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

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

Asymptotic behaviour for a time-inhomogeneous Kolmogorov type diffusion

Keywords: kinetic stochastic equation, time-inhomogeneous diffusions, explosion times, scaling transformations, asymptotic distributions, ergodicity. MSC2010 Subject Classification: Primary 60J60, Secondary 60H10, 60J65, 60F17

We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion (V, X) is considered: here X is the position of the particle and V is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form t -β F (v). The function F satisfies some homogeneity condition and β is positive. The behaviour of the process (V, X) in large time is proved by using stochastic analysis tools.

Introduction

In several domains as fluids dynamics, statistical mechanics, biology, a number of models are based on the Fokker-Planck and Langevin equations driven by Brownian motion or could be non-linear or driven by other random noises. For example, in [START_REF] Cattiaux | Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model[END_REF] the persistent turning walker model was introduced, inspired by the modelling of fish motion. An associated two-component Kolmogorov type diffusion solves a kinetic Fokker-Planck equation based on an Ornstein-Uhlenbeck Gaussian process and the authors studied the large time behaviour of this model by using appropriate tools from stochastic analysis. One of the natural questions is the behaviour in large time of the solution to the corresponding stochastic differential equation (SDE). Although the tools of partial differential equations allow us to ask of this kind of questions, since these models are probabilistic, tools based on stochastic processes could be more natural to use.

In the last decade the asymptotic study of solutions of non-linear Langevin's type was the subject of an important number of papers, see [START_REF] Cattiaux | Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case[END_REF], [START_REF] Eon | Gaussian asymptotics for a non-linear langevin type equation driven by an α-stable lévy noise[END_REF], [START_REF] Fournier | One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes[END_REF]. For instance, in [START_REF] Fournier | One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes[END_REF] the following system is studied

V t = v 0 + B t - ρ 2 t 0 F (V s ) ds and X t = x 0 + t 0 V s ds.
In other words one considers a particle moving such that its velocity is a diffusion with an invariant measure behaving like (1 + |v| 2 ) -ρ/2 , as |v| → +∞. The authors prove that for large time, after a suitable rescaling, the position process behaves as a Brownian motion or other stable processes, following the values of ρ. Results have been extended to additive functional of V in [START_REF] Béthencourt | Stable limit theorems for additive functionals of one-dimensional diffusion processes[END_REF]. It should be noticed that these cited papers use the standard tools associated with time-homogeneous equations: invariant measure, scale function and speed measure. Several of these tools will not be available when the drag force is depending explicitly on time. In [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF], a non-linear SDE driven by a Brownian motion but having time-inhomogeneous drift coefficient was studied and its large time behaviour was described. Moreover, sharp rates of convergence are proved for the 1-dimensional marginal of the solution. In the present paper, we consider the velocity process as satisfying the same kind of SDE.

Let us describe our framework: consider a one-dimensional time-inhomogeneous stochastic kinetic model driven by a Brownian motion. We denote by (X t ) t≥0 the one-dimensional process describing the position of a particle at time t having the velocity V t . The velocity process (V t ) t≥0 is supposed to follow a Brownian dynamic in a potential U (t, v), varying in time:

dV t = dB t - 1 2 ∂ v U (t, V t ) dt and X t = X 0 + t 0 V s ds.
This system can be viewed as a perturbation of the classical two-component Kolmogorov diffusion dV t = dB t and X t = X 0 + t 0 V s ds.

In the present paper the potential is supposed to grow slowly to infinity, and it will be supposed to be of the form t -β v 0 F (u) du, with β > 0 and F satisfying some homogeneity condition. It describes a one dimensional particle evolving in a force field F t -β and undergoing many small random shocks. A natural question is to understand the behaviour of the process (V, X) in large time. More precisely we look for the limit in distribution of v(ε)(V t/ε , εX t/ε ) t , as ε → 0, where v(ε) is some rate of convergence. Our results are proved on the product of path spaces and consequently contain those of [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF]. If F = 0, it is not difficult to see that the rescaled position process (ε 1 /2 V t/ε , ε 3 /2 X t/ε ) t converges in distribution towards the Kolmogorov diffusion (B t , t 0 B s ds) t . We prove that this kinetic behaviour still holds for sufficiently "small at infinity" potential. The strategy to tackle this problem is based on estimates of moments of the velocity process. The main result can then be extended for the case when the potential is equally weighted in some sense as the random noise. The potential either offsets the random noise (critical regime) or swings with it (sub-critical regime). As suggested at the beginning of the introduction, other random noises can be considered. In [START_REF] Gradinaru | Asymptotic behavior for a time-inhomogeneous stochastic differential equation driven by an α-stable Lévy process[END_REF], the case of a Lévy random noise is analysed. The case of a stochastic system in a harmonic potential is the purpose of a future work (see [START_REF] Luirard | Loi Limite de Modèles Cinétiques Inhomogènes En Temps[END_REF]).

The organisation of our paper is as follows: in the next section we introduce notations, and we state our main results. Results about existence and non-explosion of solutions are stated in Section 3. Estimates of the moments of the velocity process are given in Section 4 while the proofs of our main results are presented in Section 5.

Notations and main results

Let (B t ) t≥0 be a standard Brownian motion, β a real number and F a continuous function which is supposed to satisfy either

for some γ ∈ R, ∀v ∈ R, λ > 0, F (λv) = λ γ F (v), (H γ 1 )
or

|F | ≤ G where G is a positive function satisfying (H γ 1 ). (H γ 2 )
Each assumption implies that there exist a positive constant K such that, for all v ∈ R,

|F (v)| ≤ K |v| γ . Obviously (H γ 2
) is a generalisation of (H γ 1 ). In the following, sgn is the sign function with convention sgn(0) = 0. As an example of function satisfying (H γ 1 ) one can keep in mind F : v → sgn(v) |v| γ (see also [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF]), and as an example of function satisfying (H γ 2 ) (with γ = 0) F : v → v /(1+v 2 ) (see also [START_REF] Fournier | One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes[END_REF]).

Remark 2.1. If a function π satisfies (H γ 1 ), then for all x ∈ R, π(x) = π(sgn(x)) |x| γ .

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

dV t = dB t -t -β F (V t ) dt, V t0 = v 0 > 0, and dX t = V t dt, X t0 = x 0 ∈ R. (SKE)
Most of the convergences take place in the space of continuous functions C((0, +∞), R) endowed by the uniform topology

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

(ε) t ) t>0 ) ε>0 of continuous processes, we write (Z (ε) t ) t>0 =⇒ (Z t ) t>0 , if (Z (ε) t ) t>0 converges in distribution to (Z t ) t>0 in C((0, +∞), R), as ε → 0. We write (Z (ε) t ) t>0 f.d.d. =⇒ (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 us state our main results. Set q := β γ + 1 .

Theorem 2.2. Consider γ ≥ 0, and q > 1 2 . Assume that either (H γ 1 ) or (H γ 2 ) is satisfied. Let (V t , X t ) t≥t0 be the solution to (SKE) and (B t ) t≥0 be a standard Brownian motion. Furthermore, if γ ≥ 1, we suppose that for all v ∈ R, vF (v) ≥ 0. Then, as ε → 0,

√ εV t/ε , ε 3 /2 X t/ε t≥εt0 =⇒ B t , t 0 B s ds t≥0 .
Theorem 2.3. Consider γ ≥ 0 and q = 1 2 . Assume that (H γ 1 ) is satisfied. Let (V t , X t ) t≥t0 be the solution to (SKE). If γ ≥ 1, we suppose furthermore that for all v ∈ R, vF (v) ≥ 0. Call H the eternal ergodic process, solution to the homogeneous SDE

dH s = dW s - H s 2 ds -F H s ds,
T (u 1 , • • • , u d ) := ( √ t 1 u 1 , • • • , √ t d u d ), that is (V t ) t≥0 = ( √ t H log(t) ) t≥0 . Then, as ε → 0, √ εV t/ε , ε 3 /2 X t/ε t≥εt0 =⇒ V t , t 0 V s ds t≥0 .
Remark 2.4. The one-dimensional distribution of (V t ) t≥0 has already been explicitly computed (see Theorem 4.1 in [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF]).

Theorem 2.5. Consider γ ≥ 1 and q < 1 2 . Assume that F : v → ρ sgn(v) |v| γ with ρ > 0. Let (V t , X t ) t≥t0 be the solution to (SKE). Call H the ergodic process, solution to the homogeneous SDE dH s = dW s -F (H s ) ds,

where (W t ) t≥0 is a standard Brownian motion. Call Π F its invariant measure. We call (V t ) t≥0 the process whose f.d.d. are T * ⊗ d Π F , the pushforward measure of ⊗ d Π F by the linear map

T (u 1 , • • • , u d ) := (t 1 q u 1 , • • • , t d q u d ). Then, as ε → 0, ε q V t/ε t≥εt0 f.d.d. =⇒ (V t ) t≥0 .
Moreover, in the linear case (i.e. γ = 1) and if β > -1 2 , we define (X t ) t≥0 the centered Gaussian process with covariance function K(s, t)

:= (ρ 2 (1 + 2β)) -1 (s ∧ t) 1+2β . Then, as ε → 0, ε β+ 1 2 X t/ε t≥εt0 f.d.d. =⇒ (X t ) t≥0 . (1)
Remark 2.6. If β = 0, one can prove using the martingale method, that ( √ εX t/ε ) t≥0 converges towards a Brownian motion. Assume, by way of contradiction, that the process (ε q V t/ε ) t≥εt0 would converge (i.e. were tight), then by the continuous mapping theorem, the process (εX t/ε ) t≥0 should converge. This is a contradiction with (1). Here is why we deal only with finite-dimensional convergence for the velocity process.

