regularly at ∞ with index β.

This can be easily proved from Karamata Abelian-Tauberian Theorems (see in particular Chapters 1.6 and 1.7 of [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]). We will often use this form of the assumption (H).

To prove the forthcoming Theorem 3.1, which will then imply Theorems 1.3 and 1.6 (ii), we will need the following technical lemma, which is taken from Chow and Cuzick [START_REF] Chow | Moment conditions for the existence and nonexistence of optimal stopping rules for S n /n 1[END_REF].

Lemma 2.1. (Chow and Cuzick [10, Lemma 3]) Let f be regularly varying at infinity with index γ > 0, and suppose that for all ε > 0 there exists some x(ε) such that

λ γ-ε ≤ f (λx) f (x) ≤ λ γ+ε , ∀ λ ≥ 1, ∀x ≥ x(ε). (11)
Then for all θ > -1,

e f (t) f (t) t θ+1 ∞ t (x -t) θ e -f (x) dx → t→∞ γ -1-θ Γ(1 + θ).
We point out that Chow and Cuzick state their result for all regularly varying functions with a positive index, but that their proof strongly relies on the key point [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], which is not true for any regularly varying function (easy counter-examples can be constructed). However, the functions we are interested in, i.e.

-ln(m), and to which we will apply this result, will in general satisfy [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. In particular, see the forthcoming Lemma 3.6.

Asymptotic behavior of self-similar Markov processes

Given the construction [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF] via subordinators of solutions to the fragmentation equation, the issue of characterizing the large-time asymptotics of these solutions is equivalent to characterizing large-time behavior of distributions of time-changed subordinators.

So, let ξ be a subordinator started from 0 with Lévy measure Π and no drift. We denote by φ its Laplace exponent. Then consider α < 0 and let X(0) be a strictly positive random variable, independent of ξ. Our goal is to specify the asymptotic behavior as t → ∞ of the distributions of the random variables

X(t) := X(0) exp -ξ ρ(X(0) α t) , (12)
conditional on {X(t) > 0}, where ρ is given by

ρ(t) = inf u ≥ 0 : u 0 exp(αξ r )dr > t .
Following Lamperti [START_REF] Lamperti | Semi-stable Markov processes[END_REF], the process X belongs to the so-called family of self-similar Markov processes. This means that it is strongly Markovian and that for all x > 0, if P x denotes the distribution of X started from x, then for all a > 0, the distribution of (aX(a α t), t ≥ 0) under P x is P ax .

Moreover, X reaches 0 a.s. and it does it continuously. Conversely, Lamperti [START_REF] Lamperti | Semi-stable Markov processes[END_REF] also shows that any non-increasing càdlàg self-similar Markov processes on [0, ∞[ that reaches continuously 0 in finite time a.s. can be constructed like this via a time-changed subordinator. Note that the moment at which X reaches 0 is X(0) |α| I where I is the exponential functional defined by

I := ∞ 0 exp(αξ r )dr (13)
which is clearly a.s. finite. The distribution of the random variable I was first studied in details in [START_REF] Carmona | On the distribution and asymptotic results for exponential functionals of Lévy processes[END_REF]. In particular, it is well known that for all integers n ≥ 1,

E[I n ] = n! φ(|α|)φ(2|α|)...φ(n|α|)
and that the distribution of I is characterized by these moments [START_REF] Carmona | On the distribution and asymptotic results for exponential functionals of Lévy processes[END_REF]Prop. 3.3]). It will also be essential for us (see [6, Propositions 1 and 2]) that there exists a unique probability measure µ R on ]0, ∞[ whose entire positive moments are given by

∞ 0 x n µ R (dx) = φ(|α|)φ(2|α|)...φ(n|α|), n ≥ 1 ( 14
)
and that, moreover, if R denotes a r.v. with distribution µ R independent of I, then

RI d = e(1) (15)
where e(1) has an exponential distribution with parameter 1.

We now have the material to state the main result of this section. To be consistent with the notations used for the fragmentation equation, we denote by xµ 0 (dx), x > 0, the distribution of X(0). We recall also the definition of the function ϕ as the inverse, well-defined in a neighborhood of ∞, of t → t/φ(t).

Theorem 3.1. Suppose that ∞ u Π(dx) varies regularly at 0 with index -β, β ∈ [0, 1[ and ∞ xΠ(dx) < ∞.
(i) If the support of µ 0 is compact with a supremum equal to 1, then for all bounded continuous functions

f :]0, ∞[→ R, E f ϕ(|α|t) |α|t 1/|α| X(t) X(t) > 0 → t→∞ E f (R 1/|α| )
where R is the random variable with distribution µ R defined by ( 14).

(ii) If µ 0 possesses a density u 0 in a neighborhood of ∞ such that

ln (u 0 (x)) ∼ ∞ -Cx γ
for some γ > 0, then the function t ∈]0, ∞[→ P(X(t) > 0) is continuously differentiable. If moreover the derivative of t → ln(P(X(t) > 0)) is regularly varying at ∞ -which e.g. is true as soon as this derivative is monotone near ∞ -then, for all bounded continuous functions

f :]0, ∞[→ R, as t → ∞, E f h(t) C α,β,γ,C t 1/|α| X(t) X(t) > 0 → t→∞ E f (R 1/|α| )
where the function h is the inverse, defined in the neighborhood of ∞, of t → t 1+|α|/γ /φ(t) and C α,β,γ,C is the constant defined in the statement of Theorem 1.6.

We will see in the proof of this result that the function t → ϕ(|α|t)/|α|t in assertion (i) can be replaced by any asymptotically equivalent function. Likewise for h in the second assertion. Now, let B be the fragmentation measure related to Π by [START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF]. If (µ t , t ≥ 0) refers to the solution to the (α, B)-fragmentation equation contructed from ξ by the formula [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF], we have

m(t) = ∞ 0 xµ t (dx) = P (X(t) > 0)
and the distribution of X(t) conditional on X(t) > 0 is xµ t (dx)/m(t). The above theorem then leads directly to the statements of Theorem 1.3 and Theorem 1.

6 (ii) (note that ∞ xΠ(dx) < ∞ is equivalent to 0 | ln(x)|xB(dx) < ∞). The limit distribution xµ ∞ (dx) mentioned in these theorems is therefore the distribution of R 1/|α| . The large-time behavior of m(t) = P(X(t) > 0) is studied in Subsection 3.1 below, whereas Theorem 3.1 is established in Subsection 3.2.
Last, we finish with the following result on the quasi-stationary distributions of X, which will be proved in Subsection 3.3, and which, in terms of the fragmentation equation, will lead to Theorem 1.8. We recall that the quasi-stationary distributions of X are the distributions ς on ]0, ∞[ such that

X(0) d ∼ ς ⇒ E[f (X(t))|X(t) > 0] = E[f (X(0))]
, for all t ≥ 0 and all test-functions f defined on ]0, ∞[.

Theorem 3.2. Let µ (λ) R denote the law of λR 1/|α| , λ > 0. Then, a probability measure ς on ]0, ∞[ is a quasi-stationary distributions of X if and only if ς = µ (λ) R for some λ > 0. Moreover, if X(0) d ∼ µ (λ) R , P(X(t) > 0) = exp(-λ α t), ∀t ≥ 0.
We point out that this theorem does not directly lead to the reciprocal assertion of Theorem 1.8. However, easy manipulations of the fragmentation equation will lead to it. See Subsection 3.3 for details.

Total mass behavior

This section is devoted to the description of the behavior of the total mass

m(t) = ∞ 0 xµ t (dx) = P (X(t) > 0) = P (I > X(0) α t) .
The notations are those introduced above in the introduction of Section 3. We start with the following result, which holds for all fragmentation equations with parameters α < 0, B and all initial measure µ 0 such that

m(t) = ∞ 0 P(I > x α t)xµ 0 (dx),
it is sufficient to show that the function t ∈ [0, ∞[→ P(I > t) is strictly positive, strictly decreasing and converges to 0 as t → ∞. This last point is obvious since I < ∞ a.s.. Next, suppose that P(I ≤ t) = 1 for some t > 0. This would imply that for all n ≥ 1

n! φ(|α|)φ(2|α|)...φ(n|α|) = E[I n ] ≤ t n .
But we have seen in the Introduction that x/φ(x) → ∞ as x → ∞. In particular, 2t ≤ n/φ(n|α|) for large enough n, say n > n 0 . Hence we would have

n 0 ! φ(|α|)φ(2|α|)...φ(n 0 |α|) (2t) n-n0 ≤ t n
for all n > n 0 , which is impossible. So, P(I > t) > 0 for all t > 0.

Last, for all t > 0, using the Markov property of subordinators, we get

I = t 0 exp(αξ r )dr + exp(αξ t ) ∞ 0 exp(α(ξ r+t -ξ t ))dr ≤ t + exp(αξ t ) Ĩ,
where Ĩ is distributed as I and independent of ξ t . Consider a such that P(I ≤ a) > 0 and note, using the Poisson point process construction of the subordinator, that P(exp(αξ t ) ≤ t/a) > 0 for all t > 0. Then

0 < P exp(αξ t ) ≤ t/a, Ĩ ≤ a ≤ P(I ≤ 2t), ∀t > 0.
This leads to the fact that P(t ≥ I > s) > 0 for all 0 ≤ s < t. Indeed, the event {I > s} coincides with {ρ(s) < ∞} and when I > s,

I = s + exp(αξ ρ(s) ) ∞ 0 exp α(ξ r+ρ(s) -ξ ρ(s) dr.
Using the strong Markov property of the subordinator at the stopping time ρ(s), we get, with probability one, (I -s) + = exp(αξ ρ(s) ) Ĩ, with Ĩ independent of ξ ρ(s) and distributed as I. Hence, for all 0 ≤ s < t,

P(I > s) -P(I > t) = P(s < I ≤ t) = P exp(αξ ρ(s) ) > 0, Ĩ ≤ (t -s) exp(|α|ξ ρ(s) )
and this last probability is strictly positive since P(exp(αξ ρ(s) ) > 0) = P(I > s) > 0 and P( Ĩ ≤ a) > 0 for all a > 0.

Now we turn to the proofs of the more precise descriptions of the behavior of m stated in Proposition 1.

(I > t)) ∼ ∞ (1 -β) |α| ϕ(|α|t) ∼ ∞ (1 -β)|α| β 1-β ϕ(t)
where ϕ is the inverse of t → t/φ(t), which is well defined in the neighborhood of ∞. In particular, -ln (P(I > t)) is regularly varying at ∞ with index 1/(1 -β).

