Mihai Gradinaru
email: mihai.gradinaru@univ-rennes1.fr

Emeline Luirard
email: emeline.luirard@univ-rennes1.fr

Kinetic time-inhomogeneous Lévy-driven model

Keywords: kinetic stochastic equation, time-inhomogeneous stochastic differential equation, Lévy process, explosion times, scaling transformations, asymptotic distributions, ergodicity, tightness. MSC2020 Subject Classification: Primary 60H10, Secondary 60F17, 60G52, 60G18, 60J65

We study a one-dimensional kinetic stochastic model driven by a Lévy process with a non-linear timeinhomogeneous drift. More precisely, the process (V, X) is considered, where X is the position of the particle and its velocity V is the solution of a stochastic differential equation with a drift of the form t -β F (v). The driving process can be a stable Lévy process of index α or a general Lévy process under appropriate assumptions. The function F satisfies a homogeneity condition and β is non-negative. The behavior in large time of the process (V, X) is proved and the precise rate of convergence is pointed out by using stochastic analysis tools. To this end, we compute the moment estimates of the velocity process.

Introduction

In this paper, we consider a one-dimensional stochastic kinetic model driven by a Lévy process.

dV t = dL t -F (V t )t -β dt and X t = X 0 + t 0 V s ds. (1)
The process (V t , X t ) t≥0 may be thought of as the velocity and position processes of a particle subject to a friction force F (v)t -β and interacting with its environment.

Our purpose is to study the long-time behavior of solutions of (1) where L is an α-stable (non-symmetric) Lévy process. More precisely, we look for the convergence in distribution of the process (V t/ε , X t/ε ) t>0 , as ε → 0, with an appropriate rate.

It is a simple observation, when F = 0, to see that the rescaled process (ε

1 α V t/ε , ε 1+ 1 α X t/ε
) t>0 converges in distribution towards the Kolmogorov process (S t , t 0 S s ds) t>0 , where S has the same distribution as the driving process. The goal of the present paper is to extend the results obtained in [START_REF] Gradinaru | Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion[END_REF], where the driving process is a Brownian motion (α = 2).

The study of stochastic differential equations (SDEs) driven by a Lévy process is a topic of great interest (see [START_REF] Bass | Stochastic differential equations driven by symmetric stable processes[END_REF] for a survey). The α-stable perturbation is a generalization of the Gaussian case, and it is also motivated by some Langevin-type models in stochastic climate dynamics (see [START_REF] Peter | Observation of α-stable noise induced millennial climate changes from an ice-core record[END_REF]). So far, most of the papers present results about existence and uniqueness of solution, see for instance [START_REF] Applebaum | Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise[END_REF], [START_REF] Dong | Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients[END_REF], [START_REF] Kurenok | Stochastic Equations with Time-Dependent Drift Driven by Levy Processes[END_REF], [START_REF] Pilipenko | On existence and properties of strong solutions of one-dimensional stochastic equations with an additive noise[END_REF], [START_REF] Chen | Stochastic flows for Lévy processes with Hölder drifts[END_REF] and [START_REF] Chen | Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts[END_REF]. The coefficients of the studied SDE are often supposed to be timehomogeneous (see for instance [START_REF] Applebaum | Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise[END_REF] and [START_REF] Dong | Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients[END_REF]). Accordingly, the case of time-dependent coefficients is scarcely studied (see [START_REF] Chen | Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts[END_REF], [START_REF] Kurenok | Stochastic Equations with Time-Dependent Drift Driven by Levy Processes[END_REF] and [START_REF] Zhang | Stochastic differential equations with sobolev drifts and driven by α-stable processes[END_REF]). In this situation, the usual tools associated with time-homogeneous equation may no longer be invoked. Furthermore, few papers (see [START_REF] Applebaum | Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise[END_REF], [START_REF] Priola | Exponential ergodicity and regularity for equations with Lévy noise[END_REF], [START_REF] Reker | Short-time behavior of solutions to Lévy-driven SDEs[END_REF]) present results about the asymptotic behavior of the solution of such SDEs. For instance, in [START_REF] Applebaum | Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise[END_REF] the authors give conditions for asymptotic stability of the solutions for a SDE driven by a Brownian motion and a compensated Poisson process, with coefficients that are supposed to satisfy usual global Lipschitz and growth assumptions. In [START_REF] Priola | Exponential ergodicity and regularity for equations with Lévy noise[END_REF], the authors establish the exponential ergodicity of the solutions of a SDE driven by an α-stable process, where the drift coefficient is supposed to be the sum of two components, one linear and the other bounded. In these papers, coefficients are time-homogeneous. In a number of articles, the small noise influence of the solutions is analyzed. To our knowledge, the only works considering the long-time behavior are [START_REF] Fournier | One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes[END_REF], in a time-homogeneous setting, and the present one.

Let us explain heuristically what the intuition of our analysis is. In long-time regime, we observe three schemes, depending on the balance between the space and time coefficients of the drift function with respect to α, the parameter of stability of the driving process. When the drag force is sufficiently "small at infinity", the convergence towards the Kolmogorov process (S, S) still holds. When the two terms in the stochastic equation of the velocity process offset, we still get a kinetic process of the form (V, V), as limiting process. Though the process V no longer has the same distribution as the driving Lévy process. Alternatively, when the drift swings with the random noise, the limiting process is no longer kinetic.

Proofs are mainly based on moment estimates and the self-similarity of the driving process. By their scaling property, Lévy stable processes are natural extensions of the Brownian motion. However, the jump component of the Lévy noise brings difficulties. Indeed, by contrast with a Brownian motion, an α-stable Lévy process can only have moments of order κ ∈ [0, α). Thus, moment estimation of the velocity process stands as a significant part of our study (see Section 4). Moment estimates of Lévy and Lévy-type processes were studied in [START_REF] Luschgy | Moment estimates for Lévy Processes[END_REF], [START_REF] Kühn | Existence and estimates of moments for Lévy-type processes[END_REF] and [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF]. Nevertheless, the methods used can not be easily adapted to the solutions of a SDE. In fact, the key idea will be to make a non-homogeneous cutting of the jumps size of the driving process. As explained in [START_REF] Chaudru | On Multidimensional stable-driven Stochastic Differential Equations with Besov drift[END_REF] and references therein, the cutting threshold ξ → ξ 1 α makes appear integral terms satisfying the scaling property (see the proof of Proposition 4.5). The proof of the critical and sub-critical cases (see Theorem 2.4) significantly relies on a change in both space and time, taking advantage of the scaling property of the driving process, to be close to a stationary time-homogeneous SDE, as performed in [START_REF] Appleby | Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with applications to financial markets[END_REF] and [START_REF] Gradinaru | Existence and asymptotic behaviour of some timeinhomogeneous diffusions[END_REF]. In addition, extensions for the solution of the SDE driven by a general Lévy process are stated in Theorems 6.1 and 6.2. Let us point out that the case analyzed in Section 6.3 agrees with the equation (1) p. 1442 in [START_REF] Peter | Observation of α-stable noise induced millennial climate changes from an ice-core record[END_REF], where the described Langevin equation has two noise terms, a white noise and a pure-jump noise. To this end, we first study the asymptotic behavior of the rescaled Lévy driving process (v ε L t/ε ) t≥0 , where v ε is some rate of convergence. Under appropriate assumptions on its Lévy measure, it converges in distribution, as ε goes to zero (see Proposition 6.3 below).

Here is the structure of the paper. In Section 2, we introduce some notations and state our main results. Theorems 2.2 and 2.4 are extension for the α-stable setting of the main results stated in [START_REF] Gradinaru | Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion[END_REF] in case of a Brownian driving process. We study the existence of the solution of (1) in Section 3. In Section 4, we give estimates of the moment, which also ensure the non-explosion of the velocity process. The proofs of our main results are presented in Section 5. Finally, Section 6 is devoted to the study of the convergence of a rescaled Lévy process and to the extension of Theorem 2.2.

Notations and main results

Let (L t ) t≥0 be a Lévy process. Throughout the paper, we deal with L as an α-stable Lévy process with α ∈ (0, 2). We call ν its Lévy measure, given by ν(dz) = |z| -1-α [a + 1 {z>0} +a -1 {z<0} ] dz with a + , a -≥ 0. As a Lévy measure, it satisfies R * (1 ∧ z 2 )ν(dz) < +∞. By Lévy-Itô's decomposition, if α ∈ (0, 2), then L is a pure-jump Lévy process and there exists a Poisson point measure N and its compensated Poisson measure N such that, for all t ≥ 0,

L t =            t 0 R * zN (ds, dz) if α ∈ (0, 1), t 0 {0<|z|<1} z N (ds, dz) + t 0 {|z|≥1} zN (ds, dz) if α = 1, t 0 R * z N (ds, dz) if α ∈ (1, 2).
(

) 2
In Section 6, the case of a generalized Lévy driving process will be discussed. The space of continuous functions C((0, +∞), R) is endowed with the uniform topology

d u : f, g ∈ C((0, +∞)) → +∞ n=1 1 2 n min 1, sup [ 1 n ,n]
|f -g| .

Set Λ := {λ : R + → R + , continuous and increasing function s.t. λ(0) = 0, lim t→+∞ λ(t) = +∞} and

k n (t) =      1 if 1 n ≤ t ≤ n, n + 1 -t if n < t < n + 1, 0 if n + 1 ≤ t.
The space of right-continuous with left limits (càdlàg) functions D((0, +∞), R) is endowed with the Skorokhod topology d s defined for (f, g) ∈ D((0, +∞), R) 2 by

+∞ n=1 1 2 n min 1, inf a s.t. ∃λ ∈ Λ, sup s =t log λ(t) -λ(s) t -s ≤ a, sup t≥ 1 n |k n (t) (f • λ(t) -g(t))| ≤ a .
For simplicity, we shall write C and D for C((0, +∞), R) and D((0, +∞), R), respectively. For a family ((Z (ε) t ) t>0 ) ε>0 of càdlàg processes, we write

(Z (ε) t ) t>0 =⇒ ε→0 (Z t ) t>0 , if (Z (ε) t ) t>0 converges in distribution to (Z t ) t>0 in D, as ε → 0. We write (Z (ε) t ) t>0 f.d.d. =⇒ ε→0 (Z t ) t>0 ,
if for all finite subsets S ⊂ (0, +∞), the vector (Z

(ε) t ) t∈S converges in distribution to (Z t ) t∈S in R S , as ε → 0.
Let β a non-negative real number and F a continuous function satisfying

for some γ ∈ R, ∀v ∈ R, λ > 0, F (λv) = λ γ F (v). (H γ )
We introduce another assumption on F , which will sometimes be imposed in the sequel.