Changed-of-time processes

In the following, we suppose that γ > -1 and set Ω = C([t 0 , +∞)) the set of continuous functions, that equal +∞ after their (possibly infinite) explosion time. Following the idea used in [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF], we first perform a change of time in (SKE) in order to produce at least one time-homogeneous coefficient in the transformed equation. For every C 2 -diffeomorphism ϕ : [0, t 1 ) → [t 0 , +∞), let introduce the scaling transformation Φ ϕ defined, for ω ∈ Ω, by

Φ ϕ (ω)(s) := ω(ϕ(s)) ϕ ′ (s) , with s ∈ [0, t 1 ).
The result containing the change of time transformation can be found in [GO13], Proposition 2.1, p. 187. Let V be solution to the equation (SKE). Thanks to Lévy's characterization theorem of the Brownian motion,

(W t ) t≥0 := t 0 dB ϕ(s) ϕ ′ (s) t≥0
is a standard Brownian motion. Then, by a change of variable t = ϕ(s), one gets

V ϕ(t) -V ϕ(0) = t 0 ϕ ′ (s) dW s - t 0 F (V ϕ(s) ) ϕ(s) β ϕ ′ (s) ds.
The integration by parts formula yields

d V ϕ(s) ϕ ′ (s) = dW s - ϕ ′ (s) ϕ(s) β F (V ϕ(s) ) ds - ϕ ′′ (s) 2ϕ ′ (s) V ϕ(s) ϕ ′ (s)
ds.

As a consequence, we can state the following result in our context.

Proposition 3.1. If V is a solution to the equation (SKE), then V (ϕ) := Φ ϕ (V ) is a solution to dV (ϕ) s = dW s - ϕ ′ (s) ϕ(s) β F ( ϕ ′ (s)V (ϕ) s ) ds - ϕ ′′ (s) ϕ ′ (s) V (ϕ) s 2 ds, V (ϕ) 0 = V ϕ(0) ϕ ′ (0) , (2)
where

W t := t 0 dB ϕ(s) ϕ ′ (s) . If V (ϕ) is a solution to (2), then Φ -1 ϕ (V (ϕ) ) is a solution to the equation (SKE), where B t -B t0 := t t0 (ϕ ′ • ϕ -1 )(s) dW ϕ -1 (s) .
Furthermore, uniqueness in law, pathwise uniqueness or strong existence hold for the equation (SKE) if and only if they hold for the equation (2).

In the following, we will use two particular changes of time, depending on which term of (2) should become time-homogeneous.

• The exponential change of time: denoting ϕ e : t → t 0 e t , the exponential scaling transformation is defined by Φ e (ω) :

s ∈ R + → ω t0e s √ t 0 e s /2
, for ω ∈ Ω. Thanks to Proposition 3.1, the process V (e) := Φ e (V ) satisfies the equation

dV (e) s = dW s - V (e) s 2 ds -t 1 /2-β 0 e ( 1 /2-β)s F √ t 0 e s /2 V (e) s ds,
where (W t ) t≥0 is a standard Brownian motion.

• The power change of time:

for q = β γ+1 = 1 2 , consider ϕ q ∈ C 2 ([0, t 1 )) the solution to the Cauchy problem ϕ ′ q = ϕ 2q q , ϕ q (0) = t 0 .
Clearly, ϕ q (t) = t 1-2q 0 + (1 -2q)t 1 /(1-2q) , when 2q = 1, and ϕ q = ϕ e , when 2q = 1.

The time t 1 satisfies t 1 = +∞, when 2q ≤ 1, and t 1 = t 1-2q 0 (2q -1) -1 , when 2q > 1.

The power scaling transformation is defined by Φ q (ω) : s ∈ R + → ω(ϕ q (s)) ϕ q (s) q . The process V (q) := V (ϕq) satisfies the equation

dV (q) s = dW s -ϕ -γq q (s)F ϕ ′ q (s)V (q) s ds -qϕ 2q-1 q (s)V (q) s ds, (3)
where (W t ) t≥0 is a standard Brownian motion.

Adapting the proof of Propositions 3.2, 3.6 and 3.7 p. 188, in [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF], one can prove the following proposition.

Proposition 3.2. For γ ≥ 0, there exists a pathwise unique strong solution to (SKE), defined up to the explosion time τ ∞ of V .

• When γ ≤ 1 or for all v ∈ R, vF (v) ≥ 0, then τ ∞ is a.s. infinite. • When 2q > 1, then P(τ ∞ = +∞) > 0. • Under (H γ 1 ), if γ > 1 and (F (-1), F (1)) ∈ ((0, +∞)) × [0, +∞)) ∪ (R × (-∞, 0)), then P(τ ∞ = +∞) < 1.
Remark 3.3. Assume that (H γ 1 ) is satisfied. In the linear case (γ = 1), the drift and the diffusion terms are Lipschitz and satisfy locally linear growth condition. The existence and nonexplosion of V follow from Theorem 2.9, p. 289, in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF].

For more details, we refer to [START_REF] Luirard | Loi Limite de Modèles Cinétiques Inhomogènes En Temps[END_REF].

Moment estimates of the velocity process

In this section, we give estimates for the moment of the velocity process. It will be useful to control some stochastic terms appearing later.

Proposition 4.1. Assume that γ ≥ 0 and β ∈ R. The inequality

∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ 2
holds for

• κ ∈ [0, 1], when γ < 1 and β ≥ γ+1 2 , • κ ≥ 0, when for all v ∈ R, vF (v) ≥ 0. If κ ∈ [0, 1], γ < 1 and β < γ+1 2 , then ∀t ≥ t 0 , E [|V t | κ ] ≤ C γ,κ,β,t0 t κ 1-β 1-γ . Remark 4.2. When -1 < γ < 0, it can be proved that for all t ≥ t 0 , E [|V t |] ≤ C γ,β,t0 √ t, without hypothesis of the positivity of the function v → vF (v). Proof.
Step 1. Assume that γ ≥ 1 and that for all v ∈ R, vF (v) ≥ 0. Define, for all n ≥ 0, the stopping times T n := inf{t ≥ t 0 , |V t | ≥ n}. By Itô's formula, for all t ≥ t 0 , we have

V 2 t∧Tn = v 2 0 + t∧Tn t0 2V s dB s - t∧Tn t0 2s -β V s F (V s ) ds + (t ∧ T n -t 0 ) = v 2 0 + t t0 1 s≤Tn 2V s dB s - t∧Tn t0 2s -β V s F (V s ) ds + (t ∧ T n -t 0 ) ≤ v 2 0 + t t0 1 s≤Tn 2V s dB s + (t -t 0 ). Since t t0 41 s≤Tn V 2 s ds ≤ 4n 2 (t -t 0 ) < +∞, taking expectation yields E V 2 t∧Tn ≤ v 2 0 + (t -t 0 ) ≤ C t0 t. Set κ ∈ [0, 2],
we obtain by Jensen's inequality that

E [|V t | κ ] ≤ E |V t | 2 κ 2 ≤ lim inf n→+∞ E V 2 t∧Tn κ 2 ≤ C κ,t0 t κ 2 . ( 4
)
When κ > 2, the function v → |v| κ is C 2 , so by Itô's formula, we can write for all t ≥ t 0 ,

|V t∧Tn | κ = |v 0 | κ + t∧Tn t0 κ sgn(V s ) |V s | κ-1 dB s - t∧Tn t0 κs -β |V s | κ-1 sgn(V s )F (V s ) ds + t∧Tn t0 κ(κ -1) 2 |V s | κ-2 ds.
In addition, using the hypothesis on the sign of F , we have

|V t∧Tn | κ ≤ |v 0 | κ + t t0 1 s≤Tn κ sgn(V s ) |V s | κ-1 dB s + t∧Tn t0 κ(κ -1) 2 |V s | κ-2 ds. (5)
We observe that

t t0 κ 2 V 2κ-2 s 1 s≤Tn ds ≤ κ 2 n 2κ-2 (t -t 0 ) < +∞.
Taking expectation in (5), we obtain

E [|V t | κ ] ≤ lim inf n→+∞ E [|V t∧Tn | κ ] ≤ |v 0 | κ + t t0 κ(κ -1) 2 E |V s | κ-2 ds.
When 0 ≤ κ -2 ≤ 2, we can upper bound E |V s | κ-2 by injecting (4) and get

E [|V t | κ ] ≤ |v 0 | κ + t t0 κ(κ -1) 2 C κ,t0 s κ-2 2 ds ≤ C κ,t0 s κ 2 .
The same method is then applied inductively to prove the inequality for all κ > 2.