Proof. Note that the Laplace exponent of the subordinator |α|ξ is φ(|α|•) and that the inverse of t → t/φ(|α|t) is ϕ(|α|•)/|α|. Using this remark, we can restrict our proof to the case |α| = 1, which is supposed in the following. When β ∈]0, 1[, the statement of the lemma is exactly Proposition 2 of Rivero [START_REF] Rivero | A law of iterated logarithm for increasing self-similar Markov processes[END_REF]. When β = 0 and φ(∞) < ∞,

1 n ln E(I n ) n! = - 1 n n i=1 ln(φ(i)) ----→ n→∞ -ln φ(∞).
Then by Lemma 2.3. of König and Mörters [START_REF] König | Brownian intersection local times: upper tail asymptotics and thick points[END_REF] lim

t→∞ 1 t ln (P(I > t)) = -φ(∞).
Last, when β = 0 and φ(∞) = ∞, we can adapt König and Mörters' proof of [START_REF] König | Brownian intersection local times: upper tail asymptotics and thick points[END_REF]Lemma 2.3] to get the expected result. Indeed, first note that

1 n ln E I n φ(n) n n n = 1 n ln n! n n + ln(φ(n)) - 1 n n i=1 ln(φ(i)) → n→∞ -1, (16)
as a consequence of Stirling's formula and of the fact that

ln(φ(n)) - 1 n n i=1 ln(φ(i)) → n→∞ 0
since φ is a slowly varying function (see Section 3.2 of Rivero [START_REF] Rivero | A law of iterated logarithm for increasing self-similar Markov processes[END_REF] for a proof of this last point). Then, it is easy, using Markov's inequality, that lim sup

n→∞ 1 n ln (P (I > n/φ(n))) ≤ -1.
To get a lower bound for the limit inferior, set Y n := ln(Iφ(n)/n). For every ε > 0 and every integer m, we have that 1

E[I n ] E I n 1 {Yn≥ε} ≤ exp(-εm) E[I n+m ]φ(n) m E[I n ]n m → n→∞ exp(-εm). Letting m → ∞, this gives 1 E[I n ] E I n 1 {Yn≥ε} → n→∞ 0. (17)
Besides, for all ε > 0 and all n ≥ 1,

1 n ln (P(I > n exp(-ε)/φ(n))) ≥ 1 n ln (P(|Y n | < ε)) . But I -n > exp(-nε)n -n φ(n) n on {|Y n | < ε}, which gives 1 n ln (P(|Y n | < ε)) = 1 n ln E[I -n I n 1 {|Yn|<ε} ] E[I n ] E[I n ] ≥ 1 n ln exp(-nε) E[I n 1 {|Yn|<ε} ] E[I n ] n -n φ(n) n E[I n ] .
By ( 16) and ( 17), the last line of this inequality converges to -ε -1 as n → ∞. Thus, since the function t → t/φ(t) is increasing and ϕ(t) → ∞ as t → ∞, we have proved that lim sup

t→∞ 1 ϕ(t) ln (P (I > t)) ≤ -1 lim inf t→∞ 1 ϕ(t exp(ε)) ln (P (I > t)) ≥ -ε -1.
Using the regular variation of ϕ, we get the expected

lim t→∞ 1 ϕ(t) ln (P(I > t)) = -1.
3.1.1 µ 0 with compact support: proof of Proposition 1.2

We recall that with no loss of generality, the supremum of the support of µ 0 is supposed to be equal to 1. Then,

m(t) = 1 0 P (I > tx α ) xµ 0 (dx) ≤ P (I > t) 1 0 xµ 0 (dx) = P (I > t) .
According to Proposition 3.3 in [START_REF] Carmona | On the distribution and asymptotic results for exponential functionals of Lévy processes[END_REF], C λ := E[exp(λI)] < ∞ as soon as λ < φ(∞). Hence for such λ's, m(t) ≤ P (I > t) ≤ C λ exp(-λt), ∀t ≥ 0, which gives the first part of the statement. Now, assume (H). Then, on the one hand, since m(t) ≤ P(I > t), we get by Lemma 3.4,

lim inf t→∞ -ln (m(t)) ϕ(|α|t) ≥ 1 -β |α| .
On the other hand, for all 0 < ε < 1,

m(t) ≥ P (I > t(1 -ε) α ) 1 1-ε xµ 0 (dx).
By assumption,

1 1-ε xµ 0 (dx) > 0, hence lim sup t→∞ -ln (m(t)) ϕ(|α|t) ≤ lim sup t→∞ -ln (P(I > t(1 -ε) α )) ϕ(|α|t) = 1 -β |α| (1 -ε) α 1-β .
Then, let ε ↓ 0 to get the expected result.

3.1.2 µ 0 with unbounded support: proofs of Theorem 1.6 (i) and Proposition 1.7

Proof of Theorem 1.6 (i) 1. Suppose first that µ 0 (dx) = exp(-Cx γ )dx, γ > 0. We have

m(t) = ∞ 0 P (I > tx α ) x exp(-Cx γ )dx = t -2/α γ ∞ 0 P I > u α/γ u 2/γ-1 exp -Cut -γ/α du, (18)
using the change of variable u = (xt 1/α ) γ . Then use Lemma 3.4 and Theorem 4.12.10 (iii) of [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF] to get -ln

x 0 P I > u α/γ u 2/γ-1 du ∼ x→0 -ln P I > x α/γ ∼ x→0 (1 -β)|α| β/(1-β) ϕ(x α/γ )
which varies regularly at 0 with index α/(γ(1 -β)). Note that in a neighborhood of 0, x → 1/ϕ(x α/γ ) is the inverse of

x → xφ 1 x -γ/α
.

Hence, by de Bruijn's Tauberian Theorem [7, Th. 4.12.9], we have

-ln ∞ 0 P I > u α/γ u 2/γ-1 exp (-ut) du ∼ t→∞ C α,β,γ /h 0 (t),
where h 0 is the inverse, well-defined in the neighborhood of ∞, of x → x -1 (xφ(1/x)) γ/α and C α,β,γ is the constant defined in the statement of Theorem 1.6 (i). Together with [START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF], this leads to

-ln (m(t)) ∼ ∞ C α,β,γ C (1+(1-β)γ/|α|) -1 /h 0 (t γ/|α| ). Otherwise said, -ln (m(t)) ∼ ∞ C α,β,γ C (1+(1-β)γ/|α|) -1 h(t)
where h is the inverse of t 1+|α|/γ /φ(t).

2. Now, suppose that µ 0 possesses a density u 0 in a neighborhood of ∞ such that ln (u 0 (x)) ∼ ∞ -Cx γ , γ > 0. Fix ε > 0 and let C ε be such that u 0 (x) exists for x ≥ C ε and exp(-

(1 + ε)Cx γ ) ≤ u 0 (x) ≤ exp(-(1 -ε)Cx γ ), ∀x ≥ C ε . (19)
Then write

m(t) = Cε 0 P (I > tx α ) xµ 0 (dx) + ∞ Cε P (I > tx α ) xu 0 (x)dx.
On the one hand, following the argument developed in Section 3.1.1, we get lim sup

t→∞ ln Cε 0 P (I > tx α ) xµ 0 (dx) ϕ(t) ≤ -(1 -β)|α| β/(1-β) C α/(1-β) ε ,
which actually holds for any initial measure µ 0 . Note that ϕ(t)/h(t) → ∞ as t → ∞, where h is the function defined above, in the first part of this proof.

On the other hand, the inequalities [START_REF] König | Brownian intersection local times: upper tail asymptotics and thick points[END_REF] and the results of the first part of this proof imply that lim sup t→∞ -ln

∞ Cε P (I > tx α ) xu 0 (x)dx h(t) ≤ C α,β,γ ((1 + ε)C) (1+(1-β)γ/|α|) -1 and lim inf t→∞ -ln ∞ Cε P (I > tx α ) xu 0 (x)dx h(t) ≥ C α,β,γ ((1 -ε)C) (1+(1-β)γ/|α|) -1 .
We have therefore proved that

C α,β,γ ((1 -ε)C) (1+(1-β)γ/|α|) -1 ≤ lim inf t→∞ -ln(m(t)) h(t) ≤ lim sup t→∞ -ln(m(t)) h(t) ≤ C α,β,γ ((1 + ε)C (1+(1-β)γ/|α|) -1
for all ε > 0. The result follows by letting ε ↓ 0.

Proof of Proposition 1.7. Suppose that u 0 (x) = Cx -γ on [a, ∞[ for some a > 0 and γ > 2. Then,

m(t) = C ∞ a P(I > x α t)x 1-γ dx + a 0 P(I > x α t)xµ 0 (dx).
With the change of variables u = x α t,

∞ a P(I > x α t)x 1-γ dx = t γ-2 α |α| a α t 0 P(I > u)u 2-γ α -1 du
and this last integral converges to a finite limit as t → ∞ since P(I > u) ≤ C λ exp(-λu) for all u ≥ 0 and some λ > 0 sufficiently small (see the proof of Proposition 1.2 for this last point). Using the same upper bound for P(I > x α t), we get that

a 0 P(I > x α t)xµ 0 (dx) ≤ C λ exp(-λa α t) a 0 xµ 0 (dx).
Thus,

m(t) ∼ t→∞ C |α| t γ-2 α ∞ 0 P(I > u)u 2-γ α -1 du.
It is not hard to extend this proof to the cases when u 0 (x) ∼ ∞ Cx -γ , for some γ > 2, which is left to the reader.

Proof of Theorem 3.1

We start with the following lemma. Lemma 3.5. Suppose that -ln(m) varies regularly at ∞ with a positive index γ and satisfies [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. Then, for any function g

: [0, ∞[→]0, ∞[ such that g(t)/(-ln(m(t))) → 1 as t → ∞, we have E f γg(t) t 1/|α| X(t) X(t) > 0 → t→∞ E f (R 1/|α| ) ,
for all continuous bounded test functions f on ]0, ∞[.