When (i) α ∈ (0, 1] or (ii) α ∈ (1, 2) and γ ≥ 1, we suppose furthermore that for all v ∈ R, vF (v) ≥ 0. (H sgn )
We take an interest into the following one-dimensional stochastic kinetic model defined, for t ≥ t 0 > 0, by

dV t = dL t -t -β F (V t ) dt, with V t0 = v 0 > 0, and dX t = V t dt, with X t0 = x 0 ∈ R. (SKE)
In the following, sgn is the sign function with the convention that sgn(0) = 0. The abbreviation a.s. stands for almost surely. We denote by C some positive constants, which may change from line to line. We use the subscripts to indicate the parameters on which it depends. For instance, C t0,α denotes a constant depending on the parameters t 0 and α.

Remark 2.1. If a function π satisfies (H γ ), then for all x ∈ R, π(x) = π(sgn(x)) |x| γ .

As an example of a function satisfying (H γ ) one can keep in mind F : v → sgn(v) |v| γ (see also [START_REF] Gradinaru | Existence and asymptotic behaviour of some timeinhomogeneous diffusions[END_REF]). Let us state our main results.

Theorem 2.2. Consider γ ∈ (1 -α 2 , α). Assume that (H γ ) and (H sgn ) are satisfied, and β > 1 + γ-1 α . Let (V t , X t ) t≥t0 be the solution to (SKE) and (S t ) t≥0 be an α-stable process, having same distribution as (L t ) t≥0 . Then, in the space D, (ε

1 α V t/ε , ε 1+ 1 α X t/ε ) t≥εt0 =⇒ ε→0 S t , t 0 S s ds t>0 .
Remark 2.3. Theorem 2.2 is also true when the following hypothesis holds instead of (H γ ).

F is such that (SKE) has a unique solution up to explosion and

|F | ≤ G where G is a positive function satisfying (H γ ). (H γ bis )
For instance, the function F : v → v (1+v 2 ) (see also [START_REF] Fournier | One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes[END_REF]) satisfies (H γ bis ) (with γ = 0).

Theorem 2.4. Consider γ ∈ (1 -α 2 , α) and β = 1 + γ-1 α . Assume that (H γ ) and (H sgn ) are satisfied. Let (V t , X t ) t≥t0 be the solution to (SKE). Call H the ergodic process solving the following SDE driven by an α-stable process (R t ) t≥0 with same distribution as L and starting at its invariant measure,

dH s = dR s - H s α ds -F H s ds.
We denote by Λ F,t1,••• ,t d the finite-dimensional distributions of H. We call (V t ) t≥0 the process having as finite-dimensional distributions the pushforward measure of Λ F,log(t1),••• ,log(t d ) by the linear map

T (u 1 , • • • , u d ) := (t 1 /α 1 u 1 , • • • , t 1 /α d u d ).
Then, under (H γ ), the following convergence, in the space D, holds (ε

1 α V t/ε , ε 1+ 1 α X t/ε ) t≥εt0 =⇒ ε→0 V t , t 0 V s ds t>0 . Theorem 2.5. Consider α > 1, γ ∈ [1, α) and β < 1 + γ-1 α .
Assume that (H γ ) and (H sgn ) are satisfied. Let (V t , X t ) t≥t0 be the solution to (SKE). Define q := β α+γ-1 < 1 α . Call H the ergodic process solving the following SDE driven by an α-stable process (R t ) t≥0 with same distribution as L and starting at its invariant measure, dH s = dR s -F (H s ) ds.

Call Π F its invariant measure. We call (V t ) t≥0 the process whose finite dimensional distribution are T * Π ⊗d F : the pushforward measure of Π ⊗d F by the linear map

T (u 1 , • • • , u d ) := (t 1 q u 1 , • • • , t d q u d ). Then, ε q V t/ε t≥εt0 f.d.d. =⇒ ε→0 (V t ) t≥0 .
Remark 2.6. As we will see in Section 3, the assumption γ > 1 -α 2 is needed in order to obtain the existence up to explosion of the solution under hypothesis (H γ ).

Remark 2.7. Let us point out that, during the proof of Theorem 2.2 and Theorem 2.4, we employ some moment estimates for the solution V . We state the estimates below. Assume that (H sgn ) is satisfied. We suppose also that the hypothesis on the sign of F holds for (α, γ, κ)

∈ (1, 2) × [0, 1] × (1, α). Then, for any α ∈ (0, 2), γ ∈ R, β ∈ R and κ ∈ [0, α), there exists a constant C γ,κ,β,t0 such that, ∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ α
. Note that the above bounds are the best possible, taking F = 0.

Existence up to explosion

In this section, we study the existence of the solution to (SKE) up to explosion time.

Remark 3.1. Assume that (H γ ) holds. Pick β ≥ 0. If 0 < γ < 1, the function x → F (x)t -β is γ-Hölder and if γ ≥ 1, it is locally Lipschitz. Proposition 3.2. Assume that (H γ ) is satisfied. There exists a pathwise unique strong solution to (SKE), defined up to the explosion time, provided that (i) 1 -α 2 < γ < 1 and β ≥ 0 when α ∈ (0, 2). (ii) γ ≥ 1 when α > 1.

Proof. If γ ∈ (0, 1), the drift coefficient is γ-Hölder (see Remark 3.1) and locally bounded, thereby the conclusion of the first point follows from Remark 1.3 in [START_REF] Chen | Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts[END_REF]. Assume now that α > 1 and γ ≥ 1. The drift coefficient is locally Lipschitz (see Remark 3.1) and locally bounded, so we can apply Lemma 115 p. 78 in [START_REF] Situ | Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering[END_REF] to get the pathwise uniqueness. Thanks to Theorem 137 p. 104 in [START_REF] Situ | Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering[END_REF], it suffices to prove that there exists a weak solution. The drift coefficient is continuous with respect to its two variables, so it is a locally bounded and measurable function. By a standard localization argument, using Theorem 9.1 p. 231 in [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF], since the drift coefficient is locally Lipschitz, there is a unique solution defined up to explosion.

Moment estimates and non-explosion of the velocity process

In this section, we present estimates on moments of the velocity process V solution to (SKE). This will be useful to conclude of the non-explosion of solution to (SKE) with Lemma 4.1, and to control some terms appearing along the proofs of Theorem 2.2 and 2.4 in Section 5. Let V be the unique solution up to explosion time to (SKE). For all r ≥ 0, define the stopping time

τ r := inf{t ≥ t 0 , |V t | ≥ r}.
(3) Set τ ∞ := lim r→+∞ τ r the explosion time of V . We give first a sufficient condition for the non-explosion of a general process.

Lemma 4.1. Let (Y t ) t≥t0 be a càdlàg process and τ ∞ its explosion time. Assume that there exist two measurable and non-negative functions φ and b such that (i) φ is non-decreasing and lim r→∞ φ(r) = +∞, (ii) b is finite-valued, (iii) and for all t ≥ t 0 ,

sup r≥0 E [φ(|Y t∧τr |)] ≤ b(t). (4)
Then τ ∞ = +∞ a.s.

Proof. Pick t ≥ t 0 . Using the definition of τ r , the monotony of φ and (iii), we get, for all r ≥ 0,

φ(r)P(τ r ≤ t) ≤ E φ (Y τr ) 1 {τr≤t} ≤ E [φ (Y τr )] ≤ b(t).
Thus, by Fatou's lemma,

0 ≤ P (τ ∞ ≤ t) ≤ lim inf r→∞ P(τ r ≤ t) ≤ b(t) lim r→∞ 1 φ(r) = 0.
As a consequence,

0 ≤ P (τ ∞ < +∞) ≤ t∈Q P (τ ∞ ≤ t) = 0.
This concludes the proof.

We will show that there exists a constant C γ,κ,β,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ α . (5)
Proposition 4.2. Pick α ∈ (0, 1). Assume that (H sgn ) holds. Recall that (V t ) t≥t0 is the solution of (SKE). For any γ, β, the explosion time τ ∞ is a.s. infinite and for all κ ∈ [0, α), there exists a constant C κ,t0 such that, we have

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ α . (6)
Proof. Fix t ≥ t 0 . Since α < 1, the stable process can be written as

L t = t 0 R * zN (ds, dz) = s≤t ∆L s .
Fix κ ∈ [0, α). Pick the sequence of C 2 -functions f n : x → x 2 + 1 n , which converges uniformly to x → |x| on R. Then, for all n ≥ 1, we apply Itô's formula (see Theorem 32 p. 78 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]) to get

f n (V t∧τr ) = f n (v 0 ) - t∧τr t0 f ′ n (V s )F (V s )s -β ds + t∧τr t0 R * (f n (V s-+ z) -f n (V s-)) N (ds, dz) ≤ f n (v 0 ) + s≤t∧τr (f n (V s-+ ∆L s ) -f n (V s-)).
The term

t∧τr t0 f ′ n (V s )F (V s )s -β
ds is non-negative, since (H sgn ) holds. Hence, the previous equation can be written as

f n (V t∧τr ) ≤ f n (v 0 ) + s≤t∧τr (f n (V s ) -f n (V s-)). Since f ′ n ∞ ≤ 1, we deduce that (f n (V s ) -f n (V s-)) ≤ |∆V s | = |∆L s |, hence, |V t∧τr | ≤ f n (V t∧τr ) ≤ f n (v 0 ) + s≤t∧τr |∆L s | .
Furthermore, since κ < α < 1, we have

|V t∧τr | κ ≤ f n (v 0 ) κ +   s≤t∧τr |∆L s |   κ .
Taking the expectation, we get

E [|V t∧τr | κ ] ≤ E [f n (v 0 ) κ ] + E     s≤t |∆L s |   κ   .
Notice that the process

L + t := s≤t |∆L s | is an α-stable process. Then, since κ < α, letting n → +∞, we obtain E [|V t∧τr | κ ] ≤ |v 0 | κ + E L + t κ ≤ C t0,κ t κ α .
Thanks to Lemma 4.1, we can conclude that the explosion time of V is a.s. infinite, and (6) follows, letting r → ∞.