Step 2. Assume now that γ

∈ [0, 1[. 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 the inequality only for κ = 1. Define, for all n ≥ 0, the stopping times T n := inf{t ≥ t 0 , |V t | ≥ n} and let us recall that under both hypotheses (H γ 1 ) or (H γ 2 ), there exists a positive constant K, such that |F (v)| ≤ K |v| γ . We can write, for t ≥ t 0 and n ≥ 0,

|V t∧Tn | ≤ |v 0 -B t0 | + |B t∧Tn | + t∧Tn t0 s -β |F (V s∧Tn )| ds ≤ |v 0 -B t0 | + |B t∧Tn | + t∧Tn t0 s -β K |V s∧Tn | γ ds.
By noting that γ ∈ [0, 1[ and (B 2 t -t) t≥0 is a martingale, taking expectation we get

E [|V t∧Tn |] ≤ E [|v 0 -B t0 |] + E [|B t∧Tn |] + t t0 s -β KE [|V s∧Tn | γ ] ds ≤ E [|v 0 -B t0 |] + E B 2 t∧Tn + t t0 s -β KE [|V s∧Tn |] γ ds ≤ E [|v 0 -B t0 |] + E [t ∧ T n ] + t t0 s -β KE [|V s∧Tn |] γ ds ≤ C t0 √ t + t t0 s -β KE [|V s∧Tn |] γ ds.
The function

g n : t → E [|V t∧Tn |] is bounded by n.
Applying a Gronwall-type lemma, stated below (Lemma 4.3) and Fatou's lemma, for β = 1 and for all t ≥ t 0 , we end up with

E [|V t |] ≤ lim inf n→+∞ E [|V t∧Tn |] ≤ C γ C t0 √ t + 1 -γ 1 -β K(t 1-β -t 1-β 0 ) 1 1-γ ≤ C γ,β,t0 √ t if β ≥ γ+1 2 , t 1-β 1-γ else.
The case β = 1 can be treated similarly.

Lemma 4.3 (Gronwall-type lemma). Fix r ∈ [0, 1) and t 0 ∈ R. Assume that g is a nonnegative real-valued function, b is a positive function and a is a differentiable real-valued function. Moreover, suppose that the function bg r is continuous. If

∀t ≥ t 0 , g(t) ≤ a(t) + t t0 b(s)g(s) r ds, (6)
then,

∀t ≥ t 0 , g(t) ≤ 2 1 1-r a(t) + (1 -r) t t0 b(s) ds 1 1-r . Proof. For t ≥ t 0 , since r ≥ 0, g(t) r ≤ a(t) + t t0 b(s)g(s) r ds r , then, multiplying by b(t) > 0, b(t)g(t) r ≤ b(t) a(t) + t t0 b(s)g(s) r ds r . Now, let us make appear the derivative of H a ′ (t) + b(t)g(t) r ≤ a ′ (t) + b(t) a(t) + t t0 b(s)g(s) r ds r , that is a ′ (t) + b(t)g(t) r a(t) + t t0 b(s)g(s) r ds r ≤ b(t) + a ′ (t) a(t) + t t0 b(s)g(s) r ds r ≤ b(t) + a ′ (t) a(t) r .
Integrating, since r = 1, we obtain

(1 -r) -1 a(t) + t t0 b(s)g(s) r ds 1-r -a(t 0 ) 1-r ≤ (1 -r) -1 a(t) 1-r -a(t 0 ) 1-r + t t0 b(s) ds
or equivalently, setting H for the right-hand side of (6) and using that r < 1, we get

H(t) 1-r ≤ a(t) 1-r + (1 -r) t t0 b(s) ds.
Since 1 1-r > 0 and using ( 6)

g(t) ≤ a(t) 1-r + (1 -r) t t0 b(s) ds 1 1-r ≤ C r a(t) + (1 -r) t t0 b(s) ds 1 1-r .
This concludes the proof of the lemma.

Remark 4.4. Call H(t) the right-hand side of (6). If g is not continuous, note that the function H is continuous and satisfies (6) (since b is positive and g ≤ H). Therefore, one can apply the lemma to H and then use the inequality g ≤ H.

Proof of the asymptotic behaviour of the solution

This section is devoted to the proofs of our main results.

Asymptotic behaviour in the super-critical regime under both assumptions

In this section, we assume that γ ≥ 0 and q > 1 2 . Proof of Theorem 2.2. We split the proof into three steps.

Step 1. We note that it is enough to prove that the process

(V (ε) t ) t≥0 := ( √ εV t/ε ) t≥0
converges in distribution to a Brownian motion in the space of continuous functions C([0, +∞)) endowed by the uniform topology. In order to see V (ε) as a process of C([0, +∞)), let us state for all s ∈ [0, εt 0 ], V

:= V (ε) εt0 = √ εv 0 . (ε) s
For every ε ∈ (0, 1] and t ≥ εt 0 , we can write

ε 3 /2 X t/ε = ε 3 /2 x 0 + t εt0 V (ε) s ds.
Clearly, the theorem will be proved once we show that g ε (V (ε)

• ) := (V (ε)
• ,

• εt0 V (ε) s ds) converges weakly in C([0, +∞)) endowed by the uniform topology. Here the mapping g

ε : v → v t , t εt0 v s ds t≥0
is defined and valued on C((0, +∞)). This mapping is converging, as ε → 0, to the continuous mapping g : v → v t , t 0 v s ds t≥0 .

We have, for every ε ∈ (0, 1] and t ≥ εt 0 ,

V (ε) t = √ εV t/ε = √ ε(v 0 -B t0 ) + √ εB t/ε - √ ε t/ε t0 F (V s )s -β ds = √ ε(v 0 -B t0 ) + B (ε) t -ε β-1/2 t εt0 F (V u/ε )u -β du.
By self-similarity, B (ε) := ( √ εB t/ε ) t≥0 has the same distribution as a standard Brownian motion.

Assume that the convergence of the rescaled velocity process is proved in the strong way, that is

∀T ≥ t 0 , sup εt0≤t≤T V (ε) t -B (ε) t P -→ 0, as ε → 0. (7)
Then it suffices to prove that g ε (B (ε) ) =⇒ g(B) and d u g ε (V (ε) ), g ε (B (ε) ) P -→ 0, as ε → 0 (see Theorem 3.1, p. 27, in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]). On the one hand, the process B (ε) being a Brownian motion and |•| R 2 denoting a norm on R 2 , the first convergence follows from

∀T ≥ t 0 , sup εt0≤t≤T |g ε (B t ) -g(B t )| R 2 P -→ 0, as ε → 0. ( 8
)
Let us prove (8). For every εt 0 ≤ t ≤ T , we get

|g ε (B t ) -g(B t )| R 2 = εt0 0 B s ds ≤ εt0 0 |B s | ds.
Hence,

E sup εt0≤t≤T |g ε (B t ) -g(B t )| R 2 ≤ εt0 0 E |B s | ds ≤ C εt0 0 √ s ds -→ ε→0 0.
On the other hand, we prove that

∀T ≥ t 0 , sup εt0≤t≤T g ε (V (ε) t ) -g ε (B (ε) t ) R 2 P -→ 0, as ε → 0. (9)
For every εt 0 ≤ t ≤ T , using (7)

g ε (V (ε) t ) -g ε (B (ε) t ) R 2 = V (ε) t -B (ε) t + t εt0 V (ε) s -B (ε) s ds ≤ (1 + T -εt 0 ) sup εt0≤t≤T V (ε) t -B (ε) t P -→ 0.
Step 2. Let us prove now (7). Recall that under both hypothesis (H γ 1 ) and (H γ 2 ), there exists

a positive constant K, such that ( √ ε) γ F V (ε) u √ ε ≤ K V (ε) u γ
. Modifying the factor in front of the integral part, we get

V (ε) t = √ ε(v 0 -B t0 ) + √ εB t/ε -ε β-(γ+1) /2 t εt0 ( √ ε) γ F V (ε) u √ ε u -β du.
It follows that, for all t 0 ≤ T ,

sup εt0≤t≤T V (ε) t -B (ε) t ≤ √ ε |v 0 -B t0 | + ε β-(γ+1) /2 sup εt0≤t≤T t εt0 ( √ ε) γ F V (ε) u √ ε u -β du ≤ √ ε |v 0 -B t0 | + ε β-(γ+1) /2 T εt0 K V (ε) u γ u -β du.
Taking the expectation and using moment estimates (Proposition 4.1), we obtain, when β = γ 2 +1 and since β > γ+1 2 ,

ε β-(γ+1) /2 E T εt0 K V (ε) u γ u -β du = ε β-(γ+1) /2 T εt0 KE V (ε) u γ u -β du ≤ ε β-(γ+1) /2 T εt0 KC γ,β,t0 u γ 2 -β du ≤ C ε β-(γ+1) /2 T γ 2 -β+1 -t γ 2 -β+1 0 √ ε -→ ε→0 0.
Hence, setting r = min( 1 2 , β -(γ+1) /2) > 0

E sup εt0≤t≤T V (ε) t -B (ε) t = O(ε r ).
The case β = γ 2 + 1 can be treated similarly to get

E sup εt0≤t≤T V (ε) t -B (ε) t = O( √ ε ln(ε)).
This concludes the proof.