Proof. First note that when X(0

) |α| I > t, X(0) |α| I = X(0) |α| ρ(X(0) α t) 0 exp(αξ r )dr + X(0) |α| exp(αξ ρ(X(0) α t) ) ∞ 0 exp(α(ξ r+ρ(X(0) α t) -ξ ρ(X(0) α t) ))dr = t + X(0) |α| exp(αξ ρ(X(0) α t) ) ∞ 0 exp(α(ξ r+ρ(X(0) α t) -ξ ρ(X(0) α t) ))dr.
Then use the strong Markov property of ξ at the (randomized) stopping time ρ(X(0

) α t) to get (X(0) |α| I -t) + = X(0) |α| exp(αξ ρ(X(0) α t) ) Ĩ = X(t) |α| Ĩ, ( 20
)
where Ĩ is distributed as I and is independent of X(t). This gives, for all n ∈ N * ,

m(t) -1 E   γg(t) t 1/|α| X(t) |α|n   E[I n ] = m(t) -1 γg(t) t n E X(t) |α|n E[I n ] = m(t) -1 γg(t) t n E (X(0) |α| I -t) + n .
Then recall that m(t) = P(X(0

) |α| I > t), t ≥ 0.
Integrating by parts,

m(t) -1 γg(t) t n E (X(0) |α| I -t) + n = nm(t) -1 γg(t) t n ∞ t (x -t) n-1 m(x)dx,
which, according to Lemma 2.1 and the assumptions we have made on -ln(m) and g, converges as t → ∞ to n!. Next note that E[R n ]E[I n ] = n!, using the factorization property [START_REF] Fournier | On small particles in coagulation-fragmentation equations[END_REF] of the exponential random variable with parameter 1. Putting all the pieces together, we have proved that for all integers n ≥ 1,

E   γg(t) t 1/|α| X(t) |α|n X(t) > 0   → t→∞ E[R n ].
To sum up: let ν t denote the distribution of γt -1 g(t)X(t) |α| conditional on X(t) > 0 (ν t is a probability measure on ]0, ∞[). We have shown that for all n ≥ 1,

∞ 0 x n ν t (dx) → ∞ 0 x n µ R (dx),
where µ R is the distribution of R. Of course this still holds for n = 0. But the distribution of R is characterized by its moments. It is then well-known ([13, Chapter VIII, p.269]) that this implies that ν t converges in distribution to µ R .

Proof of Theorem 3.1 (i)

By Proposition 1.2, under the hypothesis (H), -ln(m) varies regularly at ∞ with index 1/(1 -β) and more precisely

-ln(m(t)) ∼ t→∞ (1 -β) |α| ϕ(|α|t).
Together with Lemma 3.5, this implies the statement of Theorem 3.1, provided -ln(m) satisfies [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. The goal of this section is to proof this last point when µ 0 has a compact support.

Lemma 3.6. Let f (x) = -ln(m(x)), x ≥ 0, and assume (H), that ∞ xΠ(dx) < ∞ and that µ 0 has a compact support. Then for all ε > 0, there exists some x(ε) such that

λ 1 1-β -ε ≤ f (λx) f (x) ≤ λ 1 1-β +ε , ∀λ ≥ 1 and ∀x ≥ x(ε).
This lemma is a direct consequence of the forthcoming Lemmas 3.7 and 3.10.

Lemma 3.7. Let g :]0, ∞[→]0, ∞[ be a continuously differentiable function such that xg ′ (x) g(x) → c > 0 as x → ∞.
Then for all ε > 0 there exists some x(ε) such that

λ c-ε ≤ g(λx) g(x) ≤ λ c+ε , ∀λ ≥ 1 and ∀x ≥ x(ε).
Proof. For ε > 0, let x(ε) be such that

c -ε ≤ xg ′ (x) g(x) ≤ c + ε for all x ≥ x(ε).
For such xs, and all λ ≥ 1

(c -ε) ln(λ) = (c -ε) λx x y -1 dy ≤ λx x g ′ (y) g(y) dy ≤ (c + ε) λx x y -1 dy = (c + ε) ln(λ). Since λx x g ′ (y)
g(y) dy = ln(g(λx)) -ln(g(x)), the result is proved.

Lemma 3.8. Suppose that φ is regularly varying at ∞ with index β ∈ [0, 1[ and that φ(x) → ∞ as x → ∞. Let β ′ ∈]β, 1[. Then there exists some x 1 (β ′ ) such that for x ≥ x 1 (β ′ ) and all λ ≥ 1,

1 ≤ φ(x) ≤ φ(λx) ≤ λ β ′ φ(x) and φ(x) ≤ x β ′ .
Proof. Note that φ is infinitely differentiable on ]0, ∞[, with derivative

φ ′ (x) = ∞ 0 v exp(-xv)Π(dv),
which is non-increasing. It is then a classical result on regular variation (see the Monotone Density Theorem, [7, Th.1.7.2]) that φ ′ is regularly varying with index β -1 and

xφ ′ (x) φ(x) → x→∞ β.
The first part of the lemma is then a consequence of the above Lemma 3.7 and of the fact that φ is increasing and converges to ∞. We also have that φ(x)/x β ′ converges to 0 at ∞ (since β ′ > β), hence the second assertion holds for x large enough.

Lemma 3.9. Let f (x) = -ln (P(I > x)) , x ≥ 0, which, as proved in Lemma 3.4 is regularly varying with index 1/(1 -β), under the assumption (H). Suppose moreover that ∞ xΠ(dx) < ∞. Then f is infinitely differentiable and

xf ′ (x) f (x) → 1 1 -β as x → ∞.
Proof. According to [9, Prop. 2.1], when ∞ xΠ(dx) < ∞, there exists an infinitely differentiable function

k :]0, ∞[→ [0, ∞[ such that k(x)dx is the distribution of I. Moreover, k(x) = ∞ x Π |α| -1 ln(u/x) k(u)du = ∞ 0 xe v|α| x k(u)du Π(dv).
To simplify notations, we suppose in the following that |α| = 1. The proof is identical for |α| = 1. In particular, we have,

P(I > x) = ∞ x k(u)du, and
f ′ (x) = k(x) P(I > x) = ∞ 0 (1 -exp (f (x) -f (xe v ))) Π(dv), x > 0. ( 21
)
Note that since f is regularly varying with a positive index, we have that f (x) → ∞ as x → ∞, and therefore, for all v > 0,

f (x) -f (xe v ) = f (x) 1 - f (xe v ) f (x) ∼ ∞ f (x) 1 -e v/(1-β) → x→∞ -∞.
• When Π(]0, ∞[) < ∞, this implies the expected result, since, by dominated convergence,

f ′ (x) → x→∞ Π(]0, ∞[) = lim x→∞ f (x)
x .

• The proof is much more technical when Π(]0, ∞[) = ∞, which is supposed for the rest of this proof. We proceed in two steps.

Step 1. The goal of this step is to prove that lim inf

x→∞ xf ′ (x) f (x) ≥ 1 1 -β .
To start with, suppose there exists some x 0 and some non-decreasing positive function g such that f ′ (x) ≥ g(x) for all x ≥ x 0 . Then for x ≥ x 0 and v > 0

f (xe v ) -f (x) = xe v x f ′ (u)du ≥ g(x)
x(e v -1) ≥ g(x)xv.

Using [START_REF] Laurençot | On a class of continuous coagulation-fragmentation equations[END_REF], this gives

f ′ (x) ≥ φ(g(x)x), x ≥ x 0 . (22)
Now, note that f ′ (x) → ∞ as x → ∞ since for all a > 0 lim inf

x→∞ f ′ (x) ≥ lim inf x→∞ ∞ a (1 -exp (f (x) -f (xe v ))) Π(dv) = ∞ a Π (dv)
(by dominated convergence) and this right-hand side converges to ∞ as a → 0. In particular, f ′ (x) ≥ 1 for x sufficiently large (say x ≥ x 0 ). Replacing g by 1 in ( 22), we get

f ′ (x) ≥ φ(x), ∀x ≥ x 0 .
Recall that φ is non-decreasing and then iterate the procedure to get, for all n ≥ 0

f ′ (x) ≥ h n (x), ∀x ≥ x 0 , (23)
where the functions h n :]0, ∞[→ ]0, ∞[ are defined by induction by

h 0 (x) = 1, for all x ≥ 0 h n (x) = φ(h n-1 (x)x), for all x ≥ 0.
Now the interesting fact is that for x large enough, h n (x) → ϕ(x)/x as n → ∞. Indeed, let β ′ ∈]β, 1[. With the notations of Lemma 3.8 we have for x ≥ x 1 (β ′ ), 1 ≤ φ(x) ≤ x β ′ , i.e. h 0 (x) ≤ h 1 (x) ≤ x β ′ . Using that φ is non-decreasing, we easily have, by induction, that

1 ≤ h n (x) ≤ h n+1 (x) ≤ x β ′ +....+β ′n+1 ≤ x β ′ /(1-β ′ ) < ∞,
for all n ≥ 1. Let l(x) := lim n→∞ h n (x). We have shown that 0 < l(x) < ∞. Then necessarily l(x) = φ(l(x)x), otherwise said l(x)x/φ(l(x)x) = x and finally, l(x)x = ϕ(x), ∀x ≥ x 1 (β ′ ). To conclude, for x large enough, letting n → ∞ in (23), we get f ′ (x) ≥ ϕ(x)/x, which, combined with Lemma 3.4, gives the expected liminf.