Proposition 4.3. Pick α ∈ (1, 2). Recall that (V t ) t≥t0
is the solution of (SKE). For any γ ∈ [0, 1) and any β ∈ R, the explosion time τ ∞ is a.s. infinite and for all κ ∈ [0, 1], there exists C γ,κ,β,t0 such that we have

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ α if γ-1 α + 1 ≤ β, t κ 1-β 1-γ else. (7) Remark 4.4. If γ-1 α + 1 > β, then the moment estimate (7) becomes ∀t ≥ t 0 , E [|V t |] ≤ C γ,β,t0 t 1-β 1-γ . Proof of Proposition 4.3. Assume that γ ∈ [0, 1) and fix κ ∈ [0, 1]. Then Jensen's inequality yields, for all t ≥ t 0 , E [|V t | κ ] ≤ E [|V t |]
κ , hence it suffices to verify (7) only for κ = 1. Recall that under (H γ ), there exists a positive constant K, such that for all v ∈ R, |F (v)| ≤ K |v| γ . Hence, we can write, for any t ≥ t 0 and r ≥ 0,

V (t∧τr)-≤ |v 0 -L t0 | + L (t∧τr)-+ t∧τr t0 s -β |F (V s∧τr )| ds ≤ |v 0 -L t0 | + L (t∧τr)-+ K t∧τr t0 s -β |V s∧τr | γ ds.
Since L is an α-stable process, it has a finite first moment, which can be computed. Taking the expectation in the above inequality, we get, by choosing C t0 big enough,

E V (t∧τr )-≤ E [|v 0 -L t0 |] + E L (t∧τr)-+ K t t0 s -β E [|V s∧τr | γ ] ds ≤ C t0 t 1 α + K t t0 s -β E [|V s∧τr |] γ ds.
Recalling that τ r is given by (3), the function g r : t → E V (t∧τr)-is bounded by r. Applying a Grönwall-type lemma (see Lemma A.1), we end up, for β = 1, with

∀t ≥ t 0 , E V (t∧τr)-≤ C γ C t0 t 1 α + 1 -γ 1 -β K(t 1-β -t 1-β 0 ) 1 1-γ .
The case β = 1 can be treated similarly. Thanks to Lemma 4.1, we conclude that the explosion time of V is a.s. infinite, and (7) follows from Fatou's lemma. Proposition 4.5. Pick α ∈ [1, 2). Assume here that for all v ∈ R, vF (v) ≥ 0. Recall that (V t ) t≥t0 is the solution of (SKE). For any γ ∈ R and any β ∈ R, the explosion time τ ∞ is a.s. infinite and there exists

C κ,t0 such that for κ ∈ (0, α), ∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ α . (8)
Proof. The key idea is to slice the small and big jumps in a non-homogeneous way with respect to the function ξ → ξ 1 α . We write the proof in the general setting of α ∈ (1, 2). When α = 1, the proof is similar since ν is symmetric. Pick ξ ≥ t 0 . As explained in [START_REF] Chaudru | On Multidimensional stable-driven Stochastic Differential Equations with Besov drift[END_REF] and references therein, by using this cutting threshold, the α-stable Lévy driving process can be written as

L t -L t0 = t t0 |z|≤ξ 1 α z N (ds, dz) + t t0 |z|>ξ 1 α zN (ds, dz) - t t0 |z|>ξ 1 α zν(dz) ds.
The two first integrals then satisfy the same scaling property as the α-stable Lévy driving process. We can compute

|z|>ξ 1 α zν(dz) = a + -a - α -1 ξ 1 α -1 .
Step 1. We first apply Itô's formula and estimate the expectation of each term for κ ≤ 1, in order to get (8). Fix η > 0 and define the

C 2 -function f : v → (η + v 2 ) κ/2
. For all t ≥ t 0 , by Itô's formula, using that for all v ∈ R, vF (v) ≥ 0, we have

f (V t∧τr ) ≤ f (V 0 ) - a + -a - α -1 ξ 1 α -1 t t0 1 {s≤τr} f ′ (V s ) ds + M t + R t + S t , (9)
where

M t := t t0 0<|z|<ξ 1 α 1 {s≤τr } [f (V s-+ z) -f (V s-)] N (ds, dz), (10)
R t := t t0 |z|≥ξ 1 α 1 {s≤τr} [f (V s-+ z) -f (V s-)])N (ds, dz), (11)
S t := t t0 0<|z|<ξ 1 α 1 {s≤τr } [f (V s + z) -f (V s ) -zf ′ (V s )] ν(dz) ds. (12)
Note that, since κ < 1, for all v ∈ R,

|f ′ (v)| ≤ κη κ-1 2 . ( 13
)
Moreover, remark that for all k > α,

0<|z|<ξ 1 α |z| k ν(dz) = a + + a - k -α ξ k α -1 , (14)
and for all k < α,

|z|≥ξ 1 α |z| k ν(dz) = a + + a - α -k ξ k α -1 . (15)
We estimate expectations of M , R and S.

To that end, we first show that the local martingale (M t ) t≥t0 is a martingale. Fix q ≥ 2 and r ≥ 0. Set

I t (q) := t t0 0<|z|<ξ 1 α 1 {s≤τr} |f (V s-+ z) -f (V s-)| q ν(dz) ds.
Notice that, since for all |v| ≤ r and |z| ≤ ξ

1 α , |f (v + z) -f (v)| ≤ f ′ 1 [-(r+ξ 1 α ),r+ξ 1 α ]
∞ |z|, so we have

I t (q) ≤ f ′ 1 [-(r+ξ 1 α ),r+ξ 1 α ] q ∞ t t0 0<|z|<ξ 1 α 1 {s≤τr} |z| q ν(dz) ds.
Hence, it is a finite quantity, since q ≥ 2 and (14) holds. Therefore, for q ≥ 2, by Kunita's inequality (see Theorem 4.4.23 p. 265 in [START_REF] Applebaum | Levy Processes and Stochastic Calculus[END_REF]), there exists D q such that

E sup t0≤s≤t |M s | q ≤ D q E I t (2) q/2 + E [I t (q)] < +∞.
Hence, by Theorem 51 p. 38 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF], M is a martingale.

We estimate now the finite variation part S defined in (12). We use a similar idea as in the proof of Theorem 3.1 p. 3863 in [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF]. Note that for all v ∈ R,

|f ′′ (v)| = κ(2 -κ)v 2 (v 2 + η) κ 2 -2 + κ(v 2 + η) κ 2 -1 = κ(2 -κ)v 2 (v 2 + η) -1 (v 2 + η) κ 2 -1 + κ(v 2 + η) κ 2 -1 ≤ κ(3 -κ)(v 2 + η) κ 2 -1 ≤ κ(3 -κ)η κ 2 -1 , since κ 2 -1 < 0.
Assume that |z| < ξ 1 α . Using Taylor's formula, we get a.s.

|f (V s + z) -f (V s ) -zf ′ (V s )| ≤ 1 2 κ(3 -κ)η κ 2 -1 z 2 .
Hence, we get the almost sure following bound

0<|z|<ξ 1 α (f (V s + z) -f (V s ) -zf ′ (V s )) ν(dz) ≤ 1 2 κ(3 -κ)η κ 2 -1 0<|z|<ξ 1 α z 2 ν(dz).
Injecting ( 14), we get

0<|z|<ξ 1 α (f (V s + z) -f (V s ) -zf ′ (V s )) ν(dz) ≤ 1 2 κ(3 -κ)η κ 2 -1 a + + a - 2 -α ξ 2 α -1 . ( 16
)
It remains to study the Poisson integral R defined in (11), using Theorem 2.3.7 p. 106 in [START_REF] Applebaum | Levy Processes and Stochastic Calculus[END_REF]. Pick κ ≤ 1, by Hölder property of power functions, we can write,

f (v + z) -f (v) = η + (v + z) 2 κ 2 -(v + z) 2 κ 2 + (v + z) κ -v κ + v 2 κ 2 -η + v 2 κ 2 ≤ 2η κ 2 + |z| κ .
We deduce that

|z|≥ξ 1 α f (V s + z) -f (V s )ν(dz) ≤ η κ 2 ν(|z| ≥ ξ 1 α ) + |z|≥ξ 1 α |z| κ ν(dz).
Injecting (15), this leads to

|z|≥ξ 1 α f (V s + z) -f (V s )ν(dz) ≤ η κ 2 a + + a - α ξ -1 + a + + a - α -κ ξ κ α -1 . (17)
Gathering ( 13), ( 17) and ( 16), we get

E [|V t∧τr | κ ] ≤ E [f (V t∧τr )] ≤ E [f (V t0 )] + tξ -1 × κη κ-1 2 a + -a - α -1 ξ 1 α + η κ/2 a + + a - α + a + + a - α -κ ξ κ α + 1 2 κ(3 -κ)η κ 2 -1 a + + a - 2 -α ξ 2 α . (18)
Choosing η = t 2 α and ξ = t, we get

E [|V t∧τr | κ ] ≤ E [f (V t0 )] + t κ α × κ a + -a - α -1 + a + + a - α + a + + a - α -κ + 1 2 κ(3 -κ) a + + a - 2 -α ≤ C κ,t0 t κ α .
Thanks to Lemma 4.1, we can conclude that the explosion time of V is a.s. infinite and letting r → +∞,

for all κ ∈ [0, 1], E [|V t | κ ] ≤ C κ,t0 t κ α . ( 19
)
Step 2. Pick κ ∈ (1, α). We estimate R in another way, using again Theorem 2.3.7 p. 106 in [START_REF] Applebaum | Levy Processes and Stochastic Calculus[END_REF]. By Hölder property of power function and (15), we get

|z|≥ξ 1 α |f (V s + z) -f (V s )| ν(dz) ≤ |z|≥ξ 1 α 2zV s + z 2 κ 2 ν(dz) ≤ C κ a + + a - α -κ ξ κ α -1 + |V s | κ 2 a + + a - α -κ 2 ξ κ 2α -1 . (20)
Gathering ( 16), (20) and then using that for all v ∈ R, |f

′ (v)| ≤ κ |v| κ-1 , E [|V t∧τr | κ ] ≤ E [f (V t0 )] + t C κ a + + a - α -κ ξ κ α -1 + 1 2 κ(3 -κ)η κ 2 -1 a + + a - 2 -α ξ 2 α -1 + κ a + -a - α -1 ξ 1 α -1 t t0 E |V s | κ-1 ds + C κ a + + a - α -κ 2 ξ κ 2α -1 t t0 E |V s | κ 2 ds. (21)
Injecting ( 19), and choosing η = t 2 α and ξ = t, we get

E [|V t∧τr | κ ] ≤ C κ,t0,α t κ α .
Taking r → +∞, (8) follows.

Example 4.1. Remark that the velocity process V is more explicit in the linear case (γ = 1). Choose

F (1) = ρ > 0, F (-1) = -ρ. Pick β = 1, so V t = v 0 + exp -ρ t 1-β 1 -β t t0 exp ρ s 1-β 1 -β dL s is solution of (SKE).
Hence, by an integration by parts,

V t = v 0 + L t -e ρ 1 1-β (t 1-β 0 -t 1-β ) L t0 -e -ρ t 1-β 1-β t t0 ρs -β e ρ s 1-β 1-β L s ds.
Thus,

E [|V t |] ≤ C t0 t 1 α + t 1-β+ 1 α ≤ C t0 t 1 α .
The case β = 1 can be treated similarly.