Remark 5.1. One can observe that the only moment in this proof, when we need the condition "γ < 1 or for all v ∈ R, vF (v)" is when we are proving the moment estimates.

Asymptotic behaviour in the critical regime under (H γ 1 )

Assume in this section that β = γ+1 2 and (H γ 1 ) is satisfied.

Proof of Theorem 2.3.

Step 1. As in the first step of the previous section, it suffices to prove the convergence of the rescaled velocity process ( √ εV t/ε ) t . Keeping same notations, we prove that g ε (V (ε) ) converges in distribution in C([0, +∞)) to g(V). In order to see ε) , V respectively. Then, using Pormanteau theorem (see Theorem 2.1 p.16 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]), it suffices to prove that for all function h : C([0, +∞)) × C([0, +∞)) → R bounded and uniformly continuous,

V (ε) as a process of C([0, +∞)), let us set for all s ∈ [0, εt 0 ], V (ε) s := V (ε) εt0 = √ εv 0 . Call P ε , P the distribution of V (
C([0,+∞)) 2 h(g ε (ω)) dP ε (dω) -→ ε→0 C([0,+∞)) 2 h(g(ω)) dP (dω).
Take a bounded and uniformly continuous function h. By assumption, one knows that P ε =⇒ P , hence, by Problem 4.12 p. 64, in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF], it suffices to prove that the uniformly bounded sequence (h • g ε ) of continuous functions on C([0, +∞)) converges uniformly on compact subsets of C([0, +∞)) to the continuous function h • g. Let K be a compact set of C([0, +∞)). Then, for all ω ∈ K, max [0,εt0] |ω| is uniformly bounded by a constant, called 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.
However, 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.
Step 2. We first prove the f.d.d. convergence. The exponential scaling process V (e) satisfies the time-homogeneous equation

dV (e) s = dW s - V (e) s 2 ds -F V (e) s ds, (10)
where (W t ) t≥0 is a standard Brownian motion. Using the bijection induced by the exponential change of time (Proposition 3.1), we get

V t0e t √ t 0 e t/2 t≥0 = (H t ) t≥0 ,
as solutions of the same SDE, starting at the same point. This can also be written as

V t √ t t≥t0 = (H log(t/t0) ) t≥t0 .
So, we have, for all ε > 0, and

(t 1 , • • • , t d ) ∈ [εt 0 , +∞) d , V ε -1 t1 √ ε -1 t 1 , • • • , V ε -1 t d √ ε -1 t d = H log(t1)+log((εt0) -1 ) , • • • , H log(t d )+log((εt0) -1 ) . ( 11
)
As in [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF], the scale function and the speed measure of H are respectively

p(x) := x 0 exp y 2 2 + 2 γ + 1 sgn(y)F (sgn(y)) |y| γ+1 dy and ν F (dx) := exp - x 2 2 - 2 γ + 1 sgn(x)F (sgn(x)) |x| γ+1 dx.
By the ergodic theorem (Theorem 23.15 p. 465 in [START_REF] Kallenberg | Foundations of Modern Probability. Probability and Its Applications[END_REF]), H is Λ F -ergodic, where Λ F is the probability measure associated to ν F . Call H the solution of the time homogeneous equation (10) such that the initial condition

H -∞ has the distribution Λ F . For t 1 , • • • , t d ∈ [εt 0 , +∞) 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 t ) t∈R and ( H t+s ) t∈R satisfy the same SDE, starting at the same distribution. As a consequence, for all ε > 0,

L H log(t1)+log((εt0) -1 ) , • • • , H log(t d )+log((εt0) -1 ) = Λ F,log(t1),••• ,log(t d ) . (12)
Moreover, by exponential ergodicity, for every ψ : R d → R continuous and bounded function, we can prove that

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. (13)
We postpone the proof of this convergence in Step 3.

To conclude this step, gather (11), ( 12) and (13) to get

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

√ εV t1/ε , • • • , √ εV 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 u 1 , • • • , √ t d u d ).
Step 3. Let us now prove (13). Pick εt 0 ≤ s ≤ t. Set h 0 = v 0 √ t 0 -1 . Actually we prove a more general result, which will also be useful in the last regime. The convergence (13) will be a direct consequence of this lemma.

Lemma 5.2. Let H be an exponential ergodic process with invariant measure ν, solution to a SDE driven by a Brownian motion. Pick a continuous function φ : [t 0 , +∞) → R satisfying lim s→+∞ φ(s) = +∞.

Then, for all integer d ≥ 1, every continuous and bounded function ψ :

R d → R, all h 0 ∈ R and all t 1 , • • • , t d ∈ [εt 0 , +∞) d , E ψ H φ(ε -1 t1) , • • • , H φ(ε -1 t d ) H 0 = h 0 -E ψ H φ(ε -1 t1) , • • • , H φ(ε -1 t d ) H 0 ∼ ν -→ ε→0 0.
Proof. For the 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.

We set µ ε := L H φ(ε -1 s) H 0 = h 0 . We use the generalized Markov property of solution to SDE driven by Brownian motion process (see Theorem 21.11 p. 421 in [START_REF] Kallenberg | Foundations of Modern Probability. Probability and Its Applications[END_REF]). This leads to

E ψ H φ(ε -1 s) , H φ(ε -1 t) H 0 = h 0 = E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 ∼ µ ε and, since Λ F is invariant, E ψ H φ(ε -1 s) , H φ(ε -1 t) H 0 ∼ ν = E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 ∼ ν .
Then, we are reduced to prove

E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 ∼ µ ε -E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 ∼ ν -→ ε→0 0.
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 φ(ε -1 t)-φ(ε -1 s) H 0 ∼ µ ε -E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 ∼ ν ≤ R E ψ H 0 , H φ(ε -1 t)-φ(ε -1 s) H 0 = y (µ ε (dy) -ν(dy)) ≤ ψ ∞ R p φ(ε -1 s), h 0 , dy -ν(dy) ≤ ψ ∞ p φ(ε -1 s), h 0 , • -ν T V .
We let ε → 0, using the exponential ergodicity of H.

Step 4. Let us prove now the tightness of the family of laws of continuous process V (ε) t≥εt0 = √ εV t/ε t≥εt0 on every compact interval [m, M ], 0 < m ≤ M . We prove the Kolmogorov criterion stated in Problem 4.11 p. 64, in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]. Take ε 0 small enough such that for all ε ≤ ε 0 , εt 0 ≤ m. Fix m ≤ s ≤ t ≤ M and α > 2. Recalling that B (ε) is a Brownian motion, using Jensen's inequality, moment estimates (Proposition 4.1) and the relation β = γ+1 2 , we can write

E V (ε) t -V (ε) s α ≤ C α E B (ε) t -B ε s α + C α E √ ε t/ε s/ε F (V u )u -β du α ≤ C α E [|B t -B s | α ] + C α ε 1-α 2 (t -s) α-1 E t/ε s/ε |F (V u )| α u -βα du ≤ C α E [|B t-s | α ] + C α ε 1-α 2 (t -s) α-1 t/ε s/ε u γα 2 -βα du ≤ C α (t -s) α 2 + C α ε 1-α 2 (t -s) α-1 t/ε s/ε u -α 2 du ≤ C α (t -s) α 2 + C α (t -s) α-1 (t 1-α 2 -s 1-α 2 ) ≤ C α (t -s) α 2 + C α,m,M (t -s) α-1 ≤ C α,m,M (t -s) α 2 .
Since α > 2, then α 2 > 1 and the upper bound does not depend on ε. Furthermore, by moment estimates (Proposition 4.1),

sup ε≤ε0 E V (ε) m ≤ √ m < +∞.

Conclusion.

The previous steps yields weak convergence on every compact set (Theorem 13.1 p. 139, in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]). The conclusion follows from Theorem 16.7 p. 174, in [START_REF] Billingsley | Convergence of Probability Measures[END_REF], since all processes considered are continuous.

Example 5.1. We will see that the limiting process V is more explicit in the linear case (γ = 1).

Choose F (1) = 1, F (-1) = -1, the process H solution of (10) is in fact an Ornstein Uhlenbeck process with invariant measure Λ F (dx) := e -3x 2 2 dx. It is a centered Gaussian process, hence for all s 1 , • • • , s d , its f.d.d. Λ F,s1,••• ,s d are Gaussian. As a consequence, knowing the covariance function K is enough to provide the law of the process. Since H is a stationary Ornstein-Uhlenbeck process, one has K : s, t → 1 3 e -3 2 |t-s| . Hence, the limiting process V having f.d.d T * Λ F,log(t1),••• ,log(t d ) is a centered Gaussian process with covariance function s, t → 1 3 (s∧t) 2 s∨t .