Step 2. The proof of the limsup is similar, but more technical. To start with, note that for all ε > 0 and all a < ln(1 + ε), a > 0,

lim inf x→∞ ln(1+ε) 0 (1 -exp (f (x) -f (xe v ))) Π(dv) ≥ lim inf x→∞ ln(1+ε) a (1 -exp (f (x) -f (xe v ))) Π(dv) = ln(1+ε) a Π(dv) (by dominated convergence) → a→0 ∞, whereas lim n→∞ ∞ ln(1+ε) (1 -exp (f (x) -f (xe v ))) Π(dv) = ∞ ln(1+ε) Π(dv) < ∞.
Hence, there exists some x 1 (ε) such that for x ≥ x 1 (ε)

f ′ (x) ≤ (1 + ε) ln(1+ε) 0 (1 -exp (f (x) -f (xe v ))) Π(dv). (24)
Next, fix some β ′ ∈]β, 1[ and consider some δ > 0 and ε > 0 such that (1 + δ)(1 + ε) 1/(β-1) β ′ < 1. Since f is regularly varying with index 1/(1 -β), there exists some x 2 (δ, ε) such that

f (x(1 + ε)) ≤ (1 + δ)(1 + ε) 1/(1-β) f (x), ∀x ≥ x 2 (δ, ε). (25)
We will need this later. For the moment, let x 0 = max(x 1 (β ′ ), x 1 (ε), x 2 (δ, ε)), with x 1 (β ′ ) as introduced in Lemma 3.8. Next, suppose that for all x ≥ x 0

f ′ (x) ≤ g(x)
for some non-decreasing function g s.t. g(x) ≥ 1 for all x ≥ x 0 . Note that this implies that

f (xe v ) -f (x) = xe v x f ′ (u)du ≤ g(xe v )x(e v -1). The function v → v -1 (e v -1) is increasing on [0, ∞[, hence e v -1 ≤ vγ(ε) for all v ≤ ln(1 + ε), where γ(ε) = ε/(ln(1 + ε)).
Together with [START_REF] Miermont | Self-similar fragmentations derived from the stable tree. I. Splitting at heights[END_REF] this leads to

f ′ (x) ≤ (1 + ε) ln(1+ε) 0 (1 -exp (-g(x(1 + ε))xvγ(ε))) Π(dv) ≤ (1 + ε)φ (g(x(1 + ε))xγ(ε)) (26)
for all x ≥ x 0 . Then we claim that for all n ≥ 1 and all x ≥ x 0

f ′ (x) ≤ (1 + ε) 1+β ′ +2β ′2 +...+nβ ′n γ(ε) β ′ +β ′2 +...+β ′n g (x(1 + ε) n ) β ′n h n (x), (27)
where the sequence of functions h n is that introduced in the step 1 of this proof. We will prove this by induction on n. Before doing this, let us just mention that, by an easy induction, using Lemma 3.8,

h n (x(1 + ε)) ≤ (1 + ε) β ′ +...+β ′n h n (x)
, for all x ≥ x 0 and n ≥ 1.

We now turn to the proof of [START_REF] Wagner | Explosion phenomena in stochastic coagulation-fragmentation models[END_REF]. For n = 1, use [START_REF] Rivero | A law of iterated logarithm for increasing self-similar Markov processes[END_REF] and Lemma 3.8 to get (note that γ(ε) ≥ 1, hence γ(ε)g ≥ 1)

f ′ (x) ≤ (1 + ε)g(x(1 + ε)) β ′ γ(ε) β ′ φ(x), x ≥ x 0 ,
which leads to [START_REF] Wagner | Explosion phenomena in stochastic coagulation-fragmentation models[END_REF] for n = 1. Now assume ( 27) is true for some integer n. Note that the function in the right-hand side of this inequality, which we call g 1 , is larger than 1 for all x ≥ x 0 . Note also that it is non-decreasing. Hence we get, replacing g by g 1 in (26), for x ≥ x 0 ,

f ′ (x) ≤ (1 + ε)φ (1 + ε) 1+β ′ +2β ′2 +...+nβ ′n γ(ε) β ′ +β ′2 +...+β ′n g x(1 + ε) n+1 β ′n h n (x(1 + ε))xγ(ε) ≤ (1 + ε)φ (1 + ε) 1+2β ′ +3β ′2 +...+(n+1)β ′n γ(ε) 1+β ′ +β ′2 +...+β ′n g x(1 + ε) n+1 β ′n h n (x)x ≤ (1 + ε) 1+β ′ +2β ′2 +...+(n+1)β ′n+1 γ(ε) β ′ +β ′2 +...+β ′n+1 g x(1 + ε) n+1 β ′n+1 φ (h n (x)x) ,
where for the last inequality, we have used Lemma 3.8. Hence we have ( 27) for all n ≥ 1. Now, thanks to the assumptions (H) and ∞ xΠ(dx) < ∞ and to Lemma 1 in [START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF], we know that the function k is bounded from above on ]0, ∞[, say by some constant C ≥ 1. Hence f ′ (x) = k(x)/P(I > x) ≤ C exp(f (x)) for all x > 0. Since f is non-decreasing and non-negative, the function x → C exp(f (x)) is non-decreasing, greater than 1 hence we can replace g by this function in [START_REF] Wagner | Explosion phenomena in stochastic coagulation-fragmentation models[END_REF] to get for all n ≥ 1 and all

x ≥ x 0 f ′ (x) ≤ (1 + ε) 1+β ′ +2β ′2 +...+nβ ′n γ(ε) β ′ +β ′2 +...+β ′n C β ′n exp (β ′n f (x(1 + ε) n )) h n (x). ( 28
)
Our goal now, is to let n → ∞ in this inequality. Iterating inequality (25), we get for x ≥ x 0 and for all

n ≥ 1 f (x(1 + ε) n ) ≤ (1 + δ) n (1 + ε) n/(1-β) f (x). Since (1 + δ)(1 + ε) 1/(β-1) β ′ < 1, this leads, for x ≥ x 0 , to exp (β ′n f (x(1 + ε) n )) → n→∞ 1.
As n → ∞, we also have

C β ′n → 1 and (1 + ε) 1+β ′ +2β ′2 +...+nβ ′n → (1 + ε) 1+β ′ /(1-β ′ ) 2 and γ(ε) β ′ +β ′2 +...+β ′n → γ(ε) β ′ /(1-β ′ ) .
Last, recall that for x large enough, h n (x) → ϕ(x)/x as n → ∞. Letting n → ∞ in (28), we therefore have for x large enough

f ′ (x) ≤ C ε ϕ(x)/x,
where C ε → 1 as ε → 0. This gives lim sup

x→∞ xf ′ (x) f (x) ≤ 1 1 -β . Lemma 3.10. Let f (x) := -ln (m(x)) , x ≥ 0,
and suppose (H), that ∞ xΠ(dx) < ∞ and that µ 0 has a compact support. Then f is differentiable on ]0, ∞[

and xf ′ (x) f (x) = - xm ′ (x) m(x)f (x) → 1 1 -β as x → ∞.
Proof. With no loss of generality, we suppose that the supremum of the support of µ 0 is equal to 1. Under the assumptions of the lemma, we know (see the proof of the previous lemma) that x → P(I > x) is differentiable on ]0, ∞[, with derivative -k. By Lemma 1 in [START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF], we also know that the function

x ∈]0, ∞[→ xk(x) is bounded. Let M denotes an upper bound. Recall then that m(x) = 1 0 P(I > xy α )yµ 0 (dy)
and note that for all x > a > 0 and all y ∈]0, 1[,

∂ x (P(I > xy α )) = k(xy α )y α ≤ M a .
Hence, by dominated convergence, m is continuously differentiable on ]0, ∞[, with derivative

m ′ (x) = - 1 0 k(xy α )y α yµ 0 (dy), x > 0.
Now, fix δ > 0. By Lemma 3.9, there exists some x(δ) such that for x ≥ x(δ),

1 -δ 1 -β ≤ -xk(x) P(I > x) ln (P(I > x)) ≤ 1 + δ 1 -β .
Then for x ≥ x(δ), (29) Now, let ε > 0. On the one hand, we claim that (30)

1 -δ (1 -β)x 1 0 P(I > xy α ) ln (P(I > xy α )) yµ 0 (dy) ≤ m ′ (x) ≤ 1 + δ (1 -β)x
Indeed, for all 0 < y < 1 -ε,

P(I > xy α ) ln (P(I > xy α )) P(I > x(1 -ε) α ) ln (P(I > x(1 -ε) α )) → 0 as x → ∞ since x → -ln (P(I > x)
) is regularly varying at ∞ with a positive index and α < 0. It is then not hard to see, using Lemmas 3.7 and 3.9 that for x large enough this function is bounded from above by exp

  1 - y 1 -ε α(1-ε) 1-β   y 1 -ε α(1+ε) 1-β
which, in turn, is bounded for y ∈]0, 1 -ε[. Hence, by dominated convergence, we see that where we have used for the last inequality that the function x → -x ln(x) is increasing in a neighborhood of 0. Hence (30). A similar, but simpler, argument leads to On the other hand, using that x → ln (P(I > x)) is regularly varying with index 1/(1 -β), we have for 1 -ε ≤ y ≤ 1 and x sufficiently large (say x ≥ x(ε))

(1 + ε)(1 -ε) α/(1-β) ln (P(I > x)) ≤ ln (P(I > x(1 -ε) α )) ≤ ln (P(I > xy α )) ≤ ln (P(I > x)) .
Thus

≤ (1+ε)(1-ε) α/(1-β) 1 1-ε P(I > xy α )yµ 0 (dy).
Which, taking x(ε) larger if necessary and using ( 30) and (31), gives for x ≥ x(ε)

(1 -δ)m(x) ≤ 1 0 P(I > xy α ) ln (P(I > xy α )) yµ 0 (dy) ln (P(I > x))

≤ (1 + δ)(1 + ε)(1 -ε) α/(1-β) m(x).
Plugging this in (29) and letting first ε → 0 and then δ → 0, we get the expected convergence, since f (x) ∼ -ln(P(I > x)) as x → ∞.

Proof of Theorem 3.1 (ii)

The fact that the function

x ∈]0, ∞[→ f (x) := -ln(m(x)) = -ln (P(X(x) > 0))
is continuously differentiable on ]0, ∞[ can be proved exactly as when the support of µ 0 is compact. See the beginning of the proof of Lemma 3.10. Next, by Karamata's Theorem (Th. 1.5.11 of [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]), if f varies regularly at ∞ with index λ > 0 and if its derivative is also regularly varying at ∞, then

xf ′ (x) f (x) → λ as x → ∞.
Together with Theorem 1.6 (i), Lemma 3.7 and Lemma 3.5, this implies Theorem 3.1 (ii).

Quasi-stationary distributions

Proof of Theorem 3.3. When X(0) ∼ µ

(λ)
R , the distribution of X(0) |α| I is that of λ |α| RI, with R independent of I, i.e. that of an exponential random variable with parameter λ α . We then immediately have that for n ≥ 1 and t ≥ 0,

E (X(0) |α| I -t) + n = λ |α|n n! exp(-λ α t), and
P(X(t) > 0) = P(X(0) |α| I > t) = exp(-λ α t).
Following the beginning of the proof of Lemma 3.5 this gives

E (X(t)) |α|n E[I n ] = E (X(0) |α| I -t) + n = λ |α|n n! exp(-λ α t)
and then

E (X(t)) |α|n |X(t) > 0 = E[λ |α|n R n ] = E X(0) |α|n . Hence µ (λ)
R is a quasi-stationary distribution, since the distribution of R is characterized by its entire positive moments. Note there is no other quasi-stationary distribution. Indeed, let ς be a quasi-sationary distribution and suppose X(0) ∼ ς. Then necessarily, by the Markov property of X, P(X(t + s) > 0) = P(X(t) > 0)P(X(s) > 0) which implies that X(0) |α| I has an exponential distribution. Say with parameter ℓ, i.e. ℓX(0) |α| I has a exponential distribution with parameter 1. Since the factorization (15) characterizes the distribution of R, we get that ς = µ

(ℓ 1/α ) R .
Proof of Theorem 1.8. The first part of this theorem is an obvious consequence of Theorem 3.3. The reverse cannot be directly deduced from Theorem 3.3, since we do not know if uniqueness holds for the fragmentation equation when the initial measure has an unbounded support.