Proof of the asymptotic behavior of the solution

This section is devoted to the proofs of our main results, Theorems 2.2 and 2.4. Notice that it suffices to prove the convergence of the rescaled velocity process (ε

1 α V t/ε
) t≥εt0 in the space D endowed with the Skorokhod topology. Assume for a moment that this convergence is proved. For ε ∈ (0, 1] and t ≥ εt 0 we can write

ε 1+ 1 α X t/ε = ε 1+ 1 α x 0 + t εt0 V (ε) s ds.
Let us introduce the mapping g ε : V → V t , t εt0 V s ds t>0 defined and valued on D. Clearly, the theorem will be proved once we show that g ε (V (ε)

• ) converges weakly in D endowed by the Skorokhod topology. This mapping is converging, as ε → 0, to the continuous mapping g :

V → V t , t 0 V s ds t>0 .
In order to see

V (ε) as a process of D([0, +∞)), let us state for all s ∈ [0, εt 0 ], V (ε) s := V (ε) εt0 = ε 1 α v 0 .
Call P ε , P the distribution of V (ε) , S, respectively. Invoking the Portmanteau theorem (see Theorem 2.1 p. 16 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]), it suffices to prove that for all bounded and uniformly continuous function h :

D([0, +∞)) × D([0, +∞)) → R, D([0,+∞)) 2 h(g ε (ω)) dP ε (dω) -→ ε→0 D([0,+∞)) 2 h(g(ω)) dP (dω).
Pick such a function h. By assumption, the convergence P ε =⇒ ε→0 P holds, hence, using Lemma A.4, it suffices to prove that the uniformly bounded sequence (h • g ε ) of continuous functions on D([0, +∞)) converges to the continuous function h • g uniformly on compact subsets of D([0, +∞)). Let K be a compact set of D([0, +∞)). Then, for all ω ∈ K, max [0,εt0] |ω| is uniformly bounded by a constant, say M . Fix η > 0. By the uniform continuity of h, there exists δ > 0 such that for all ω ∈ K,

d u (g ε (ω), g(ω)) ≤ δ =⇒ |h • g ε (ω) -h • g(ω)| ≤ η.
There exists ε 1 > 0 small enough, such that for all ε ≤ ε 1 , for all ω ∈ K,

d u (g ε (ω), g(ω)) ≤ C εt0 0 ω(s) ds ≤ Cεt 0 M ≤ δ.
Therefore, we proved that it suffices to prove the convergence of the rescaled velocity process (ε 1 α V t/ε ) t≥εt0 in order to prove Theorems 2.2 and 2.4. In Sections 5.1 and 5.2, the aim is to prove the convergence of the velocity process.

Asymptotic behavior above the critical line

In the remainder of this section, we assume that γ ≥ 0 and β > 1 + γ-1 α . Proof of Theorem 2.2. Thanks to a change of variables, we have, for all ε ∈ (0, 1] and t ≥ εt 0 ,

ε 1 α V t/ε =ε 1 α (v 0 -L t0 ) + ε 1 α L t/ε -ε 1 α t/ε t0 F (V s )s -β ds =ε 1 α (v 0 -L t0 ) + ε 1 α L t/ε -ε β-1+ 1 α t εt0 F (V u/ε )u -β du.
By self-similarity, L (ε) := (ε 1 α L t/ε ) t≥0 has the same distribution as an α-stable process. As a consequence, thanks to Theorem 3.1 p. 27 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF] and Lemma A.3, it suffices to prove

∀T ≥ t 0 sup εt0≤t≤T V (ε) t -L (ε) t P → 0, as ε → 0. ( 22
)
Recall that under (H γ ), there exists a positive constant K, such that

ε γ α F V (ε) • ε 1 α ≤ K V (ε) • γ . (23)
Modifying the factor in front of the integral, we get

V (ε) t = ε 1 α (v 0 -L t0 ) + L (ε) t -ε β-1+ 1-γ α t εt0 ε γ α F V (ε) u ε 1 α u -β du. (24)
Gathering ( 24) and ( 23), for all T ≥ εt 0 , we have,

sup εt0≤t≤T V (ε) t -L (ε) t ≤ε 1 α (v 0 -L t0 ) + ε β-1+ 1-γ α sup εt0≤t≤T t εt0 ε γ α F V (ε) u ε 1 α u -β du ≤ε 1 α (v 0 -L t0 ) + ε β-1+ 1-γ α T εt0 K V (ε) u γ u -β du.
Taking the expectation and using the moment estimates on V (see Remark 2.7), we obtain, when β = γ α +1,

ε β-1+ (1-γ) α E T εt0 K V (ε) u γ u -β du = ε β-1+ 1-γ α T εt0 KE V (ε) u γ u -β du = ε β-1+ 1 α T εt0 KE V u/ε γ u -β du ≤ ε β-1+ 1-γ α T εt0 KC α,β,t0 u γ α -β du = C ε β-1+ 1-γ α T γ α -β+1 -t γ α -β+1 0 ε 1 α . Hence, setting q := min(β -1 + 1-γ α , 1 α ) which is positive, since β > 1 + γ-1 α , we get E sup εt0≤t≤T V (ε) t -L (ε) t = O ε→0 (ε q ).
The case β = 1 + γ α can be treated similarly to get

E sup εt0≤t≤T V (ε) t -L (ε) t = O ε→0 (ε 1 α ln(ε)).
This concludes the proof.

Remark 5.1. Observe that we did not use the condition "γ < 1 or for all v ∈ R, vF (v) ≥ 0" in this proof, except to get moment estimates.

Asymptotic behavior on the critical line

We adapt the Proposition 2.1, p. 187 of [START_REF] Gradinaru | Existence and asymptotic behaviour of some timeinhomogeneous diffusions[END_REF] to the α-stable Lévy case. Pick a C 2 -diffeomorphism ϕ : [0, t 1 ) → [t 0 , +∞). Let V be the solution to the equation (SKE). Thanks to Proposition 3.4.1 p. 124 in [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF], the following process is also an α-stable process

(R t ) t≥0 := t 0 dL ϕ(s) ϕ ′ (s) 1 α t≥0 . ( 25
)
Then, by the change of variables t = ϕ(s), we get

V ϕ(t) -V ϕ(0) = t 0 ϕ ′ (s) 1 α dR s - t 0 F (V ϕ(s) ) ϕ(s) β ϕ ′ (s) ds.
Thanks to an integration by parts, we get

d V ϕ(s) ϕ ′ (s) 1 α = dR s - ϕ ′ (s) 1-1 α ϕ(s) β F (V ϕ(s) ) ds - ϕ ′′ (s) αϕ ′ (s) V ϕ(s) ϕ ′ (s) 1 α ds.
Set Ω = D([t 0 , ∞)) the set of càdlàg functions, that equal ∞ after their (possibly infinite) explosion time.

Introduce the scaling transformation Φ ϕ defined, for ω ∈ Ω, by

Φ ϕ (ω)(s) := ω(ϕ(s)) ϕ ′ (s) 1 α
, with s ∈ [0, t 1 ).

As a consequence, we obtain the following result.

Proposition 5.2. If V is a solution to the equation (SKE), then V (ϕ) := Φ ϕ (V ) is a solution to

dV (ϕ) s = dR s - ϕ ′ (s) 1-1 α ϕ(s) β F (ϕ ′ (s) 1 α V (ϕ) s ) ds - ϕ ′′ (s) ϕ ′ (s) V (ϕ) s α ds, with V (ϕ) 0 = V ϕ(0) ϕ ′ (0) 1 α , ( 26
)
where R is given by (25).

Conversely, if V (ϕ) is a solution to (26), then Φ -1 ϕ (V (ϕ)
) is a solution to the equation (SKE), where

L t -L t0 := t t0 (ϕ ′ • ϕ -1 ) 1 α (s) dR ϕ -1 (s) .
Furthermore, uniqueness in law, pathwise uniqueness, strong existence hold for the equation (SKE) if and only if they hold for the equation (26).

In the following, we will focus on an exponential change of time ϕ e : 0 ≤ t → t 0 e t . This scaling is convenient since it allows to produce a time-homogeneous term in (26). Thanks to Proposition 5.2, the process V (e) := Φ e (V ) satisfies the SDE driven by an α-stable process (R t ) t≥0 ,

dV (e) s = dR s - V (e) s α ds -t 1-1 α -β 0 e (1-1 α -β)s F t 1 α 0 e s α V (e) s ds. ( 27
)
Proof of Theorem 2.4. Assume in the sequel that β = 1 + γ-1 α .

Step 1. Firstly we prove the finite-dimensional convergence. To that end, we reduce the problem to the convergence of a time-homogenous process. Since (H γ ) holds, (27) becomes

dV (e) s = dR s - V (e) s α ds -F V (e) s ds. (28)
Using the bijection Φ e induced by the exponential change of time (see Proposition 5.2), and the unique strong existence of the velocity process V (see Proposition 3.2 and Remark 2.7), there exists a pathwise unique strong solution H to the time-homogeneous equation (28). Hence, we have the equality

V t0e t (t 0 e t ) 1 /α t≥0 = (H t ) t≥0 ,
as two solutions of the same SDE, starting from the same point. We can write the above equality as

V t t 1 α t≥t0 = (H log(t/t0) ) t≥t0 .
So, we have, for all ε > 0, d ∈ N * , and

(t 1 , • • • , t d ) ∈ [εt 0 , +∞) d , V ε -1 t1 (ε -1 t 1 ) 1 /α , • • • , V ε -1 t d (ε -1 t d ) 1 /α = H log(t1)+log((εt0) -1 ) , • • • , H log(t d )+log((εt0) -1 ) . ( 29
)
Since lim sup |x|→+∞ -F (x)-x/α x < 0, it follows from Proposition 0.1 in [START_REF] Alexey | Exponential ergodicity of the solutions to SDE's with a jump noise[END_REF] that the process (H t ) t≥0 is exponentially ergodic. We denote its invariant measure by Λ F . Call H the solution of the time homogeneous equation (28

) starting from Λ F . For t 1 , • • • , t d ∈ R d , let Λ F,t1,••• ,t d := L( H t1 , • • • , H t d ) be the distribution of ( H t1 , • • • , H t d ). Then, for all s ≥ 0, Λ F,t1,••• ,t d = Λ F,t1+s,••• ,t d +s .
Indeed, thanks to the invariance property of Λ F , ( H • ) and ( H •+s ) satisfy the same SDE, starting from the same point. As a consequence, we get the stationary limit

lim ε→0 L H log(t1)+log((εt0) -1 ) , • • • , H log(t d )+log((εt0) -1 ) = Λ F,log(t1),••• ,log(t d ) . (30)
Moreover, by exponential ergodicity, for every ψ : R d → R continuous and bounded function,

E ψ H log(t1/(t0ε)) , • • • , H log(t d /(t0ε)) -E ψ H log(t1/(t0ε)) , • • • , H log(t d /(t0ε)) -→ ε→0 0. (31)
We postpone the proof of this limit in Step 2.