Asymptotic behaviour in the subcritical regime under (H γ 1 )

Assume in this section that β < γ+1 2 and F : v → ρ sgn(v) |v| γ with γ ≥ 1. For simplicity, we shall write ϕ instead of ϕ q .

Proof of Theorem 2.5. Step 1. We first prove the f.d.d. convergence of the velocity process (V (ε) t ) t≥εt0 := (ε q V t/ε ) t≥εt0 . Again we give a proof only for d = 2, since the general case d ≥ 2 is similar. The power scaling process V (q) , solution to (3) satisfies

dV (q) s = dW s -F V (q) s ds -qϕ 2q-1 (s)V (q) s ds.
We call H the ergodic process solution to the SDE

dH s = dW s -F H s ds, with H 0 = h 0 := v 0 t -q 0 . (14)
We denote by Π F (dx) := e -2ρ γ+1 |x| γ+1 dx its invariant measure. Using the bijection induced by the power change of time (Proposition 3.1), 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) R 2 -→ ε→0 0. • 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 2q-1 F (H (ϕ,ε) t ) -F (V (ϕ,ε) t ) t -2q dt + qt -1 V (ϕ,ε) t dt.
Pick δ > 0. By straightforward differentiation, we can write

d H (ϕ,ε) t -V (ϕ,ε) t 2 ≤ - 2ε 2q-1 t 2q F (H (ϕ,ε) t ) -F (V (ϕ,ε) t ) H (ϕ,ε) t -V (ϕ,ε) t 1 H (ϕ,ε) t -V (ϕ,ε) t >δ dt + 2t -1 qV (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t 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 2 ≤ - 2ε 2q-1 t 2q C γ δ γ-1 H (ϕ,ε) t -V (ϕ,ε) t 2 1 H (ϕ,ε) t -V (ϕ,ε) t >δ dt + 2t -1 qV (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t dt. (15) We set g ε (t) = E H (ϕ,ε) t -V (ϕ,ε) t 2 and g ε (t) = E H (ϕ,ε) t -V (ϕ,ε) t 2 1 H (ϕ,ε) t -V (ϕ,ε) t
>δ . Taking expectation in (15), we get

g ′ ε (t) ≤ - 2ε 2q-1 t 2q C γ δ γ-1 g ε (t) + b ε (t), with g ε (εt 0 ) = 0 (16) where b ε (t) := 2t -1 qE V (ϕ,ε) t H (ϕ,ε) t -V (ϕ,ε) t .
Using Cauchy-Schwarz inequality and moment estimates (Proposition 4.1), we have

|b ε (t)| ≤ 2t -1 |q| E V (ϕ,ε) t 2 g ε (t) ≤ C2t -1 |q| ε 2q-1 t 1-2q g ε (t). Set h(t) := 1 1 -2q C γ δ γ-1 t 1-2q
.We use the comparison theorem for ordinary differential equation on (16) to get

g ε (t) ≤ t εt0 b ε (s) exp(-2ε 2q-1 (h(t) -h(s))) ds.
As a consequence, we deduce that

g ε (t) ≤ δ 2 + g ε (t) ≤ δ 2 + exp(-2ε 2q-1 h(t))C t εt0 2s -1 √ ε 2q-1 s 1-2q g ε (s) exp(2ε 2q-1 h(s)) exp(ε 2q-1 h(s)) ds.
Applying a Gronwall-type lemma (Lemma 4.3) to the function g ε exp(2ε 2q-1 h), we obtain

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

ε→0 g ε (t) ≤ δ 2 . ( 17
)
To prove the domination hypothesis, notice that by optimization of the function x → √ x exp(-Ax),

1 εt0≤s≤t s -1 √ ε 2q-1 s 1-2q exp(-ε 2q-1 (h(t) -h(s))) ≤ 1 0≤s≤t s -1 2 -q 1 h(t) -h(s) .
This function is integrable, since 1 -2q > 0.

We let δ → 0 in (17) to conclude that for all t > 0, lim ε→0 g ε (t) = 0.

Step 3. Pick s, t ∈ [εt 0 , +∞) 2 . We prove that the solution H to ( 14) satisfies

H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) =⇒ ε→0 Π F ⊗ Π F . (18)
Observe that ϕ -1 (ε -1 t) -ϕ -1 (ε -1 s) = t 1-2q -s 1-2q ε 1-2q -→ ε→0 0.

By Lemma 5.2, for every continuous and bounded function ϕ, we can write

E ψ H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) H 0 = h 0 -E ψ H ϕ -1 (ε -1 s) , H ϕ -1 (ε -1 t) H 0 ∼ Π F -→ ε→0 0.
Hence, it suffices to prove that for every bounded continuous functions f, g : R → R, the following convergence holds

lim ε→0 E f H ϕ -1 (ε -1 s) g H ϕ -1 (ε -1 t) H 0 ∼ Π F = Π F (f )Π F (g).
The following reasoning is inspired from the proof of Lemma 3.2 p. 7-8 in [START_REF] Cattiaux | Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model[END_REF]. Since H 0 is starting from the invariant measure, up to considering f -Π F (f ) and g -Π F (g), we can assume that f and g have zero Π F -mean. We call (P t ) t≥0 the semigroup of H, then we get, by invariance property of Π F , E f H ϕ -1 (ε -1 s) g H ϕ -1 (ε -1 t) H 0 ∼ Π F = P ϕ -1 (ε -1 s) f P ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s) g dΠ F = f P ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s) g dΠ F .

Note that U : v → |v| 1+γ 1+γ is a convex function, thus a λ-Poincaré inequality holds for the process H (see [START_REF] Bobkov | Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures[END_REF] p. 1904). This implies the exponential decay of the variance (see Theorem 4.2.5 p. 183, in [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]), i.e. there exists a constant C > 0 such that, since Π F is a probability measure, f P ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s) g dΠ F ≤ f P ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s) g 2 ≤ f ∞ P ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s) g 2 ≤ C f ∞ g ∞ e -λ(ϕ -1 (ε -1 t)-ϕ -1 (ε -1 s)) .

We deduce (18) from (19).

Step 4. We prove the f.d.d. convergence of the position process (X (ε) t ) t≥εt0 := (ε β+ 1 2 X t/ε ) t≥εt0 . Take γ = 1 and β ∈ (-1 2 , 1). Pick t ≥ εt 0 . By Itô's formula applied to t β V t , we get ρX

(ε) t = ε β+ 1 2 (t β 0 v 0 + x 0 ) -ε 1-β 2 t β V (ε) t + ε β+ 1 2 t/ε t0 s β dB s + ε β+ 1 2 t/ε t0 βs β-1 V s ds.
Since β > -1 2 , the first term converges to 0 in probability as ε → 0. Moreover, by Itô's formula, for all t ≥ t 0 ,

d dt E V 2 t = -2ρs -β E V 2 s + 1.
Hence, by comparison theorem for ordinary differential equation,

E V 2 t ≤ exp(-2ρ t 1-β 1 -β ) v 2 0 + t t0 exp(2ρ s 1-β 1 -β ) ds .
Using an integration by parts, we deduce that there exists a positive constant C such that, for all t ≥ t 0 , E V 2 t ≤ Ct β .

As a consequence, we obtain Since the convergence of centered Gaussian processes is characterized by the convergence of their covariance functions, the conclusion follows from Theorem 3.1, p. 27, in [START_REF] Billingsley | Convergence of Probability Measures[END_REF].

E -ε 1-β 2 t β V (ε) t + ε β+ 1 2 t/ε t0 βs β-1 V s ds ≤ ε 1-β 2 t β E V (ε) t + ε

  s β dB s . By Itô's isometry and since β > -1 2 , for all εt 0 ≤ s ≤ t, we can write

								β+ 1 2	t/ε t0	βs β-1 E [|V s |] ds
		≤ Cε	1 2 t	3β 2 + Cε	1-β 2 t	3β 2 -Cε β+ 1 2 t	3β 2 0 -→ ε→0	0.
	It remains to study the centered Gaussian process M	(ε) t	:= ε β+ 1 2	t/ε
	Cov(M (ε) s , M	(ε) t ) = ε 2β+1	s/ε t0		u 2β ds ∼ ε→0	s 1+2β 1 + 2β	.

t0

Acknowledgements The authors would like to thank Jürgen Angst for valuable exchanges and suggestions about this work and Thomas Cavallazzi for his careful reading of the manuscript. We would also like to thank the anonymous referees for her/his careful reading of the manuscript and useful advices.


domain: sdv sde

text:
Tarek Hattab
email: tarek.hattab@ifremer.fr

Ali Gucu

Ana Ventero

Andrea De Felice

Athanassios Machias

Claire Saraux

Denis Gašparević

Gualtiero Basilone

Ilaria Costantini

Iole Leonori

Jean-Hervé Bourdeix

Magdalena Iglesias

Marco Barra

Marianna Giannoulaki

Rosalia Ferreri

Salah El Ayoubi

Sara Malavolti

Simona Genovese

Stelios Somarakis

Tea Juretić

Vjekoslav Tičina

Grégoire Certain

Temperature strongly correlates with regional patterns of body size variation in Mediterranean small pelagic fish species

Keywords: small pelagic fish, body size, Bergmann's rule, temperature-size rule, Mediterranean Sea

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Species traits often vary substantially across geographical gradients [START_REF] Gaston | Ecogeographical rules: elements of a synthesis[END_REF]). Among them, body size is a key functional trait that influences the fitness of organisms [START_REF] Brown | Toward a metabolic theory of ecology[END_REF]) and is of central importance because it relates directly to several ecological and physiological processes, such as respiration, growth, maturation, reproduction and longevity [START_REF] Blueweiss | Relationships between body size and some life history parameters[END_REF][START_REF] Calder | Size, function, and life history[END_REF].