So, consider (µ t , t ≥ 0) a quasi-stationary solution to the fragmentation equation ( 2). We want to prove that this solution belongs to the family of solutions (µ

(λ)
∞,t , t ≥ 0), λ > 0 as defined in Theorem 1.8. Replacing µ t by m(t)µ 0 in equation ( 2), we get that

(1 -m(t)) < µ 0 , f >= - t 0 m(s)ds < µ 0 , G(f ) >, ∀f ∈ C 1 c ,
where

G(f )(x) = x α 1 0 (f (xy) -f (x)y)B(dy).
Otherwise said, there exists some constant C > 0 such that m(t) = exp(-Ct), ∀t ≥ 0, and

< µ 0 , f >= -C -1 < µ 0 , G(f ) >, ∀f ∈ C 1 c . When f ∈ C 1 c , the function x → xf (x) is also in C 1 c .
Hence the above identity rewrites

< xµ 0 , f >= -C -1 < xµ 0 , Ã(f ) >, ∀f ∈ C 1 c , (32)
where Ã(f )(x) = x α 1 0 (f (xy) -f (x))yB(dy). To show that this characterizes µ 0 , we need the following fact: for all β > 0, there exists a non-decreasing sequence of functions

f β,n :]0, ∞[→ [0, ∞[ such that f β,n (x) → x β as n → ∞, ∀x > 0; and f β,n ∈ C 1 c and |f ′ β,n (x)| ≤ βx β-1
for all x > 0 and all n ≥ 1. This sequence can, for example, be constructed by considering first a non-decreasing sequence of continuous functions g

β,n :]0, ∞[→ [0, ∞[ such that g β,n (x) ≤ βx β-1 , ∀x > 0, n ≥ 1, g β,n (x) = βx β-1 for x ∈ [n -1 , n] and g β,n (x) = 0 for x ∈]0, (2n) -1 ] ∪ [2n, ∞[. Then set f β,n (x) := x 0 g β,n (u)du for x ∈]0, 2n] and extend these functions on ]2n, ∞[ so that f β,n ∈ C 1 c and |f ′ β,n (x)| ≤ βx β-1
, for all x > 0 and all n ≥ 1, and the sequence (f β,n , n ≥ 1) is non-decreasing. For all β > 0, this implies that for all x > 0

Ã(f β,n )(x) → n→∞ x α+β 1 0 (y β -1)yB(dy) = -x α+β φ(β), (33)
together with

| Ã(f β,n )(x)| ≤ (2 + β)x α+β 1 0 (1 -y)yB(dy). (34)
Indeed, this is obvious when β ≥ 1: we just need that

|f β,n (xy) -f β,n (x)| ≤ sup z∈[xy,x] |f ′ β,n (z)|x(1 -y) ≤ βx β (1 -y), for y ∈]0, 1[, x > 0,
and then use the dominated convergence theorem. The case 0 < β < 1 needs more care. Using the above mentioned properties of f β,n and also that f β,n (x) ≤ x β we obtain, for x > 0 and y ∈]0, 1[

x|f β,n (xy) -f β,n (x)| ≤ xf β,n (xy)(1 -y) + |xyf β,n (xy) -xf β,n (x)| ≤ x 1+β (1 -y) + sup z∈[xy,x] |(idf β,n ) ′ (z)|x(1 -y) ≤ x 1+β (1 -y) + (1 + β)x 1+β (1 -y),
which leads to (33) and (34). Now, take β = |α|. Then use (33), (34) and the dominated convergence theorem in the right-hand side of (32) (recall that xµ 0 (dx) is a probability measure), together with the monotone convergence theorem in the left-hand side of (32) to get

∞ 0 x |α| xµ 0 (dx) = C -1 φ(|α|) < ∞.
Then, by an obvious induction, taking successively β = 2|α|, β = 3|α|, etc., we get for all n ≥ 1

∞ 0 x n|α| xµ 0 (dx) = C -1 φ(n|α|) ∞ 0 x (n-1)|α| xµ 0 (dx) = C -n φ(n|α|)...φ(|α|).
We recognize the moments formula [START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF]. Hence xµ 0 (dx) = µ

(C 1/α ) R (dx) = xµ (C 1/α ) ∞
(dx) and for all t ≥ 0,

µ t = m(t)µ 0 = exp(-Ct)µ (C 1/α ) ∞ = µ (C 1/α ) ∞,t .
4 Different speeds of decrease: proof of Proposition 1.4

Proof of Proposition 1.4 (i)

Recall that the support of µ 0 is supposed compact with supremum 1. The goal of this section is to prove the forthcoming Corollory 4.3, which is the statement of Proposition 1.4 (i) translated in terms of the process X defined by [START_REF] Ethier | Markov processes[END_REF], provided the Lévy measure Π of the subordinator ξ involved in the construction of X is related to the fragmentation measure B by (6) and X(0) is distributed according to xµ 0 (dx). We recall that the distribution of X(t) conditional on X(t) > 0 is then xµ t (dx)/m(t), t ≥ 0, where (µ t , t ≥ 0) denotes the solution to the fragmentation equation starting from µ 0 . We start with some preliminary lemmas. Lemma 4.1. Suppose (H) and that 0 | ln(x)|xB(dx) < ∞. Consider some r.v. I independent of X, with distribution that of ∞ 0 exp(αξ r )dr. Then (i) there exists some t 0 > 0 such that sup t≥t0,a>0

a α P ϕ(|α|t) |α|t 1/|α| X(t)I 1/|α| ≤ a X(t) > 0 < ∞ (ii) for all positive function g : [0, ∞[→]0, ∞[ converging to → 0 at ∞, we have, as t → ∞, g(t) α P ϕ(|α|t) |α|t 1/|α| X(t)I 1/|α| ≤ g(t) X(t) > 0 → 1.
Proof. To simplify notations, suppose α = -1 (the proof is identical for all α < 0). Recall then the key equality in law [START_REF] Lamperti | Semi-stable Markov processes[END_REF], which leads to the following identities for all a > 0

a -1 P ϕ(t) t X(t)I ≤ a X(t) > 0 = 1 am(t) P 0 < X(0)I -t ≤ at ϕ(t) = m(t) -m(t + at/ϕ(t)) am(t) (35) = 1 a 1 -exp -ln(m(t)) 1 - ln (m(t(1 + a/ϕ(t)))) ln(m(t)) .
Use then the regular variation of -ln (m) with index 1/(1 -β) (Prop. 1.2) and Lemma 3.6 to see that for all ε > 0, there exists a real t(ε) such that for all t ≥ t(ε) and all a > 0,

1 -(1 + a/ϕ(t)) 1 1-β +ε ≤ 1 - ln (m(t(1 + a/ϕ(t)))) ln(m(t)) ≤ 1 -(1 + a/ϕ(t)) 1 1-β -ε . (36) Now, let 0 < ε < 1 -β. Since 1 -(1 + x) 1 1-β +ε ≥ -x 1 1 -β + 2ε
for all x > 0 sufficiently small, since moreover ϕ(t) → ∞ as t → ∞ and -ln(m) ∼ ∞ (1 -β)ϕ, we have that for all 0 < a ≤ 1 and all t ≥ t ′ (ε) (for some t ′ (ε) depending on ε but not on 0 < a ≤ 1),

-ln(m(t)) 1 -(1 + a/ϕ(t)) 1 1-β +ε ≥ ln(m(t)) ϕ(t) a 1 1 -β + 2ε ≥ -a(1 -β + ε) 1 1 -β + 2ε .
Together with the identities (35) and inequalities (36) this implies that for all t ≥ max(t(ε), t ′ (ε))

sup 0<a≤1 a -1 P ϕ(t) t X(t)I ≤ a X(t) > 0 < ∞.
This is enough to get (i), since for a ≥ 1, a -1 times a probability is bounded by 1.

The proof of (ii) relies on the same idea. Since

g(t)/ϕ(t) → 0 as t → ∞, -ln(m(t)) 1 -(1 + g(t)/ϕ(t)) 1 1-β +ε ∼ ∞ -g(t)(1 -β) 1 1 -β + ε ,
and a similar result holds by replacing ε by -ε. Together with the inequalities (36) and the identities (35) (replacing there a by g(t)), using also that g(t) → 0 as t → ∞, we get (ii).

Lemma 4.2. Suppose κ :=

1 0 | ln(x)|xB(dx) < ∞. Then, (i) I possesses a density k ∈ C ∞ (]0, ∞[), (ii) E[I -1 ] = κ|α| < ∞,
(iii) if moreover the support of B is not included in a set of the form {a n , n ∈ N} for some a ∈]0, 1[, then the function

x ∈ R → E[I ix-1 ] = ∞ 0 y ix-1 k(y)dy
is well-defined and non-zero for all real x.

Proof. If Π is the Lévy measure associated with the fragmentation equation, the assumption κ < ∞ is equivalent to ∞ 0 xΠ(dx) < ∞, which, by Propositions 3.1 and 2.1 of [START_REF] Carmona | On the distribution and asymptotic results for exponential functionals of Lévy processes[END_REF] implies (i) and (ii). Next, it was proved in the proof of Theorem 2 of [START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF] that E[I ix-1 ] = 0 for all x ∈ R under the additional assumption that the support of Π is not included in a set of the form {rn, n ≥ 0} for some r > 0.