To conclude this step, gather (29), ( 30) and (31) to get

V ε -1 t1 (ε -1 t 1 ) 1 /α , • • • , V ε -1 t d (ε -1 t d ) 1 /α =⇒ ε→0 Λ F,log(t1),••• ,log(t d ) .
This can also be written as

ε 1 α V t1/ε , • • • , ε 1 α V t d /ε =⇒ ε→0 T * Λ F,log(t1),••• ,log(t d ) ,
where

T * Λ F,log(t1),••• ,log(t d ) is the pushforward of the measure Λ F,log(t1),••• ,log(t d ) by the linear map T (u 1 , • • • , u d ) := (t 1 /α 1 u 1 , • • • , t 1 /α d u d ).
Step 2. Let us now prove (31). For sake of clarity, let us give a proof for d = 2, the general case d ≥ 2 is similar. Let ψ : R 2 → R be a continuous and bounded function.

Pick εt 0 ≤ s ≤ t. Set h 0 = v 0 t -1 α 0 , (31) is now equivalent to E ψ H log(s/(t0ε)) , H log(t/(t0ε)) H 0 = h 0 -E ψ H log(s/(t0ε)) , H log(t/(t0ε)) H 0 ∼ Λ F -→ ε→0 0.
We set µ ε := L H log(s/(t0ε)) H 0 = h 0 . We now use the generalized Markov property of solution to SDE driven by Lévy process. For the sake of completeness, we state and prove it in our context in Appendix (see Lemma A.6). This leads to

E ψ H log(s/(t0ε)) , H log(t/(t0ε)) H 0 = h 0 = E ψ H 0 , H log(t/s) H 0 ∼ µ ε and, since Λ F is invariant, E ψ H log(s/(t0ε)) , H log(t/(t0ε)) H 0 ∼ Λ F = E ψ H 0 , H log(t/s) H 0 ∼ Λ F .
Then, we are reduced to prove

E ψ H 0 , H log(t/s) H 0 ∼ µ ε -E ψ H 0 , H log(t/s) H 0 ∼ Λ F -→ ε→0 0.
The left-hand side can be written as,

R E ψ H 0 , H log(t/s) H 0 = y (µ ε (dy) -Λ F (dy)) .
Hence, setting p(t, x, dy) := P x (H t ∈ dy) and . T V for the total variation norm, we get

E ψ H 0 , H log(t/s) H 0 ∼ µ ε -E ψ H 0 , H log(t/s) H 0 ∼ Λ F ≤ ψ ∞ R |p (log(s/(t 0 ε)), h 0 , dy) -Λ F (dy)| ≤ ψ ∞ p (log(s/(t 0 ε)), h 0 , •) -Λ F T V .
This converges to 0, as ε → 0 by exponential ergodicity of H.

Step 3. Let us prove now the tightness of the family of distributions of càdlàg process

V (ε) t≥εt0 = ε 1 α V t/ε t≥εt0 on every compact interval [m, M ], 0 < m ≤ M .
We check the Aldous's tightness criterion stated at Theorem 16.10 p.178 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]. Let a, η, T be positive reals. Let τ be a discrete stopping time with finite range T , bounded by T . Choose δ > 0 and ε > 0 small enough.

We have, by Jensen's inequality, for r = α 2 ,

E V (ε) τ +δ -V (ε) τ r ≤ E L (ε) τ +δ -L (ε) τ r + E τ +δ τ K V (ε) u γ u -β du r .
Since L (ε) is an α-stable process, by the strong Markov property,

E L (ε) τ +δ -L (ε) τ r = E [E Lτ [|L δ -L 0 | r ]] ≤ Cδ r α .
The stopping time has a finite range T . Hence, we can write

E τ +δ τ K V (ε) u γ u -β du = E E τ +δ τ K V (ε) u γ u -β du τ = E τi∈τ 1 P(τ = τ i ) E 1 {τ =τi} τi+δ τi K V (ε) u γ u -β du 1 {τ =τi} ≤ E τi∈τ 1 P(τ = τ i ) E τi+δ τi K V (ε) u γ u -β du 1 {τ =τi} .
For each τ i ∈ T , using the relation

β = 1 + (γ-1)
α and the moment estimates on V (see Remark 2.7), we obtain

E τi+δ τi K V (ε) u γ u -β du = τi+δ τi KE V (ε) u γ u -β du ≤ K τi+δ τi u γ α -β du = K (τ i + δ) 1 α -τ 1 α i ≤ Kδ 1, 1 α .
The term δ 1, 1 α has to be read as δ or δ 1 α depending on the fact that x → x 1 α is a Lipschitz continuous function on [0, T + δ], if α < 1, or a 1 α -Hölder function, if α > 1. By Markov's inequality, for δ small enough, we have

P V (ε) τ +δ -V (ε) τ ≥ a ≤ Kδ r, r α a r ≤ η.
Furthermore, by moment estimates (see Propositions 4.2, 4.3 and 4.5), for all t ≥ εt 0 ,

sup ε V (ε) t r ≤ Ct r α .
Hence, using again Markov's inequality, by Corollary and Theorem 16.8 p. 175 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF], this concludes the proof of the tightness of the velocity process and therefore the proof of Theorem 2.4.

Asymptotic behavior below the critical line

Assume in this section that β < 1 + γ-1 α and α > 1. As a consequence, αq < 1. We take an interest into the power change of time ϕ q : t → t 1-αq 0 + (1 -αq)t 1 1-αq . Thanks to Proposition 5.2, the process V (q) := Φ q (V ) satisfies the SDE driven by an α-stable process R having the same distribution as L,

dV (q) s = dR s -F V (q) s ds -qϕ αq-1 q V (q) s ds. ( 32
)
Proof of Theorem 2.5.

Step 1. We first prove the finite dimensional convergence of the velocity process (V (ε) t ) t≥εt0 := (ε q V t/ε ) t≥εt0 . We give a proof for d = 2, the general case d ≥ 2 is similar. We call H the ergodic process solution to

dH s = dR s -F H s ds, with H 0 = h 0 := v 0 t -q 0 . (33)
We denote by Π F its invariant measure. Using the bijection induced by the power change of time (Proposition 5.2), as solutions of the same SDE starting at the same point, we have, for all ε > 0, and (s, t)

∈ [εt 0 , +∞) 2 , ε q V ε -1 s s q , ε q V ε -1 t t q = V (q) ϕ -1 (ε -1 s) , V (q) ϕ -1 (ε -1 t)
Using Theorem 3.1 p. 27 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF], it suffices to prove that for all (s, t)

∈ [εt 0 , +∞) 2 • H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) , V (q) ϕ -1 (ε -1 s) , V (q) ϕ -1 (ε -1 t) -→ ε→0 0, where • is a metric on R 2 . • H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) =⇒ ε→0 Π F ⊗ Π F .
Step 2. We prove that for all t ≥ εt 0 ,

E H ϕ -1 (ε -1 t) -V (q) ϕ -1 (ε -1 t) 2 -→ ε→0 0.
Pick t ≥ εt 0 . For simplicity of notation, we write

H (ϕ,ε) t := H ϕ -1 (ε -1 t) and V (ϕ,ε) t := V (q) ϕ -1 (ε -1 t) . We have d H (ϕ,ε) t -V (ϕ,ε) t = -ε αq-1 F (H (ϕ,ε) t ) -F (V (ϕ,ε) t ) t -αq dt + qt -1 V (ϕ,ε) t dt.
Pick δ > 0 and κ ∈ (1, α), by straightforward differentiation,

d H (ϕ,ε) t -V (ϕ,ε) t κ ≤ - κε αq-1 t αq F (H (ϕ,ε) t ) -F (V (ϕ,ε) t ) H (ϕ,ε) t -V (ϕ,ε) t κ-1 1 H (ϕ,ε) t -V (ϕ,ε) t >δ dt + κt -1 qV (ϕ,ε) t sgn H (ϕ,ε) t -V (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t κ-1 dt Since γ ≥ 1, the function F -1 is 1 γ -Hölder, therefore there exists C γ > 0 such that, d H (ϕ,ε) t -V (ϕ,ε) t κ ≤ - κε αq-1 t αq C γ δ γ-1 H (ϕ,ε) t -V (ϕ,ε) t κ 1 H (ϕ,ε) t -V (ϕ,ε) t >δ dt + κt -1 qV (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t κ-1 dt. We set g ε (t) = E H (ϕ,ε) t -V (ϕ,ε) t κ and g ε (t) = E H (ϕ,ε) t -V (ϕ,ε) t κ 1 H (ϕ,ε) t -V (ϕ,ε) t >δ
. Taking expectation, we get,

g ′ ε (t) ≤ - κε αq-1 t αq C γ δ γ-1 g ε (t) + b ε (t), with g ε (εt 0 ) = 0 (34)
where, using Hölder's inequality and moment estimates (Proposition 4.2), we have

b ε (t) := κt -1 qE V (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t κ-1 ≤ κt -1 |q| E V (ϕ,ε) t κ 1 κ g ε (t) κ-1 κ ≤ Ct -1 t ε 1 α -q g ε (t) κ-1 κ .
We use comparison theorem for ordinary differential equation on (34) to get, setting h(t

) := 1 1 -αq C γ δ γ-1 t 1-αq g ε (t) ≤ t εt0 b ε (s) exp(-κε αq-1 (h(t) -h(s))) ds.
As a consequence,

g ε (t) ≤ δ 2 + g ε (t) ≤ δ 2 + exp(-κε αq-1 h(t))C t εt0 s -1 s ε 1 α -q g ε (s) exp(κε αq-1 h(s)) κ-1 κ exp(ε αq-1 h(s)) ds.
Applying a Grönwall-type lemma (Lemma A.1) to g ε (s) exp(κε αq-1 h(s)), we have

g ε (t) ≤ Cδ 2 + C t εt0 s -1 s ε 1 α -q exp(-ε αq-1 (h(t) -h(s))) ds κ .
We conclude, using the dominated convergence theorem, since 1 -αq > 0, that for all δ > 0 0 ≤ lim sup

ε→0 g ε (t) ≤ δ 2 . ( 35
)
To prove the domination hypothesis, note that by optimization of the function x → x 1 α exp(-Ax),

1 {εt0≤s≤t} s -1 s -1 s ε 1 α -q exp(-ε αq-1 (h(t) -h(s)) ≤ C1 {0≤s≤t} s -1+ 1 α -q 1 (h(t) -h(s)) 1 α .
This function in integrable, since 1 -αq > 0 and α > 1. We let δ → 0 in (35) to conclude that for all t > 0, lim ε→0 g ε (t) = 0.