Geographical gradients of body size are evident across multiple faunal taxa and have been observed in both terrestrial and aquatic environments (Walters and Hassall 2006;[START_REF] Blanck | Large-scale intraspecific variation in life-history traits of European freshwater fish[END_REF]Fisher et al. 2010b;Fisher et al. 2010a). Such gradients are thought to rely on a relation between size and temperature. Bergman's rule (Bergmann 1848) states that larger individuals should be favored in cold environment through decreased heat perdition (reduced surface to volume ratio), resulting in a macroecological gradient of increasing size at higher latitudes. While this was originally developed for endotherms, it has quickly expanded to all organisms [START_REF] Blanckenhorn | Bergmann and converse Bergmann latitudinal clines in arthropods: two ends of a continuum?[END_REF]. This pattern was particularly prevalent in the strongly body size-structured aquatic communities, such as pelagic fishes and zooplanktons [START_REF] Blueweiss | Relationships between body size and some life history parameters[END_REF][START_REF] Angilletta | Temperature, growth rate, and body size in ectotherms: fitting pieces of a life-history puzzle[END_REF][START_REF] Blanchard | How does abundance scale with body size in coupled size-structured food webs?[END_REF][START_REF] Ohlberger | Climate warming and ectotherm body size-from individual physiology to community ecology[END_REF][START_REF] Evans | Temperature-mediated changes in zooplankton body size: large scale temporal and spatial analysis[END_REF]. Furthermore, according to temperature-size rule, ectotherms should achieve smaller size at maturity at higher temperature through phenotypic plasticity [START_REF] Atkinson | Temperature and organism size: a biological law for ectotherms?[END_REF][START_REF] Blackburn | Geographic gradients in body size: a clarification of Bergmann's rule[END_REF]. Laboratory experiments and meta-analyses have revealed that across a broad range of taxonomic groups (from bacteria to vertebrates), body sizes of aquatic ectotherms strongly co-vary with temperature [START_REF] Atkinson | Temperature and organism size: a biological law for ectotherms?[END_REF][START_REF] Angilletta | Temperature, growth rate, and body size in ectotherms: fitting pieces of a life-history puzzle[END_REF][START_REF] Forster | Warming-induced reductions in body size are greater in aquatic than terrestrial species[END_REF][START_REF] Hoefnagel | Is the temperature-size rule mediated by oxygen in aquatic ectotherms?[END_REF][START_REF] Horne | Temperature-size responses match latitudinal-size clines in arthropods, revealing critical differences between aquatic and terrestrial species[END_REF] and decline by about 3% per 1°C. The general trend for animal sizes to increase with cold temperature or high latitude, has been extensively studied from a biogeographic perspective and has been validated to marine fishes such as blennies (Gilligan 1991), syngnathids (Wilson 2009) and mesopelagic fishes [START_REF] Saunders | Southern Ocean mesopelagic fish comply with Bergmann's rule[END_REF] at broad taxonomic levels and large spatial scales (Fisher et al. 2010b).

Recently, [START_REF] Audzijonyte | Fish body sizes change with temperature but not all species shrink with warming[END_REF] analyzed the body sizes of 335 reef fish species using a multidecadal dataset spanning the whole Australian continent and multiple decades confirming that temperature is a major determinant of reef fish body sizes in the wild. Additionally, this study shows that the direction of a species' response to warming through space was generally consistent with its response to warming through time at any given location, suggesting that spatial trends could help forecast fish responses to long-term warming. However, many other studies considering adult body sizes in other marine fish species have found little evidence of Bergmann's rule (Macpherson and overfishing through size-selective harvesting that eliminates the faster-growing genotypes [START_REF] Conover | Sustaining fisheries yields over evolutionary time scales[END_REF] and climate change homogenizing latitudinal patterns (Fisher et al. 2010a).

In this study, we consider the applicability of Bergmann's rule to the populations of 10 small pelagic fish species in the Mediterranean Sea. These species represent key components of Mediterranean marine ecosystems and fisheries and they have recently experienced a lasting decrease in individual size and weight in several regions (e.g. anchovy Engraulis encrasicolus and sardine Sardina pilchardus see [START_REF] Brosset | Spatio-temporal patterns and environmental controls of small pelagic fish body condition from contrasted Mediterranean areas[END_REF]), making the study of geographical and temporal patterns of variation of their body size an important study topic to unravel the influences of fisheries and climate change. Indeed, the adherence of small pelagic fish to Bergmann's rule enables such species to act as sentinels for identifying signs and consequences of warming for the Mediterranean ecosystem. Here we employ empirical data on pelagic fish assemblages from a large number of locations to investigate the intraspecific body size structure over a regional scale. Specifically, we aim to: 1) establish whether latitudinal body size clines can be observed on 10 pelagic fish species across the Mediterranean sea; 2) quantitatively examine size-temperature trends in space and their intensities (i.e. slope estimates) while accounting for potential confounding variables (i.e. extraneous determinant such us bathymetric differences, which could influence both dependent and independent variable) and sampling bias (i.e. differences in sample sizes and timing of surveys).

Material and methods

Biological sampling

Small pelagic fish were collected during acoustic surveys carried out on continental shelves of the Mediterranean Sea in the June-September period. The surveys were conducted along acoustic transects over the continental shelf from 15/20 m to 200 m (the border of the continental shelf, where the abundance of most small pelagic fish species drops critically). These surveys enabled us to monitor fish body size in the following geographical subareas (GSA, as established by GFCM 2009): the northern (GSA 01) and southern (GSA 03) Alboran Sea, the northern Spain (GSA 06), the Gulf of Lions (GSA 07), the Ligurian and Tyrrhenian Seas (GSA 9-10), southern Sicily and Malta (GSA , the northern (GSA 17) and southern (GSA 18) Adriatic Sea, the western Ionian Sea (GSA 19), the eastern Ionian Sea (GSA 20), the Aegean Sea (GSA 22) and the south of Turkey (GSA 24). Length-frequency distributions (in terms of abundance and mass) of all species in the pelagic community were measured to 0.5 cm total size classes on board the research vessels. Biological sampling was made using pelagic trawl gears suitable to catch representative samples of the fish population in terms of species composition and size distribution. Trawl hauls were conducted during daytime in all GSAs except for the western GSA 17, GSA 18, 20 and 22 were daytime/ night time sampling was carried out (i.e. as it has been shown that there is no significant differences between day and night sampling [START_REF] Machias | Catch of pelagic hauls in Mediterranean acoustic surveys: Is it the same between day and night?[END_REF]). Still, trawl nets were not exactly the same between areas and surveys, e.g. some used a covered codend and accessed smaller fish than others. To obtain comparable size distributions among areas, we consequently set a common minimal total length of 72 mm for all areas (i.e. defined as the maximum value of the minimum sizes caught in each survey) above which we kept data.

In the present analysis we selected only the most abundant pelagic species in catches (those representing 98% of total catches). Thus, 10 species were selected, namely anchovy (Engraulis encrasicolus), sardine (Sardina pilchardus), Atlantic chub mackerel (Scomber colias), sprat (Sprattus sprattus), Atlantic horse mackerel (Trachurus trachurus), round sardinella (Sardinella aurita), Mediterranean horse mackerel (Trachurus mediterraneus), Atlantic mackerel (Scomber scombrus), bogue (Boops boops), and blue jack mackerel (Trachurus picturatus).

A georeferenced dataset was compiled and included 1721 pelagic trawl hauls collected in 7 annual surveys carried out between 2012 and 2018 where catch weights and mean body size of all fish species were recorded (Fig. 1).

Temperature data

Temperature data were extracted from the Mediterranean Sea physical reanalysis system [START_REF] Simoncelli | Mediterranean Sea Physical Reanalysis (CMEMS MED-Physics)[Data Set[END_REF]) which includes 3D monthly fields at 1/16° horizontal resolution (which for the Mediterranean basin ranges from 5 km at 45°N to 6 km at 30°N) and 72 unevenly spaced vertical layers. These reanalyzes include a variational data assimilation scheme of in-situ and satellite observations [START_REF] Simoncelli | Mediterranean Sea Physical Reanalysis (CMEMS MED-Physics)[Data Set[END_REF]). These data were averaged vertically by calculating mean temperature value in each grid cell by considering only vertical layers located in the upper 200 meters (i.e. the vertical layers covered by survey data) (Fig. 1). A spatio-temporal match-up was subsequently performed between monthly fields of temperature and pelagic trawl hauls based on geographic coordinates of hauls and their corresponding year and month (Fig. 2).