Corollary 4.3. Suppose (H), that κ = 1 0 | ln(x)|xB(dx) < ∞ and that the support of B is not included in a set of the form {a n , n ∈ N} for some a ∈]0, 1[. Then, for all measurable functions g :

[0, ∞[→]0, ∞[ converging to → 0 at ∞, g(t) α P ϕ(|α|t) |α|t 1/|α| X(t) g(t) ≤ 1 X(t) > 0 → 1 |α|κ . Proof. Set for x ≥ 0 U t (x) := g(t) α P ϕ(|α|t) |α|t 1/|α| X(t) ≤ xg(t) X(t) > 0
and note this quantity increases in x when t is fixed. Then consider some r.v. I independent of X, with distribution that of ∞ 0 exp(αξ r )dr. Consider b such that P(I ≤ b) > 0. Then

U t (x)P(I ≤ b) ≤ g(t) α P ϕ(|α|t) |α|t 1/|α| X(t)I 1/|α| ≤ b 1/|α| xg(t) X(t) > 0 ,
which, according to Lemma 4.1 (i), is bounded from above by some constant (independent of t and x) times bx |α| for all x ≥ 0 and t ≥ t 0 . I.e., there exists some finite constant C such that for all t sufficiently large and all x ≥ 0,

x α U t (x) ≤ C. (37)
Now, consider an increasing function l : N → N. For all x ≥ 0, the sequence (U l(n) (x), n ≥ 0) is bounded. Hence there exist some non-decreasing right-continuous function

U : [0, ∞[→ [0, ∞[, with U (0) = 0, and a subsequence (U l(n) , n ≥ 0) of (U l(n) , n ≥ 0) such that U l(n) (x) → U (x)
for a.e. x > 0. See e.g. [Theorem 2, Section VIII.7] [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. Hence if we prove that the limit U is given by

U (x) = x |α| |α|κ , ∀x ≥ 0, (38)
for all sequences (l(n), n ≥ 0), ( l(n), n ≥ 0) as defined above, we will have the expected result (note that the continuity of the function involved in (38) implies that the convergence will hold for every x ≥ 0). To prove (38), recall that by Lemma 4.2 (ii),

∞ 0 x -1 k(x)dx < ∞.
Hence by dominated convergence, for all a > 0,

∞ 0 U l(n) (ax 1/α )k(x)dx → ∞ 0 U (ax 1/α )k(x)dx. By Lemma 4.1 (ii), we therefore have ∞ 0 U (ax 1/α )k(x)dx = a |α| , ∀a > 0. ( 39
)
We claim that this equation characterizes U under the additional assumption that the support of B is not included in a set of the form {a n , n ∈ N} for some a ∈]0, 1[. Indeed, note first that by setting V (x) := exp(x)U (exp(x/α)) and k(x) := k(exp(-x)) for all x ∈ R, the above equation rewrites

∞ -∞ V (x) k (y -x) dx = 1, ∀y ∈ R.
But the function V is bounded a.e. on R, by (37). Moreover, by Lemma 4.2 (ii), k ∈ L 1 (R) and by Lemma 4.2 (iii), the Fourier transform of k is non-zero on R. We conclude with the Wiener approximation Theorem for L 1 (R) ( [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]Th.4.8.4]) that the above equation in V has a unique bounded solution (in the sense that two solutions are equal a.e.). This determines V , hence U , almost everywhere. Since U is right-continuous, it is determined for all x ≥ 0. Last, it is not hard to check that the expression of U given by (38) indeed satisfies equation (39).

Proof of Proposition 1.4 (ii)

We just have to prove the second part of Proposition 1.4 (ii), the first one being obvious since µ t (]1, ∞[) = 0 for all t ≥ 0 and g(t)(ϕ(|α|t)/|α|t) Suppose moreover that h(t) → ∞ as t → ∞ and that β < 1, which implies that h(t) |α| /φ(h(t) |α| ) → ∞. By Lemma 3.4, we have, for all real c > 0,

-ln P I ≥ h(t) |α| /cφ(h(t) |α| ) ∼ ∞ 1 -β |α| ϕ |α|h(t) |α| cφ(h(t) |α| ) ∼ ∞ (1 -β)|α| β/(1-β) c 1/(1-β) h(t) |α| ,
using both the regular variation of ϕ and the fact that ϕ is the inverse of t → t/φ(t) near ∞. Now let ε ∈]0, 1[ and c be such that c 1/(1-β) > (1 -β)|α| β/ (1-β) . We have proved that ln m(t)

-1 P ϕ(|α|t) |α|t 1/|α| X(t) ≥ cφ(h(t) |α| ) 1/|α| ≤ -(1 -ε) 1 - (1 -β)|α| β/(1-β) c 1/(
X(t) ≥ g h,c (t) ≤ -(1 -ε) 1 - (1 -β)|α| β/(1-β) c 1/(1-β) φ -1 (g h,c (t) |α| /c) ∼ ∞ -(1 -ε) 1 - (1 -β)|α| β/(1-β) c 1/(1-β) c -1/β φ -1 (g h,c (t) |α| ).
It is not hard to check that the maximum of

1 - (1 -β)|α| β/(1-β) c 1/(1-β) c -1/β , c > (1 -β) 1-β |α| β is equal to β/|α| and is reached at c = |α| β . Finally, if g h = g h,|α| β , letting ε → 0, we have proved that lim sup t→∞ 1 φ -1 (g h (t) |α| ) ln m(t) -1 P ϕ(|α|t) |α|t 1/|α| X(t) ≥ g h (t) ≤ - β |α| . (40)
To conclude, to get the second part of the statement of Proposition 1.4 (ii), suppose 0 < β < 1 and consider some positive function g that converges to ∞ at ∞, such that g(t) |α| t/ϕ(t) → 0. Set h(t) = (φ -1 (g(t) |α| /|α| β ) 1/|α| , t ≥ 0. Then, h(t) converges to ∞ as t → ∞ and it is easily seen that h(t) |α| /ϕ(t) → 0 as t → ∞. Since g = g h with the notations above, the result follows from (40).

5 Some properties of the limit measure µ ∞ Recall that the distribution xµ ∞ (dx) on ]0, ∞[ is that of R 1/|α| , where R denotes a random variable with entire positive moments

E[R n ] = φ(|α|)....φ(n|α|), n ≥ 1, (41)
that characterize its distribution. Using this particular moments' shape, wet get the following description of the measure µ ∞ near 0 and ∞. Some of these properties are then used at the end of this section to prove Proposition 1.5.

Proposition 5.1. (Behavior at ∞)

(i) Suppose (H) for some β ∈]0, 1[. Then, -ln ∞ t xµ ∞ (dx) = -ln P(R > t |α| ) ∼ ∞ β |α| φ -1 (t |α| )
where φ -1 denotes the inverse of φ (and is therefore a function regularly varying at ∞ with index 1/β).

(ii) Suppose φ(∞) :=

1 0 xB(dx) < ∞. Then µ ∞ has a compact support with supremum φ(∞) 1/|α| and µ ∞ {φ(∞) 1/|α| } > 0 ⇔ 1 B(dx) 1 -x < ∞.
∞ s x 1+α µ ∞ (dx) = E[1 {R>s |α| } R -1 ] ∼ 1 (|α| γ Γ(1 + γ)φ(-1/ ln(s |α| ))) and s 0 xµ ∞ (dx) = P(R < s |α| ) = • s |α| φ(-1/ ln(s |α| ))
.

Proof. This is a direct consequence of Corollary 1 of Caballero-Rivero [START_REF] Caballero | [END_REF], which gives these results in terms of the random variable R.

Proof of Proposition 5.1. (i) Our proof strongly relies on the proof of Proposition 2 of Rivero [START_REF] Rivero | A law of iterated logarithm for increasing self-similar Markov processes[END_REF]. Rivero shows there that if a positive random variable Y has entire moments satisfying

E[Y n ] = n i=1 ψ(i)
for some function ψ regularly varying at ∞ with index γ ∈]0, 1[, then

-ln (P(Y > t)) ∼ ∞ γψ ← (t),
where ψ ← is the right-inverse of ψ. Apply then this result to the r.v. R, by taking ψ = φ(|α|•) and γ = β.

(ii) Using (41) and that φ is increasing, we get for all n ≥ 0,

E R φ(∞) n ≤ 1. (42)
Besides, writing

E R φ(∞) n = E R φ(∞) n 1 {R>φ(∞)} + E R φ(∞) n 1 {R<φ(∞)} + P (R = φ(∞)) ,
and using the monotone and dominated convergence theorems, we see that

E R φ(∞) n → n→∞ ∞ if P(R > φ(∞)) > 0 P(R = φ(∞)) otherwise.
In particular, from (42), we see that P(R > φ(∞)) = 0. Similarly, it is easy to show, using (41), that for all 0

< ε < φ(∞), E R φ(∞) -ε n → ∞,
which implies that P(R > φ(∞) -ε) > 0. Last, to get the remaining part of the statement, note that

ln φ(n|α|) φ(∞) ∼ n→∞ φ(n|α|) φ(∞) -1 = -1 φ(∞) 1 0 x n|α|+1 B(dx). Therefore ln E R φ(∞) n = n i=1 ln φ(i|α|) φ(∞) converges to -∞ as n → ∞ if and only if 1 0 ∞ i=1 x i|α|+1 B(dx) = 1 0 x 1+|α| 1 -x |α| B(dx) = ∞. Since 0 xB(dx) < ∞, E R φ(∞) n → n→∞ 0 i.i.f 1 1 1 -x B(dx) = ∞,
which ends the proof.

Proof of Proposition 1.5. From the construction (7) of µ t , we see that µ t ({1}) = µ 0 ({1})P (ξ(ρ(t)) = 0) = µ 0 ({1})P (ξ(t) = 0) , and from the Poisson point process construction of a pure-jump subordinator with Lévy measure Π we have that P (ξ(t) = 0) = exp(-tΠ(]0, ∞[)) = exp (-tφ(∞)). Next, we get from the factorization [START_REF] Fournier | On small particles in coagulation-fragmentation equations[END_REF], that exp (-tφ(∞)) = P (RI > tφ(∞)) ≥ P(I > t)P(R ≥ φ(∞)).