Step 3. Pick (s, t) ∈ [εt 0 , +∞) 2 . Similarly, as in [START_REF] Gradinaru | Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion[END_REF], one can prove that the solution H to (33) satisfies

H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) =⇒ ε→0 Π F ⊗ Π F . ( 36
)
6 Extended results in the Lévy case

In this section, the driving process of (SKE) is supposed to be a general Lévy process L. We denote by (A, ν, b) its generating triplet, with respect to the truncation function h : z → -1 ∨ (z ∧ 1). Here ν is the Lévy measure, A > 0 and b ∈ R. This means that, in virtue of the Lévy-Khintchine formula (see Theorem 8.1 p. 37 in [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]), the characteristic function of L t is given by u) , where ψ(u) := -

E e iuLt = e tψ(
Au 2 2 + ibu + R (e iux -1 -iuh(x))ν(dx).
Pick α ∈ (0, 2) \ {1}. The Lévy measure ν of the driving process is supposed to satisfy either

ν(z) = g(z)
|z| 1+α 1 {z =0} , where g is a non-negative measurable function such that

c + := lim z→+∞ g(z) ≥ 0, c -:= lim z→-∞ g(z) ≥ 0, (H ν,α 1 )
or for some α 0 > 1,

|z|≥1 |z| α0 ν(dz) < +∞. (H ν,α0
2 ) (ii) Suppose that the Lévy measure satisfies the condition (H ν,α 1 ) with 1 < α < 2 and (H g ). If

b * := b + +∞ 1 g(z) -g(-z) z 1+α (z -h(z)) dz = 0,
then the process εL t/ε t≥0 converges in distribution to the deterministic Lévy process L generated by (0, 0, b * ).

(iii) If the Lévy measure satisfies (H ν,α0

2

) and b * := b + |z|≥1 (z -h(z)) ν(dz) = 0, then the process εL t/ε t≥0 converges in distribution to the deterministic Lévy process L * generated by (0, 0, b * ).

(iv) Assuming that 0 < α < 1, if the Lévy measure satisfies the condition (H ν,α 1 ), then the process ε

1 α L t/ε t≥0
converges in distribution to the α-stable Lévy process L with

ν * (dz) := c + 1 {z>0} + c -1 {z<0} |z| 1+α dz.
(v) Assuming that 1 < α < 2 and b = 0, if the Lévy measure satisfies the condition (H ν,α 1 ), then the process ε

1 α L t/ε t≥0
converges in distribution to the Lévy process L with measure and center

ν * (dz) := c + 1 {z>0} + c -1 {z<0 } |z| 1+α dz, b * := R * zν * (dz).
Proof. From the Lévy-Khintchine's formula, the generating triplet of (L (ε) ) t≥0 := (v ε L t/ε ) t≥0 is given by

A ε = v 2 ε ε A, (37)
for all B ∈ B(R), ν ε (B) = ε -1 ν({z, zv ε ∈ B}), (38)
b ε = v ε ε b + R * v -1 ε h(v ε z) -h(z) ν(dz) . (39)
Call (A * , ν * , b * ) the generating triplet of the limiting process L. By Corollary 3.6 p. 415 in [START_REF] Jacod | The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals[END_REF] and Theorem 14.7 p. 81 in [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF], we have to check that

b ε -→ ε→0 b * , (40)
A ε + R * h 2 (z)ν ε (dz) -→ A * + R * h 2 (z)ν * (dz), (41)
and that for any continuous and bounded function f which is zero in a neighborhood of zero,

R * f (z)ν ε (dz) -→ ε→0 R * f (z)ν * (dz). (42)
(i) Recall that α = 1 and assume that the Lévy measure ν satisfies the conditions (H ν,α 1 ) and (H g ). To prove (40), we write b ε as

b ε = b + +∞ 0 g(z) -g(-z) z 1+α ε -1 h(εz) -h(z) dz.
The dominated convergence theorem can be applied, and we show that it converges to

b * = b + +∞ 1 g(x) -g(-x) x 2 (x -h(x)) dx.
Observe that the condition (H g ) was only required for this step. Afterwards, note that, using a change of variables,

R * h 2 (z)ν ε (dz) = v α ε ε -1 R * h 2 (z) g(zv -1 ε ) |z| 1+α dz,
and thus, we can apply the dominated convergence theorem to prove that the last integral converges to

R * h 2 (z)ν * (dz).
Let f be a continuous and bounded function which is zero in a neighborhood of zero, then, using again a change of variables and applying the dominated convergence theorem,

R * f (z)ν ε (dz) -→ ε→0 R * f (z)ν * (dz).
(ii) The proof is similar to the previous one and thus is left to the reader.

(iii) In this point, we assume that (H ν,α0

2 ) holds. The convergence (40) follows from the dominated convergence theorem. Using the explicit form of the truncation function, we get, for

v ε < 1, R * h 2 (z)ν ε (dz) = R * ε -1 h 2 (v ε z)ν(dz) = ε 0<|z|<1 z 2 ν(dz) + 1≤|z|≤ε -1 εz 2 ν(dz) + |x|>ε -1 ε -1 ν(dz). (43)
Using the property of a Lévy measure, the first term in (43) converges to zero, as ε → 0, by assumption. Then, the last two terms in (43) are lower than

ε α0-1 1<|z|<ε -1 |z| α0 ν(dz) + ε α0-1 |z|≥ε -1 |z| α0 ν(dz),
which converges to zero, when ε goes to zero, since α 0 > 1.

Let f be a continuous and bounded function and assume that there exists δ > 0 such that f (z) = 0 for all |z| ≤ δ, thus,

R * f (z)ν ε (dz) = |z|> η ε ε -1 f (v ε z)ν(dz) ≤ Cε α0-1 |z|> η ε |z| α0 ν(dz).
This vanishes as ε → 0.

(iv) Take 0 < α < 1.

Using the explicit form of h, giving in Proposition 6.3, we have,

b ε = ε 1 α -1 b + -ε -1 α -∞ [ε 1 α -1 -ε -1 ]ν(dz) + +∞ ε -1 α [ε -1 -ε 1 α -1 ]ν(dz) + ε 1 α -1 -1 -ε -1 α (z + 1)ν(dz) + ε 1 α -1 ε -1 α 1 (z -1)ν(dz).
Since g has finite limits at infinity, for any δ > 0, we can choose η > 1 big enough so that |g(z) -

c + | < δ for z ≥ η, and |g(z) -c -| < δ for z ≤ -η. Hence, lim sup ε→0 ε 1 α -1 ε -1 α 1 (z -1)ν(dz) = lim sup ε→0 ε 1 α -1 ε -1 α η (z -1)ν(dz) ≤ lim sup ε→0 ε 1 α -1 (c + + δ) ε -1 α η x -1 x 1+α dx = c + + δ 1 -α .
Similarly,

lim inf ε→0 ε 1 α -1 ε -1 α 1 (z -1)ν(dz) = lim inf ε→0 ε 1 α -1 ε -1 α η (z -1)ν(dz) ≥ lim inf ε→0 ε 1 α -1 (c + -δ) ε -1 α η z -1 z 1+α dz = c + -δ 1 -α .
The choice of δ being arbitrary, we get

lim ε→0 ε 1 α -1 ε -1 α 1 (z -1)ν(dz) = c + 1 -α .
If ε is small enough, then we can upper bound

lim sup ε→0 +∞ ε -1 α [ε -1 -ε 1 α -1 ]ν(dz) ≤ lim sup ε→0 (ε -1 -ε 1 α -1 )(c + + δ)ν([ε -1 α , +∞)) = lim sup ε→0 (1 -ε 1 α ) c + + δ α = c + + δ α .
Moreover,

lim inf ε→0 +∞ ε -1 α [ε -1 -ε 1 α -1 ]ν(dz) ≥ lim inf ε→0 (ε -1 -ε 1 α -1 )(c + -δ)ν([ε -1 α , +∞]) = lim inf ε→0 (1 -ε 1 α ) c + -δ α = c + -δ α .
Similarly, this leads to

lim ε→0 ε 1 α -1 -1 -ε -1 α (z -1)ν(dz) = - c - 1 -α ,
and,

lim ε→0 -ε -1 α -∞ [ε 1 α -1 -ε -1 ]ν(dz) = - c - α .
Hence, we obtain

lim ε→0 b ε = c + -c - α(1 -α) . Since, R * h(z)ν * (dz) = c + -c - α(1-α)
, the drift coefficient of the limiting process equals zero. The proof of (41) and (42) are identical to the one done in (i).

(v) Take 1 < α < 2 and assume that b = 0.

After a change of variables, we can apply the dominated convergence theorem to

b ε = R * h(y) -ε 1 α h(yε -1 α ) g(yε -1 α ) |y| 1+α dy.
The proof of (41) and (42) are identical to the one done in (i).

6.2 Proofs of Theorems 6.1 and 6.2

In this section, we suppose that β > 1 + p α (γ) -θ, where θ is the exponent of the rate of convergence given in Proposition 6.3 and p α (γ) is given in the statement of Theorems 6.1 and 6.2. Recall that (V t ) t≥t0 is a solution of (SKE).

Moment estimates of the velocity process

As in Section 4, we will show that there exists a constant C γ,κ,β,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0,b t pα(γ,κ) , (44)
where p α (γ, κ) has to be detailed.

Proposition 6.4. Pick α ∈ (0, 1). Assume (H ν,α 1 ) and (H sgn ). For any γ, β, the explosion time τ ∞ is a.s. infinite and for all κ ∈ [0, α), there exists C κ,t0,b such that, we have

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0,b t κ α , or equivalently p α (γ, κ) = κ α .
Proof. The proof is analogous to the proof of Proposition 4.2. The term

t t0 bf ′ n (V s ) ds is bounded by |b| t since f ′ n ∞ ≤ 1.
Moreover, since ν satisfies (H ν,α 1 ) with α < 1, the process L + t := s≤t |∆L s | satisfies the conditions of Theorem 3.1 c) in [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF]. Thus, for all κ ∈ [0, α), for all t ≥ t 0 ,

E L + t κ ≤ C t0,κ t κ α .
The estimates for V follows. Proposition 6.5. Assume either (H ν,α0

2

) or (H ν,α 1 ) with α ∈ (1, 2). For any γ ∈ [0, 1) and any β ∈ R the explosion time τ ∞ is a.s. infinite and for all κ ∈ [0, 1], there exists C γ,κ,β,t0 and C γ,t0 such that under (H ν,α 1 ), we have

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ α + C γ,t0 |b| κ t κ . ( 45
)
Or equivalently p α (γ, κ) = κ α if b = 0, and p α (γ, κ) = κ, else. And under (H ν,α0 2 ), there exists C γ,κ,β,t0,b such that we have

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0,b t κ .
Proof. We explain the differences with respect to the proof of Proposition 4.3. Under each hypothesis on the Lévy measure ν, the Lévy process has a first finite moment. We write L t as the sum bt + L t , where L is the Lévy process without the drift part. We get,

E V (t∧τr)-≤ C t0 |b| t + E L (t∧τr)-+ K t t0 s -β E [|V s∧τr |] γ ds.
The proof of Theorem 3.1 (a) p. 3861 in [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF] can be adapted to estimate the moment of the Lévy process stopped at the stopping time τ r , given by (3).