Statistical analyses

Bergmann's rule is traditionally evaluated by linearly correlating body size with temperature and latitude. Significant negative correlations with temperature or positive correlations with latitude can be interpreted as evidence for Bergmann's rule. Given that body size dataset was compiled from different surveys, there were some limitations that needed to be considered in the analyses. First, even though all surveys were carried out in the same season, they spread over 4 months, between June and September, a variability that might create a source of bias (i.e. a four-month period may result in important ontogenetic variations in body sizes of these fast growing and short-lived species). Second, the nature of acoustic surveys prevented from a balanced sampling design. Indeed, hauls number per survey varied across regions and years as this largely depended on fish spatial distribution and abundance.

Consequently, some temperature values had a lower or higher sampling probability than others, which could potentially lead to distortions of the outcomes since the considered sample will not be equally representative of all populations' body sizes at all environmental conditions. Finally, the size-specific bathymetric patterns of distribution, frequently observed in pelagic fish, may cause spurious associations in temperature-size relationships because both dependent and independent variables are confounded by depth (i.e. given the strong correlation between the mean temperature of the water column and the bathymetry on one side (Fig. S1), and the known ontogenetic differences in the bathymetric distribution between different ages and sizes in pelagic species on the other side). The presence of confounding variables may affect the variables under study so that the results might not reflect the actual relationships.

These three limitations could not be tackled simultaneously with a single linear mixed model because the structure of the random effects for such a model was too complex to be supported by the data and produced singular fits which is often an indication of over-fitting. We then addressed these limitations in separate steps using more parsimonious models. First, to account for the variability in the sampling period, linear mixed models were fitted using restricted maximum likelihood estimation, by considering mean body size in trawls hauls as a response variable, temperature and latitude as fixed effect predictor variables, and season (i.e. month) as a random effect variable which includes a random intercept and a random slope of temperature. These mixed effects models were fitted using the lmer function in lme4 R package [START_REF] Bates | Fitting linear mixed models in R[END_REF]. By comparing the residual variance and the variance associated with the random effect, we identified species for which survey related variability (i.e. month) was not significant, and those were retained for further analyses.

Thereafter, in order to verify whether the observed patterns were due to spatial distribution along bathymetric gradient, a second set of linear models were fitted including temperature and depth as predictor variables to control for bathymetric variation that may confound the temperature-size relationships. In these models predictor variables were scaled in order to make the coefficient of each variable indicative of its influence in the regression equation. In addition, collinearity among the predictor variables was checked using the variance inflation factor (VIF), a measure of severity of multicollinearity. VIF values were calculated using the car R package [START_REF] Fox | An R Companion to Applied Regression Third[END_REF].

Finally, to cope with the unbalanced sampling, we implemented as a third step a bootstrap sampling procedure among trawl hauls from over-sampled temperature conditions. The temperature range of the dataset was split into 13 equal bins of 1°C. For each bin we selected a random subsample of trawl hauls so as to get an equal number of observations in all temperature bins. The number of subsamples was set as the minimum number of trawl hauls observed by temperature bins. Resampling was repeated 999 times, where at each run a linear model was fitted by considering mean size as the response variable and temperature and depth as the predictor variable.

Results

Biogeographic patterns of body size in 10 small pelagic species

Among the 10 considered species, only bogue showed positive correlation with latitude. Five other species showed an opposite trend, where a significant pattern of decreasing size with latitude was observed for anchovy, sardine, Atlantic chub mackerel, Atlantic mackerel and blue jack mackerel (Table 1).

Size-temperature relationships in 10 small pelagic species

Linear mixed models results indicate that survey related variability did not allow identifying clear spatial patterns of body size variation for sprat, Atlantic horse mackerel and Atlantic mackerel for which no reliable conclusion could be drawn. For example, in the case of sprat, temperature negatively affected body size, lowering it by 0.4 mm. However, the residual variance (i.e., variance outside our measured parameters) was considerably low (132 mm²) in comparison with the random effect of seasonality which accounted for a variance of 446 mm² (Table 1, Fig. 3). Thus only the 7 remaining species were selected for further analysis. Among the latter, only 5 species (anchovy, sardine, Atlantic chub mackerel, bogue and blue jack mackerel) showed significant negative effect of temperature on mean body size with a low random effect of seasonality compared with other possible factors (Table 1, Fig. 3). For round sardinella and Mediterranean horse mackerel, the residual variance was higher compared to the random effect of seasonality, however no significant negative effect of temperature was observed, suggesting that these two species do not follow a temperature-size rule.

Among the 5 species showing a negative correlation with temperature, 3 species (anchovy, sardine and bogue) showed bathymetric patterns of distribution with a significant increase in body size with depth (Table 2, Fig. 4) and 1 species (Atlantic chub mackerel) showed a negative correlation between body size and depth. However the negative effect of temperature remained significant for all 5 species when controlling for depth in the model (Table 2), suggesting that the temperature-size rule observed was not an artifact resulting solely from their bathymetric distribution. Indeed, we observed a decrease in size with an increase in temperature at each given depth range (Fig. 5). Even more, the effect size of temperature was stronger than that of bathymetry in all species but bogue (Fig. 4 andTable 2).

Finally, the bootstrap-sampling procedure of trawl hauls within over-sampled temperature conditions showed that temperature-body size relationships were relatively stable against the sampling size.

Ninety-nine (%) of the slope values were still negative for sardine, Atlantic chub mackerel, bogue and blue jack mackerel while 98 % of the slope values remained negative for anchovy (Fig. 6). The bootstratp results confirmed that body size was affected by temperature, lowering it on average by -2.07 (± 0.09), -1.62 (± 0.03), -8.26 (± 0.24), -2.74 (± 0.04) and -6.80 (± 0.25) mm.°C -1 for anchovy, sardine, Atlantic chub mackerel, bogue and blue jack mackerel respectively (Fig. 6), corresponding to a decline on the average body size by about 3.01, 3.43, 3.67, 3.82 and 3.76 % per 1°C of warming.

Discussion

Small pelagic fish are fast growing, migratory and short-lived species relative to most marine fish species, often presenting rapid shifts in distribution related to oceanographic changes [START_REF] Checkley | Synthesis and perspective[END_REF]). Consequently, despite the fact that we used in this study a dataset collected during the summertime, the variability associated with the sampling period was relatively high for 3 species out of 10 (i.e. sprat, Atlantic horse mackerel and Atlantic mackerel) for which no reliable conclusion could be drawn. For round sardinella and Mediterranean horse mackerel, no significant and negative effect of temperature was observed. For the 5 remaining species, we demonstrated that they follow the Bergmann's rule, with larger individuals being more common at lower temperature. In the case of anchovy, these results are consistent with previous investigations of the effect of temperature on their growth which showed a significant negative effect of temperature on asymptotic length (i.e. L ∞ the length at which growth rate is theoretically zero) [START_REF] Basilone | Linking habitat conditions and growth in the European anchovy (Engraulis encrasicolus)[END_REF]). These negative correlations did not seem to be confounded by the sampling period or by the bathymetric variation of fish sizes. Indeed, when controlling for depth by adding it as a covariate in the model, all negative size-temperature relationships remained significant. Furthermore, except for bogue, the effect of temperature was stronger than the effect of depth. The output of the bootstrap-sampling procedure also showed that the negative correlations between temperature and body sizes were stable against sample-size variation. As a whole, these analyses strongly suggest that temperature is a major determinant of pelagic fish body sizes. The rates of body size changes with temperature through space are in the same order of magnitude as those estimated by [START_REF] Audzijonyte | Fish body sizes change with temperature but not all species shrink with warming[END_REF] for reef fishes in the wild (~4% body size change per 1°C change through space) and those measured for marine ectothermic organisms in controlled experiments (-3.0% °C-1 ; Forster et al 2012).