On the one hand, from the proof of Proposition 5.1, we see that when φ(∞) < ∞, P(R ≥ φ(∞)) = P(R = φ(∞)) and that this quantity is non-zero i.f.f. 1 (1 -x) -1 B(dx) < ∞. On the other hand, under (H), we get from the regular variation of -ln(P(I > t)) that P(I > x α t)/P(I > t) → 0 for all 0 < x < 1, as t → ∞; and then, from the dominated convergence theorem that

m(t) P(I > t) = 1 0 P(I > x α t) P(I > t) xµ 0 (dx) → µ 0 ({1}) as t → ∞.
Otherwise said, we have proved that under the hypothesis (H), when µ 0 ({1}) > 0 and φ(∞) < ∞,

lim inf t→∞ µ t ({1}) m(t) ≥ P(R = φ(∞)) = φ(∞) 1/|α| µ ∞ ({φ(∞) 1/|α| }). Next, suppose (H), that 0 | ln(x)|xB(dx) < ∞ and that φ(∞) < ∞. According to Theorem 1.3, for all ε ∈]0, 1[ such that (1 -ε)φ(∞) 1/|α| is not an atom of µ ∞ , µ t ({1}) m(t) ≤ 1 1-ε xµ t (dx) m(t) → t→∞ φ(∞) 1/|α| (1-ε)φ(∞) 1/|α| xµ ∞ (dx).
Letting ε → 0, we get lim sup

t→∞ µ t ({1}) m(t) ≤ φ(∞) 1/|α| µ ∞ ({φ(∞) 1/|α| }).

Examples

Below is a list of standard examples where the main quantities involved in our results can be computed explicitly. More precisely, for each of these examples, we specify the distributions of I (defined in [START_REF] Feller | An introduction to probability theory and its applications[END_REF]) and R (defined in ( 14)), which lead to explicit expressions of the limit measure µ ∞ (since R 1/|α| d ∼ xµ(dx)) and of the mass m 1 (t) = P(I > t), t ≥ 0, which is the mass of the solution to the fragmentation equation starting from µ 0 = δ 1 . We also specify the behavior as t → ∞ of the quantity ϕ(|α|t)/|α|t, involved in the statement of Theorem 1.3. For all these examples, we give the main tools to get the distributions of I and R, but we leave the calculation details to the reader. We recall that β denotes the index of regular variation of hypothesis (H) and that when (µ t , t ≥ 0) is a solution to the equation with parameters (α, B), (µ ct , t ≥ 0) is a solution to the equation with parameters (α, cB). For this reason, in the examples below, given a measure B we choose its "representative" among the measures cB, c > 0, which is the most convenient for the statement of the results.

The first four examples concerns absolutely continuous measures B(du) = b(u)du, where b is a function defined on ]0, 1[. The Lévy measure is therefore also absolutely continuous and we denote by π its density. It turns out that the limit distribution µ ∞ is also absolutely continuous. We denote by u ∞ its density.

Ex.1. b(u) = bu b-2 , b > 0; α < 0. • β = 0 • ϕ(t) ∼ t as t → ∞ • I d ∼ Γ(b/|α| + 1, 1) • m 1 (t) = 1 Γ(b/|α|+1) ∞ t x b/|α| exp(-x)dx, t ≥ 0 • R d ∼ β(1, b/|α|) • u ∞ (x) = bx |α|-2 1 -x |α| b |α| -1 , 0 < x < 1
The notations Γ(x, y) (resp. β(x, y)) refer to the classical Gamma distribution with parameters x, y > 0 (resp. Beta distribution). In these examples, the density of the Lévy measure associated to B is π(x) = b exp(-bx), x > 0, hence the Lévy measure associated to the subordinator |α|ξ (where ξ has Lévy measure Π) has a density given by b exp(-bx/|α|)/|α|, x > 0. According to the Example B, p.5 of [START_REF] Carmona | On the distribution and asymptotic results for exponential functionals of Lévy processes[END_REF], the density of I is then proportional to x b/|α| exp(-x), x > 0. Last we refer to the formula (4), Section 3 of [START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF], to get the distribution of R.

We point out that the solutions to the fragmentation equation with this measure B are studied in [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF]. In particular, when α = -b/2 and µ 0 = δ 1 the solutions (µ t , t ≥ 0) has the explicit expression

µ t (dx) = exp(-t) δ 1 (dx) + bx b-2 t - 1 2 t 2 (1 -x -b/2 )) 1 {0<x<1} dx , which gives m 1 (t) = exp(-t) 1 + t + t 2 2
and for all bounded test functions

f :]0, ∞[→ R 1 m 1 (t) 1 0 f (x)xµ t (dx) → t→∞ 1 0 f (x)bx b-1 (x -b/2 -1)dx,
which is consistent with the above expressions of m 1 and u ∞ .

Ex.2. b(u) = |α|Γ(1 -γ) -1 u |α| γ -2 (1 -u |α| γ ) -γ-1 ; 0 < γ < 1; α < 0. • β = γ • ϕ(t) ∼ γ |α| γ 1-γ t 1 1-γ as t → ∞ • I d ∼ τ -γ γ • m 1 (t) = ∞ t g γ (x)dx, t > 0. • R d ∼ e(1) γ • µ ∞ (x) = |α|γ -1 x |α| γ -2 exp -x |α|/γ , x > 0
Here, e(1) denotes a r.v. with exponential distribution with parameter 1 and τ γ a γ-stable random variable, i.e. with Laplace transform t ∈ [0, ∞[→ exp(-t γ ). Hence τ -γ γ has the so-called Mittag-Leffler distribution. We recall that it possesses a density given by

g γ (x) = 1 πγ ∞ i=0 (-x) i-1 i! Γ(γi + 1) sin(πγi), x > 0
and its entire positive moments are equal to n!/Γ(γn + 1), ∀n ≥ 1 (see e.g. [25, Section 0.3]). The Lévy measure associated to B has a density given for x > 0 by

π(x) = |α| exp(-|α|x/γ) Γ(1 -γ)(1 -exp(-|α|x/γ)) γ+1 .
Using formula (5) and the following discussion in [START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF], we get that

I d ∼ τ -γ γ and R d ∼ e(1) γ . Ex.3. b(u) = |α|γ 2 ((1 -γ)Γ(2 -γ)) -1 u γ|α| 1-γ -2 (1 -u |α|/(1-γ) ) -γ-1 ; 0 < γ < 1; α < 0. • β = γ • ϕ(t) ∼ (1 -γ) -1 |α| γ γ-1 t 1 1-γ as t → ∞ • I d ∼ e(1) 1-γ • m 1 (t) = exp(-t 1/(1-γ) ), t ≥ 0 • R d ∼ τ γ-1 1-γ • µ ∞ (x) = |α|x |α|-2 g 1-γ (x |α| ), x > 0,
where g 1-γ is the Mittag-Leffler density given in the previous example. Note the duality with this previous example. In the present example,

π(x) = |α|γ 2 exp(|α|x/(1 -γ)) (1 -γ)Γ(2 -γ)(exp(|α|x/(1 -γ)) -1) 1+γ , x > 0,
and we again refer to the formula (5) and the following discussion in [START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF] to get the distributions of I and R.

Ex.4. b(u) = |α|Γ(2 + α) -1 u |α|-2 (1 -u) α-1 ; -1 < α < 0. • β = |α| • ϕ(t) ∼ t 1+α 1 1-|α| as t → ∞ • I/(1 + α) is a size-biased
f E[f (I)] = E f (1 + α)τ -|α| |α| τ -|α| |α| E τ -|α| |α| • m 1 (t) = Γ(|α| + 1) ∞ t/(1+α) xg |α| (x)dx, t > 0 • ((1 + α)R) 1/|α| d ∼ Γ(|α|, 1) • µ ∞ (x) = (1+α) Γ(|α|) x |α|-2 exp(-(1 + α) 1/|α| x), x > 0.
Indeed, here

π(x) = |α| exp(x) Γ(2 + α)(exp(x) -1) 1-α , x > 0.
Following the end of the proof of Lemma 4 of Miermont [START_REF] Miermont | Self-similar fragmentations derived from the stable tree. I. Splitting at heights[END_REF], we get that I has its moment of order k equal to

k!(1 + α) k Γ(|α|) Γ ((k + 1)|α|) , for all k ∈ N. Hence E[R k ] = k! E[I k ] = Γ ((k + 1)|α|) (1 + α) k Γ(|α|) = 1 (1 + α) k Γ(|α|) ∞ 0 x (k+1)|α|-1 exp(-x)dx.
Remark. Note that the Examples 2, 3 and 4 give, for all 0 < γ < 1,

-if b(u) = u -1 (1 -u) -γ-1 and α = -γ, xµ ∞ (dx) d ∼ c 1 (γ)e(1) -if b(u) = u γ-2 (1 -u) -γ-1 and α = γ -1, xµ ∞ (dx) d ∼ c 2 (γ)τ -1 1-γ , -if b(u) = u γ-2 (1 -u) -γ-1 and α = -γ, xµ ∞ (dx) d ∼ c 3 (γ)Γ(γ, 1)
where c 1 (γ), c 2 (γ) and c 3 (γ) are reals that depend on γ. Hence, both α and the behavior of b near 0 play a significant role on the shape of the limit measure µ ∞ .

Last we turn to the case where B is a Dirac measure.

Ex.5. B = a -1 δ a for some a ∈]0, 1[; α < 0.

• β = 0

• ϕ(t) ∼ t as t → ∞

• I has a density k on ]0, ∞[ given by k(x) = i≥0 exp (α ln(a)i -x exp(α ln(a)i))

p =i

(1 -exp(α ln(a)(i -p)))

-1

• m 1 (t) = ∞ t k(x)dx, t ≥ 0 In this case, Π = δ -ln(a) , i.e. the associated subordinator is a Poisson process. We then refer to [9, Prop. 6.5 (ii)] for the expression of the density k. Note that φ(t) = (1-a t ) for all t ≥ 0, hence E[R n ] = n i=1 (1-(|α|a) i ) for all n ≥ 1.

Appendix: Existence and uniqueness of solutions

This appendix is devoted to the proof of Theorem 1.3 on existence and uniqueness of solutions to the fragmentation equation [START_REF] Bertoin | Lévy processes[END_REF]. So in this section, α ∈ R. The proof follows the main lines of that of Theorem 1 in [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF], which states existence and uniqueness of solutions to a slightly restricted form of the fragmentation equation [START_REF] Bertoin | Lévy processes[END_REF], and which was concentrated on solutions starting from µ 0 = δ 1 . We note that it was implicit in the statement of this theorem that a solution should satisfy assumptions (4) and [START_REF] Bertoin | Entrance from 0+ for increasing semi-stable Markov processes[END_REF].