(i) If the Lévy measure ν satisfies (H ν,α 1 ) with 1 < α < 2, then it satisfies the conditions of Theorem 3.1 (a) and (c) in [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF]. Thus, for all κ ∈ [0, 1], there exists C t0,κ such that, for all t ≥ t 0 ,

E L (t∧τr)- κ ≤ C t0,κ t,
and,

E L t κ ≤ C t0,κ t κ α .
(ii) If the Lévy measure ν satisfies (H ν,α0

2 ), then it satisfies the conditions of Theorem 3.1 (a) in [START_REF] Deng | On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes[END_REF]. Thus, for all κ ∈ [0, 1], there exists C t0,κ such that, for all t ≥ t 0 ,

E L (t∧τr)- κ ≤ C t0,κ t.
Applying the Grönwall-type lemma again (see Lemma A.1) and Fatou's lemma, for β = 1, we end up with

∀t ≥ t 0 , E V (t∧τr)-≤ C γ C t0,b t + 1 -γ 1 -β K(t 1-β -t 1-β 0 ) 1 1-γ .
The case β = 1 can be done in a similar manner. We conclude that the explosion time of V is a.s. infinite. To refine the estimates under the hypothesis (H ν,α 1 ), we apply again the Grönwall-type lemma to

E [|V t |] ≤ C t0 |b| t + E L t + K t t0 s -β E [|V s |] γ ds.
This proves (45) since γ-1 α + 1 -β ≤ 0. Proposition 6.6. Assume that for all v ∈ R, vF (v) ≥ 0. Pick γ ∈ R and β ∈ R. For each of the following cases, the explosion time τ ∞ is a.s. infinite.

(i) Assume that (H ν,α 1 ) holds with a bounded function g and α ∈ [1, 2), there exists C κ,t0 such that

for κ ∈ (0, α), ∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ α , or equivalently p α (γ, κ) = κ α . (46)
(ii) Assume that (H ν,α 1 ) holds with α ∈ [1, 2). Then, for κ ∈ [0, 1] (resp. κ ∈ [0, 1), if α = 1), there exists C κ,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ , or equivalently p α (γ, κ) = κ; (47)
for κ ∈ (1, α), γ ∈ [0, 1) and b = 0, there exists C κ,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t 3κ 2α , or equivalently p α (γ, κ) = 3κ 2α ; (48)
for κ ∈ (1, α), there exists C κ,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ α + κ 2 , or equivalently p α (γ, κ) = κ α + κ 2 . ( 49
) (iii) Assume (H ν,α0 2
). Then, for κ ∈ [0, 1], there exists C κ,t0 such that,

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ , or equivalently p(γ, κ) = κ; (50)
for κ ∈ [0, α 0 ], there exists C κ,t0 such that

∀t ≥ t 0 , E [|V t | κ ] ≤ C κ,t0 t κ α 0 + κ 2 , or equivalently p(γ, κ) = κ α 0 + κ 2 . ( 51
)
Proof. We highlight only the differences with respect to the proof of Proposition 4.5. In the following, we assume that α = 1, the proof is similar for α = 1.

Step A. Assume that (H ν,α 1 ) holds with a bounded function g. Step A1. Pick κ ∈ [0, 1]. We adapt the estimates of the Itô's formula's terms. There is an additional term in (9), given, for t ≥ t 0 , by t t0

1 {s≤τr } f ′ (V s )b ds.
If g is a bounded function, the other terms of (9) can be estimated in the same way, and (4) becomes

E [|V t∧τr | κ ] ≤ E [f (V t0 )] + t κ α +1-1 α κ |b| + t κ α sup R |g| × κ a + -a - α -1 + a + + a - α + a + + a - α -κ + 1 2 κ(3 -κ) a + + a - 2 -α ≤ C κ,t0,b t κ α . (52)
This gives the proof of (46) for κ ∈ [0, 1].

Step A2. Pick κ ∈ (1, α). We estimate R, given by (11) in another way. The inequality (21) becomes

E [|V t∧τr | κ ] ≤ E [f (V t∧τr )] ≤ E [f (V t0 )] + t C κ a + + a - α -κ ξ κ α -1 + 1 2 κ(3 -κ)η κ 2 -1 a + + a - 2 -α ξ 2 α -1 + κ a + -a - α -1 ξ 1 α -1 + |b| t t0 E |V s | κ-1 ds + C κ a + + a - α -κ 2 ξ κ 2α -1 t t0 E |V s | κ 2 ds. ( 53
)
The inequality (46) follows as in the proof of Proposition 4.5.

Step B. When working under (H ν,α0

2 ), we will pick κ ≤ α 0 during the proof. Assume that either (H ν,α 1 ) holds with an unbounded function g or (H ν,α0 2 ) holds, then we are not able to estimate the following terms Hence, we apply the same proof scheme with small and big jumps sliced at 1.

Step B1. This leads to a similar bound as in (4):

E [|V t∧τr | κ ] ≤ E [f (V t∧τr )] ≤ E [f (V t0 )] + t t0 |b| E [|f ′ (V s∧τr )|] ds + E [t ∧ τ r ] × η κ/2 ν(|z| ≥ 1) + |z|≥1 |z| κ ν(dz) + 1 2 κ(3 -κ)η κ 2 -1 0<|z|<1 z 2 ν(dz) ≤ C κ,t0 t.
By Jensen's inequality, we can deduce (47) and (50).

Step B2. Pick κ ∈ (1, α). We estimate R, given by (11) in another way. By classical Hölder inequality,

|z|≥1 |f (V s + z) -f (V s )| ν(dz) ≤ |z|≥1 2zV s + z 2 κ 2 ν(dz) ≤ C(1 + |V s | κ 2 ) |z|≥1 |z| κ ν(dz) (54)
The last integral is finite. Gathering ( 16), (54) and then using (47), (50) or Proposition 6.5,

E [|V t∧τr | κ ] ≤ E [f (V t∧τr )] ≤ C κ,t0 t + C t t0 E |V s | κ 2 ds + t t0 |b| E |V s | κ-1 ds.
Taking r → +∞, we can conclude that p α (γ, κ) = 1 + p α (γ, κ/2).

Step B3. We refine the estimates. Fix κ ∈ [0, α). There exists ε 1 such that, κ ≤ α -ε and α -ε > 1.

Hence, we can write

E |V t | α-ε ≤ C κ,t0 t 1+pα(γ,(α-ε)/2) .
Using Jensen's inequality, we get

E [|V t | κ ] ≤ C κ,t0 t κ α-ε (1+pα(γ,(α-ε)/2)) ,
and it suffices to let ε → 0 to conclude. This concludes the proof of ( 48) and (49) The last step is identical under (H ν,α 1 ) and can be done with α 0 instead of α -ε, under (H ν,α0

2

). This concludes the proof of (51).

Proofs of Theorems 6.1 and 6.2

We are now in position to give the proofs of Theorems 6.1 and 6.2. Assume that either (H ν,α 1 ) or (H ν,α0

2 ) is satisfied. As in the α-stable case (see Section 5), it suffices to prove the convergence of the rescaled velocity process (V

(ε) t ) t>0 := (v ε V t/ε ) t>0 .
Thanks to Proposition 6.3, there exists θ ∈ {1, 1 α } such that L (ε) := (ε θ L t/ε ) t≥0 converges in distribution. For sake of simplicity, we omit the dependencies of p with respect to α and α 0 . Assume that γ ≥ 0 and β > 1 + p(γ) -θ. We can write, for T ≥ εt 0 ,

sup εt0≤t≤T V (ε) t -L (ε) t ≤ v ε (v 0 -L t0 ) + v 1-γ ε ε β-1 T εt0 K V (ε) u γ u -β du.
Using the moment estimates on V (see Section 6.2.1), this leads, with β = p(γ) + 1, to

v 1-γ ε ε β-1 E T εt0 K V (ε) u γ u -β du ≤ C v ε ε β-1-p(γ) T p(γ)-β+1 -t p(γ)-β+1 0 v ε .
Hence, setting q := min(β -1 + θ -p(γ), θ), which is a positive number since β > 1 + p(γ) -θ, we obtain

E sup εt0≤t≤T V (ε) t -L (ε) t = ε→0 O(ε q ).
The case β = 1 + p(γ) can be treated similarly. This concludes both of the proofs.

Lévy driving process with a Brownian component

We extend the results given in Theorems 2.2, 2.4, 6.1 and 6.2 to a driving process having a Brownian component. To this end, we decompose the Lévy noise as the sum of a Brownian part (B t ) t≥0 and a Lévy part without a Brownian component (L t ) t≥0 .

Remark 6.7. The reasoning is the same with a Lévy process L instead of the Brownian driving process. Indeed, let v ε be the rate such that ( v ε L t/ε ) t≥0 converges. The conclusion follows, provided that v -1 ε v ε tends to 0.

Pick ρ ≥ 0. We consider the following one-dimensional stochastic kinetic model, for t ≥ t 0 > 0, dV

(2) t = dL t + dB t -t -β ρ sgn(V (2) t ) V (2) t γ dt, V (2) t0 = v 0 ∈ R, and dX (2) t = V (2) t dt, X (2) t0 = x 0 ∈ R. (SKE2)
We compare this solution with the solution of the equation driven by a Lévy process L without a Brownian component. From now on, for sake of convenience, the solutions of (SKE) with F (v) = ρ sgn(v) |v| γ , will be denoted by (V

t , X

t ) t≥t0 instead of (V t , X t ) t≥t0 . The asymptotic behavior of the latter SDE is given by Theorems 2.2, 2.4, 6.1 and 6.2. We will show that the Brownian part has no contribution. Let us first point out some results about existence up to explosion of (SKE2). Proposition 6.8.

(i) When γ ≥ 1, (SKE2) admits a unique solution up to explosion.

(ii) Pick α ∈ (1, 2). If (SKE2) is driven by the sum of a Brownian motion and an α-stable process, then, if 0 ≤ γ < 1, there exists a solution of (SKE2).

Proof.

(i) Since the drift coefficient is locally Lipschitz, by a standard localization argument, using Theorem 3.1 and Remark 2 p. 338-339 in [START_REF] Kunita | Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms[END_REF], there is a unique solution defined up to explosion.

(ii) Since the drift coefficient is a continuous function, using a standard localization argument, we can apply Theorem 3.1 p. 866 in [START_REF] Kurenok | Stochastic Equations with Time-Dependent Drift Driven by Levy Processes[END_REF] to conclude.

Assume in the following that (SKE2) admits a unique solution up to explosion.

Theorem 6.9. Consider γ ∈ [0, α) and β ≥ 0. Let (V

t , X

t ) t≥t0 be a solution to (SKE2). The Lévy noise is supposed to be the sum of a Brownian noise (B t ) t≥0 and a Lévy process (L t ) t≥0 without Brownian component. Suppose also that (SKE) satisfies the conditions of either Theorems 2.2, 2.4, 6.1 or 6.2. Then (V (2) , X (2) ) has the same asymptotic behavior as (V (1) , X (1) ), which is given in Theorems 6.1 and 6.2.

Proof. For i ∈ {1, 2}, let us introduce the following rescaled processes:

(V (i,ε) t ) t≥εt0 := (v ε V (i) t/ε ) t≥t0 , (X (i,ε) t ) t≥εt0 := (εv ε X (i) t/ε ) t≥t0 , and (B (ε) t ) t≥0 := ( √ εB t/ε ) t≥0 .
We write, for all t ≥ εt 0 ,

V (1,ε) t -V (2,ε) t = v ε B t0 -v ε ε -1 2 B (ε) t -v 1-γ ε ε β-1 t εt0 v γ ε F V (1,ε) u v γ ε -F V (2,ε) u v γ ε u -β du.
Hence, by Itô's formula, we get

V (1,ε) t -V (2,ε) t 2 = v 2 ε B 2 t0 -2 t εt0 V (1,ε) s -V (2,ε) s v ε ε -1 2 dB (ε) s + v 2 ε ε -1 (t -εt 0 ) -2v 1-γ ε ε β-1 t εt0 V (1,ε) s -V (2,ε) s v γ ε F V (1,ε) u v γ ε -F V (2,ε) u v γ ε u -β du.
The last term on the right-hand side of the upper equality is positive since F is an increasing function. Moreover, we can apply BDG inequalities (see Theorem 73 p. 222 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]) to the local martingale, defined by

M t := -2 t εt0 V (1,ε) s -V (2,ε) s v ε ε -1 2 dB (ε) s .
This yields, for each T > 0,

E sup εt0≤t≤T M 2 t ≤ E 4v 2 ε ε -1 T εt0 V (1,ε) s -V (2,ε) s 2 ds .
For the sake of clarity, given x and y, we introduce x y to mean that there exists a positive constant C such that x ≤ Cy. We get

E sup εt0≤t≤T V (1,ε) t -V (2,ε) t 4 v 4 ε E B 4 t0 + v 4 ε ε -2 T 2 + E sup εt0≤t≤T M 2 t v 4 ε E B 4 t0 + v 4 ε ε -2 T 2 + 4v 2 ε ε -1 T εt0 E V (1,ε) s -V (2,ε) s 2 ds.
Hence, using that for all real a, a 2 ≤ a 4 + 1, we deduce that

E sup εt0≤t≤T V (1,ε) t -V (2,ε) t 4 v 4 ε E B 4 t0 + v 4 ε ε -2 T 2 + 4v 2 ε ε -1 T + 4v 2 ε ε -1 T εt0 E sup εt0≤t≤s V (1,ε) t -V
(2,ε) t 4 ds.

Applying Grönwall's lemma, up to stopping times, we get

E sup εt0≤t≤T V (1,ε) t -V (2,ε) t 4 v 4 ε E B 4 t0 + v 4 ε ε -2 T 2 + 4v 2 ε ε -1 T exp 4v 2 ε ε -1 (T -εt 0 ) .
We deduce also that

E sup εt0≤t≤T X (1,ε) t -X (2,ε) t 4 ≤ T 4 E sup εt0≤t≤T V (1,ε) t -V (2,ε) t 4 T 4 v 4 ε E B 4 t0 + v 4 ε ε -2 T 2 + 4v 2 ε ε -1 T exp 4v 2 ε ε -1 (T -εt 0 ) . Hence, V (1,ε) t -V (2,ε) t , X (1,ε) t -X
(2,ε) t t≥εt0

converges uniformly towards 0 in probability on compacts, as ε goes to zero. The conclusion follows from Theorems 6.1, 6.2, Theorem 3.1 p. 27 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF] and Lemma A.3.

A Some technical results

Let us state first a Grönwall-type lemma which has been used to get moment estimates. The proof can be found in [START_REF] Gradinaru | Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion[END_REF].

Lemma A.1 (Grönwall-type lemma). Fix r ∈ [0, 1) and t 0 ∈ R. Assume that g is a non-negative realvalued function, b is a positive function and a is a differentiable real-valued function. Moreover, suppose that the function bg r is a continuous function. Assume that ∀t ≥ t 0 , g(t) ≤ a(t) + Remark A.2. Call H(t) the right-hand side of (55). If g is not continuous, note that the function H is still continuous and satisfies (55) (since b is positive and g ≤ H). So, one can apply the lemma to H and thereafter use the inequality g ≤ H.

We state now a technical lemma about the convergence in the spaces C and D. We recall that the space of continuous functions C is endowed with the uniform topology Set Λ := {λ : R + → R + , continuous and increasing function such that λ(0) = 0, lim t→+∞ λ(t) = +∞} and

k n (t) =      1 if 1 n ≤ t ≤ n, n + 1 -t if n < t < n + 1, 0 if n + 1 ≤ t.
The space of càdlàg functions D is endowed with the Skorokhod topology d s defined for (f, g) ∈ D((0, +∞)) , that

P (d (f ε , h ε ) > η) ≤ N n=1 P sup [ 1 n ,n] |f ε -h ε | > η ′ -→ ε→0 0.
For the sake of completeness, we state and improve the result of Problem 4.12 p. 64 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF], on a general metric space.

Lemma A.4. Let S be a Polish metric space endowed with a Borel σ-field S. Suppose that (P n ) n≥1 is a sequence of probability measures on (S, S) which converges weakly to a probability measure P . Suppose, in addition, that the sequence (f n ) n≥1 of real-valued continuous functions on S is uniformly bounded and converges to a continuous function f , the convergence being uniform on compact subsets of S. Then, we have lim Proof. Notice that, since (P n ) n≥1 converges weakly thus, it is tight. So, for each ε > 0, there exists a compact subset K of S such that for any n ≥ 1, P n (K) ≥ 1 -ε.

  ∀t ≥ t 0 , g(t) ≤ C r a(t) + (1 -r)

  d u : f, g ∈ C((0, +∞)) → +∞ n=1 1 2 n min 1, sup [ 1 n ,n] |f -g| .

  |f -g| .Let us now prove the second part. Assume that for all n ≥ 1, sup[ 1 n ,n] |f ε -h ε | P -→ 0, as ε → 0. Fix η > 0 and choose N > 0 such that +∞ n=N +1 1 2 n ≤ η 2 . Then, d s (f ε , h ε ) ≤ d u (f ε , h ε ) |f ε -h ε | .It follows, by setting η ′ := η

  n→+∞ S f n (ω) dP n (ω) = S f (ω) dP (ω).

  Let us decomposeS f n dP n -S f dP = A + B + C + D,

  The uniform distance is finer than the Skorokhod one i.e. d s ≤ d u .(ii) Let (f ε ) ε≥0 , (h ε ) ε≥0 be two sequences of functions of D. If for all n ≥ 1, Proof. Let f, g be two càdlàg functions. The first point is true using the definition of the metrics d s and d u and noting that inf a s.t. ∃λ ∈ Λ, sup

								2
	by
	+∞ n=1	1 2 n min 1, inf a s.t. ∃λ ∈ Λ, sup s =t	log	λ(t) -λ(s) t -s	≤ a, sup t≥ 1 n	|k n (t) (f • λ(t) -g(t))| ≤ a	.
	Lemma A.3.
	(i) sup
				t∈[ 1 n ,n]
		s =t	log	λ(t) -λ(s) t -s	≤ a, sup t≥ 1		[ 1 n+1 ,n+1]

|f ε (t) -h ε (t)| P → 0 as ε → 0, then d(f ε , h ε ) P → 0, where d ∈ {d u , d s }. n |k n (t) (f (λ(t) -g(t))| ≤ a ≤ sup

Acknowledgements The authors would like to thank Nicolas Fournier and Thomas Cavallazzi for helpful discussions about moment estimates. We would like to thank Vlad Bally for his suggestion which gives rise to the Section 6. We are grateful to Thomas Cavallazzi for his careful reading of the manuscript.

Remark that if ν satisfies (H ν,α 1 ) with α > 1, then it satisfies (H ν,α0

2

). Note that any tempered stable process satisfies (H ν,α 1 ), and any truncated α-stable process satisfies (H ν,α0

2

). For clarity, we recall the stochastic kinetic model: for t ≥ t 0 > 0, dV t = dL t -t -β F (V t ) dt, with V t0 = v 0 > 0, and dX t = V t dt, with X t0 = x 0 ∈ R.

(SKE)

We work under the assumption that there exists a unique solution to (SKE). We will show that Theorem 2.2 can be extended with this general Lévy driving process. We suppose first the Lévy process to be without a Brownian component. This case will be discussed in Section 6.3. We obtain the two following theorems, depending on which hypothesis is satisfied by the Lévy measure ν.

Theorem 6.1. Consider γ ∈ [0, α) and β ≥ 0. Assume that (H ν,α 1 ) and (H sgn ) are satisfied and define

Let (V t , X t ) t≥t0 be the solution to (SKE).

Then there exist a rate of convergence ε θ and a Lévy process L, given in Proposition 6.

) is satisfied and define

Suppose that β > 1 + p α0 (γ) -1. Let (V t , X t ) t≥t0 be the solution to (SKE). Then there exists a Lévy process L, given in Proposition 6.3 (iii), such that, as ε → 0, in the space D,

Large time behavior of the Lévy driving process

Since the Lévy noise is no longer self-similar, we need to study its large-time behavior. We dedicate this subsection to the study of the convergence in distribution of the rescaled noise (L

for a suitable rate v ε , tending to 0. We introduce another assumption on ν, which will sometimes be imposed in the sequel:

, where g is a non negative measurable function with

Inspired from [START_REF] Rosenbaum | Asymptotic results for time-changed Lévy processes sampled at hitting times[END_REF], we get the following result.

Proposition 6.3. Let (L t ) t≥0 be a Lévy process having generating triplet (A, ν, b), with respect to the truncation function h : z → -1 ∨ (z ∧ 1).

(i) Assume that the Lévy measure satisfies the condition (H ν,α 1 ) with α = 1, c + = c -= c and (H g ). Then the process εL t/ε t≥0 converges in distribution to the strictly 1-stable Lévy process L generated where

Let M be a bound for the sequence (f n ). Thus, by the choice of K,

The third integral can be treated analogously. Besides, since the sequence (f n ) converges uniformly on K to f , there exists n ε such that for all n ≥ n ε , sup K |f n -f | ≤ ε. Thereby, we get

The last integral is smaller than ε for n large enough, since P n converges weakly to P , and this concludes the proof.

Remark A.5. Lemma A.4 could be applied with S = C([0, +∞)) or D([0, +∞)). However, the result for S = C([0, +∞)) is already contained in Problem 4.12 p. 64 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF].

Lemma A.6. Let (Y y t ) t≥0 be the solution of a time-homogeneous SDE driven by an α-stable process,

The measurable function b is supposed to be such that (56) has a pathwise unique strong solution. Then (Y t ) t≥0 is a Markov process. Namely, for any

Proof. We give a proof for d = 2, the general case being similar. Call (Y s,y t ) the solution dY t = dL