Let ξ denotes a subordinator with Lévy measure Π and zero drift, such that ξ 0 = 0. We recall that its semigroup possesses the Feller property and that the domain of its infinitesimal generator contains at least all functions f that are continuously differentiable on R and such that f and f ′ tend to 0 at infinity. See e.g. Chapter 1 of [START_REF] Bertoin | Lévy processes[END_REF]. As a consequence, the domain of the infinitesimal generator of exp(-ξ) contains continuously differentiable functions f on ]0, ∞[ with compact support.

One can easily checked that when f :]0, ∞[→ R is bounded and continuous, the function

x → E f (x exp(-ξ ρ(x α t) ))

is also bounded and continuous on ]0, ∞[. This mainly relies on the càdlàg and quasi-left-continuity ([2, Prop.7, Chapter 1]) of subordinators. Now, for every 0 < a < b, let C a,b be the set of continuous functions f :]0, b] → R that are null on ]0, a], and C 1 a,b be the set of continuously differentiable functions f :]0, b] → R that are null on ]0, a]. It is clear from the remark above that for all 0 < a < b, the linear operators T t and Tt , t ≥ 0, defined by

T t (f )(x) = E [f (x exp(-ξ t ))]
and Tt (f )(x) = E f (x exp(-ξ ρ(x α t) ))

send C a,b into C a,b . Following the proof of Theorem 1 of [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF] (see also [START_REF] Lamperti | Semi-stable Markov processes[END_REF]), we see that both family of operators define strongly continuous contraction semigroups on C a,b , and that the domains of their infinitesimal generators are identical and contain C Existence of solutions to [START_REF] Bertoin | Lévy processes[END_REF]. With the above remarks, and Kolmogorov's backward equation (see Prop.15, p.9 of [START_REF] Ethier | Markov processes[END_REF]), we have that Tt (f )(x) = f (x) + t 0 Ts ( Ã(f ))(x)ds, (43)

∀x ∈]0, b], ∀f ∈ C 1 a,b , ∀0 < a < b, ∀b > 0. Otherwise said: let f :]0, ∞[→ R be null near 0 and continuously differentiable. Then, considering its restriction to ]0, b] and x ≤ b, we have that f and x satisfy (43). Now, consider ν 0 , a probability measure on ]0, ∞[, and set < ν t , g >:=< ν 0 , Tt (g) > for all bounded, measurable functions g on ]0, ∞[. Note that for all t ≥ 0, ν t (]0, ∞[) ≤ 1 and ν t (x ≥ M ) = 0 as soon as ν 0 (x ≥ M ) = 0 for some M > 0. Then let f be some continuously differentiable function on ]0, ∞[ with compact support. It is clear that Ã(f ) is null near 0 and it is easy to see, using Fubini's theorem, that there exists some constants b, c > 0 such that | Ã(f )|(x) ≤ cx α Π(ln(x/b)) for large enough x (here Π(y) = ∞ y Π(dx)). In particular, Ã(f ) is bounded on ]0, ∞[ when x α Π(ln(x)) is bounded near ∞ (hence when α ≤ 0). It is then clear that in such case we can apply Fubini's theorem when integrating (43) with respect to ν 0 to get

< ν t , f >=< ν 0 , f > + t 0 < ν s , Ã(f ) > ds.
This holds for all continuously differentiable functions f on ]0, ∞[ with compact support. Therefore, defining the measures µ t on ]0, ∞[ by < µ t , g >:=< ν t , g >, where g denotes any test-function on ]0, ∞[ and g(x) = g(x)/x, x > 0, we have proved that (µ t , t ≥ 0) is a solution to the fragmentation equation, as defined in the Introduction. To sum up: provided that the function x → x α Π(ln(x)) is bounded near ∞, for all measure µ 0 on ]0, ∞[ such that ∞ 0 xµ 0 (dx) = 1, there exists a solution constructed via subordinators to the fragmentation equation.

When α > 0, the function x → x α Π(ln(x)) may not be bounded near ∞. Another way to tackle the problem in this case is to use the definition of ρ to get that x 0 E[exp(-sR)]ds, where R is a random variable with distribution µ R defined by [START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF]. Let then I be a random variable defined by [START_REF] Feller | An introduction to probability theory and its applications[END_REF], independent of R, and consider a real a such that P(I ≤ a) > 0. Using the factorization property ( 15 It this then possible to apply Fubini's theorem when integrating (43) with respect to ν 0 and we conclude as above to the existence of a solution to [START_REF] Bertoin | Lévy processes[END_REF].

Uniqueness of solutions to [START_REF] Bertoin | Lévy processes[END_REF]. Let ν 0 be a probability measure with support included in ]0, b] for some b > 0 and suppose that (ν t , t ≥ 0) is a family of measures with support included in ]0, b] such that

< ν t , f >=< ν 0 , f > + t 0 < ν s , Ã(f ) > ds, ∀f ∈ ∪ 0<a<b C 1 a,b .
Suppose moreover that ν t (]0, ∞[) ≤ 1, ∀t ≥ 0. Our goal is to prove that (ν t , t ≥ 0) is uniquely determined. Using that the total weight of ν t is less or equal to 1, we get that sup t≥0 < ν t , |f | >< ∞ and sup t≥0 < ν t , |A(f )| >< ∞ for each f ∈ ∪ 0<a<b C 1 a,b . It is then possible to follow the proof of Proposition 18, Sect. 4.9 of [START_REF] Ethier | Markov processes[END_REF] to deduce that uniqueness holds provided that, for all λ > 0, (λid -Ã(f ))(C 1 a,b ) is dense (for the uniform norm) in C a,b , for all 0 < a < b. Following the proof of Theorem 1 in [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF], we see that C 1 a,b is a core for the strongly contraction semi-group Tt : C a,b → C a,b , t ≥ 0. Hence the result.

Theorem 1 . 1 . 1 x

 111 Let µ 0 be a measure on ]0, ∞[ such that ∞ 0 xµ 0 (dx) = 1. Then (i) there exists a solution to (2) starting from µ 0 as soon as α ≤ 0 or α > 0 and either ∞ ln(x)µ 0 (dx) < ∞ or x ∈]0, 1[→ x |α| x 0 yB(dy) is bounded near 0.

  , t ≥ 0) denote the solution to the fragmentation equation (2) starting from µ (λ)

∞0

  xµ 0 (dx) = 1. Proposition 3.3. The total mass m is strictly positive and strictly decreasing on [0, ∞[. Moroever m(t) → 0 as t → ∞.

1 0P

 1 (I > xy α ) ln (P(I > xy α )) yµ 0 (dy).

1 0P

 1 > xy α ) ln (P(I > xy α )) yµ 0 (dy) ∼ x→∞ (I > xy α ) ln (P(I > xy α )) yµ 0 (dy).

  > xy α ) ln (P(I > xy α )) yµ 0 (dy) ≪ x→∞ P(I > x(1 -ε) α ) ln (P(I > x(1 -ε) α )) > xy α ) ln (P(I > xy α )) yµ 0 (dy) ,

1 0P

 1 > xy α )yµ 0 (dy) ∼ x→∞ (I > xy α )yµ 0 (dy).(31)

1 1

 1 > xy α )yµ 0 (dy) ≤ -ε P(I > xy α ) ln (P(I > xy α )) yµ 0 (dy) ln (P(I > x))

Proposition 5 . 2 .

 52 (Behavior at 0) Suppose that exp(-u) 0 xB(dx) varies regularly at ∞ with index -γ, γ ∈ [0, 1]. Then, as s → 0

1 0

 1 1 a,b . These generators are respectively given, for f ∈ C 1 a,b and x ∈]0, b], byA(f )(x) = ∞ 0 (f (x exp(-y)) -f (x)) Π(dy) and Ã(f )(x) = x α A(f )(x).Note that when B is a measure on ]0, 1[ defined from Π by[START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF], we have Ã(f )(x) = x α (f (xy) -f (x)) yB(dy).

  Ã(f )|(x)dsν 0 (dx) = ∞ 0 E ρ(x α t) 0 |A(f )|(x exp(-ξ u ))du ν 0 (dx),the function f being still supposed continuously differentiable on ]0, ∞[ with compact support. For such f , the function A(f ) is bounded on ]0, ∞[. Hence the double integral involved in the identity above is bounded by a constant times ∞ 0 E [ρ(x α t)] ν 0 (dx), which is finite as soon as ∞ ln(x)ν 0 (dx) < ∞: indeed, according to Proposition 2 of[START_REF] Bertoin | On subordinators, self-similar Markov processes and some factorizations of the exponential variable[END_REF], for all x ≥ 0, E[ρ(x)] =

  ), we getE[ρ(x)]P(I ≤ a) = x 0 E[exp(-sR)1 {I≤a} ]ds ≤ x 0 E[exp(-sa -1 e(1))]ds = a ln(1 + a -1 x).

  2, Theorem 1.6 (i) and Proposition 1.7. The crucial point is the following lemma, which is basically a consequence of Rivero [26, Prop. 2] and König and Mörters [19, Lemma 2.3]. Lemma 3.4. Assume (H), or, equivalently, that φ varies regularly at ∞ with index β ∈ [0, 1[. Then

	-ln (P

  1/α → ∞ as t → ∞. We keep the notations of the previous section and we recall that we work under the assumption (H). From the proof of Lemma 4.1, we get that

				ln	 	m t 1 + |α|h(t) |α| ϕ(|α|t) m(t)	  ∼ t→∞	-h(t) |α| ,
	for all positive functions h such that h(t) |α| /ϕ(t) → 0 as t → ∞. Otherwise said, for such functions h,
		ln m(t) -1 P		ϕ(|α|t) |α|t	1/|α|	X(t)I 1/|α| ≥ h(t)	∼ t→∞	-h(t) |α| .
	Note that for all t ≥ 0 and all c > 0, since X is independent of I,
	ln m(t) -1 P	ϕ(|α|t) |α|t	1/|α|	X(t) ≥ cφ(h(t) |α| )	1/|α|	+ ln P I 1/|α| ≥ h(t)/ cφ(h(t) |α| )	1/|α|
							≤ ln m(t) -1 P	ϕ(|α|t) |α|t	1/|α|	X(t)I 1/|α| ≥ h(t)	.

  version of the Mittag-Leffler distribution with parameter |α|, i.e. for all test functions

Acknowledgements. I warmly thank Stéphane Mischler who is at the origin of this paper, asking whether a probabilistic approach is possible to study the large-time behavior of solutions to the fragmentation equation with shattering. Many thanks also to Victor Rivero who pointed out the reference [19] to generalize his Proposition 2 in [26].