Several partially competing hypotheses exist regarding the occurrence of thermal size clines in ectotherms [START_REF] Atkinson | From cells to colonies: at what levels of body organization does the 'temperature-size rule'apply?[END_REF]Walters & Hassall 2006;[START_REF] Delong | Experimental demonstration of a 'rate-size'trade-off governing body size optimization[END_REF]Zuo et al. 2012;[START_REF] Audzijonyte | Is oxygen limitation in warming waters a valid mechanism to explain decreased body sizes in aquatic ectotherms?[END_REF]. Oxygen limitation was originally proposed as a key mechanism to explain smaller ectotherm body size at higher temperatures (see e.g. [START_REF] Atkinson | From cells to colonies: at what levels of body organization does the 'temperature-size rule'apply?[END_REF][START_REF] Audzijonyte | Is oxygen limitation in warming waters a valid mechanism to explain decreased body sizes in aquatic ectotherms?[END_REF] for a review). The general body size optimization hypothesis "maintain aerobic scope and regulate oxygen supply" states that ectotherms grow to a smaller final body size at higher temperatures to maintain their aerobic scope as oxygen concentrations decline with increasing temperature [START_REF] Atkinson | From cells to colonies: at what levels of body organization does the 'temperature-size rule'apply?[END_REF]. For marine fishes, the gill oxygen limitation hypothesis [START_REF] Pauly | The relationships between gill surface area and growth performance in fish: a generalization of von Bertalanffy's theory of growth[END_REF] proposes that body size in fish is limited by the inability of gills (whose surface area is limited) to supply enough oxygen to meet the requirements of increasing metabolic costs that scale with body volume. According to [START_REF] Pauly | Sound physiological knowledge and principles in modeling shrinking of fishes under climate change[END_REF], the gill oxygen limitation hypothesis provides the most parsimonious explanation for a range of responses including temperature dependence of maximal attainable body masses in ectotherms, prevalence of small fish in tropical waters, higher sensitivity of larger individuals to temperature, and lower food assimilation efficiency in larger individuals. However, it is important to note that this hypothesis is strongly debated, as recent criticisms were expressed about its validity and universality (see [START_REF] Lefevre | In modelling effects of global warming, invalid assumptions lead to unrealistic projections[END_REF][START_REF] Lefevre | The role of mechanistic physiology in investigating impacts of global warming on fishes[END_REF][START_REF] Pauly | The gill-oxygen limitation theory (GOLT) and its critics[END_REF]. Another main hypothesis used to explain the temperature-size rule is the mismatch, or difference in developmental rates versus growth rates thermal responses (Van der Have & De Jong 1996;[START_REF] Forster | Warming-induced reductions in body size are greater in aquatic than terrestrial species[END_REF]Zuo et al. 2012). This hypothesis considers that, temperature responses of growth and development during ontogeny differ. An increase in development rate relative to growth rate at high temperature leads to smaller body sizes as a result of lower somatic growth following earlier sexual maturity (Zuo et al. 2012). Furthermore, resource supply models state that the proximate cause for optimal body size is determined by the temperaturedependent interplay of resource supply from the environment versus demand, leading to smaller body sizes (higher resource demands) at higher temperatures (DeLong 2012). Evidence of abiotic and biotic predictors of body size structure has been obtained from several studies showing that differences in productivity and taxonomic composition contribute to size differences [START_REF] Jeppesen | Trophic structure, species richness and biodiversity in Danish lakes: changes along a phosphorus gradient[END_REF][START_REF] Emmrich | Size spectra of lake fish assemblages: responses along gradients of general environmental factors and intensity of lake-use[END_REF][START_REF] Emmrich | Geographical patterns in the body-size structure of European lake fish assemblages along abiotic and biotic gradients[END_REF]. Thus, the realized size in marine ectotherms populations integrates growth, mortality, competition, predation risk and food availability simultaneously, and the interplay among these factors [START_REF] Audzijonyte | Fish body sizes change with temperature but not all species shrink with warming[END_REF]). Several of these factors are modulated by temperature, which result in the occurrence of thermal body size clines in ectotherms, such as the 5 pelagic species of the present study.

Our results indicate that a positive correlation between body size and latitude can be observed only in bogue. The presence of negative correlations between body sizes and latitude can be explained by the geographical gradient of the temperature in the Mediterranean Sea which does not solely follow a latitudinal gradient. The averaged zonal surface temperature gradient over most of the Mediterranean Sea increases from north to south especially in the Gulf of Lions and in the northern Adriatic Sea where the temperature minima are located. However, a meridional gradient of temperature, partly due to the Mediterranean surface circulation, occurs also over the northern Tyrrhenian and the Levantine subbasins where temperature increases from west to east [START_REF] Shaltout | Recent sea surface temperature trends and future scenarios for the Mediterranean Sea[END_REF]. Moreover, the annual temperature in the northern Aegean sub-basin is much lower than in the northern Ionian sub-basin, although the two areas are at the same latitude. This is largely due to the input of less saline and colder water of Black Sea origin entering the northern Aegean Sea through the Dardanelles strait. Besides, here, we used a vertically integrated temperature dataset rather than only surface temperature. Thus the local bathymetric variations of temperature may further increase the decoupling between temperature and latitude. The increase in fish body size with increasing latitude, reported in several other studies at larger spatial scales, may thus be partially due to a greater latitudinal gradient of temperature than that encountered in the Mediterranean Sea (e.g. Fisher et al. 2010b;[START_REF] Saunders | Southern Ocean mesopelagic fish comply with Bergmann's rule[END_REF]. In addition, it should be noted that, as the Mediterranean is classified as an oligotrophic Sea (d'Ortenzio & Ribera d'Alcalà 2009), primary production can be considered as a limiting factor for fish growth in this area as demonstrated for anchovy [START_REF] Basilone | Linking habitat conditions and growth in the European anchovy (Engraulis encrasicolus)[END_REF]). Thus the Mediterranean west-east gradient of primary productivity (d'Ortenzio & Ribera d'Alcalà 2009) could also explain the absence of positive correlations between body sizes and latitude observed for planktivorous fish species, namely anchovy, sardine and blue jack mackerel.

In addition to the temperature, fishing is also an external driver that may alter the size structure of a fish population (Fisher et al. 2010a;[START_REF] Tu | Fishing and temperature effects on the size structure of exploited fish stocks[END_REF]. Fishing represents size-selective removal of larger individuals that can truncate the size structure of a fish population [START_REF] Berkeley | Fisheries sustainability via protection of age structure and spatial distribution of fish populations[END_REF][START_REF] Barnett | Old-growth fishes become scarce under fishing[END_REF], which in turn may cause recruitment failure, reduce the reproductive outputs, and increase the size variability of fish populations [START_REF] Rouyer | Does increasing mortality change the response of fish populations to environmental fluctuations?[END_REF]. This may also lead to evolutionary consequences and selection of slow-growing fish [START_REF] Heino | Fisheries-induced evolution, Annual review of ecology[END_REF]) that start first maturation at a younger age and smaller size due to both genetic and plastic responses [START_REF] Conover | Sustaining fisheries yields over evolutionary time scales[END_REF][START_REF] De Roos | Evolutionary regime shifts in age and size at maturation of exploited fish stocks[END_REF]. Unfortunately, the exploitation status was available only for sardine and anchovy and not in all areas due to the absence of stock assessment for the other 8 species included in this study.

The exploitation status of sardines and anchovies in the Mediterranean Sea varies greatly from one area to another. Stocks assessed were either sustainably exploited (for sardine in GSA 9 and anchovy in GSAs 6, 9 and 22) or in overexploitation (sardine in GSAs 1,3,6,[START_REF] Gfcm | Report of Working Group on Stock Assessment of Small Pelagic Species (WGSAS)[END_REF][17][18] or ecologically unbalanced (sardine in GSA 7) or at a low biomass level (anchovy in GSA 7) (GFCM 2019). Although, body size variations can be related to exploitation, we believe that the effects of the spatial patterns of exploitation on body size clines are unlikely to affect our overall conclusions.

Fisheries targeting small pelagic in the Mediterranean Sea, which consists mainly of purse seine fleets (and pelagic trawler in GSA 7 and GSA 17), cannot be considered as large size-biased exploitation in comparison to other types of fisheries. For instance, in the Gulf of Lions (GSA 7), where the most drastic decrease in growth, condition and size of the two main exploited small pelagic fish species (e.g. sardine and anchovy) was observed [START_REF] Van Beveren | Rapid changes in growth, condition, size and age of small pelagic fish in the Mediterranean[END_REF][START_REF] Brosset | Spatio-temporal patterns and environmental controls of small pelagic fish body condition from contrasted Mediterranean areas[END_REF], it has been shown that these decreasing trends did not appear to be mediated by a high fishing pressure [START_REF] Saraux | Small pelagic fish dynamics: A review of mechanisms in the Gulf of Lions, Deep Sea Research Part II[END_REF] but rather by a bottom-up control [START_REF] Feuilloley | Concomitant changes in the environment and small pelagic fish community of the Gulf of Lions[END_REF].

the research vessels that carried out the acoustic surveys. We are very grateful to Angelo Bonanno for 1 his help in editing the manuscript. 2 Table 1: Parameter estimates for the linear mixed models relating mean body size to temperature and latitude while considering the season as a random effect variable (including a random intercept and a random slope of temperature). N is number of observations.

2 22Figure 1 :

 21 Figure 1: Map of the Mediterranean Sea geographical sub areas (GSAs): northern (GSA 01) and southern (GSA 03) Alboran Sea, northern Spain (GSA 06), Gulf of Lions (GSA 07), Ligurian and Tyrrhenian Seas (GSA 9-10), southern Sicily and Malta (GSA 16-15), northern (GSA 17) and southern (GSA 18) Adriatic Sea, western Ionian Sea (GSA 19), eastern Ionian Sea (GSA 20), Aegean Sea (GSA 22) and the south of Turkey (GSA 24). Black dots represent the location of the trawl hauls. The color scale represents summertime climatology of mean temperature of the water column along the continental shelf (0-200m depth).

Figure 2 :

 2 Figure 2: Maps indicating the location of the trawl hauls and their corresponding temperature value and showing the geographic patterns of body size for five species. The color gradient indicates the mean total body size in mm (left panels) and temperature in °C (right panels).

Figure 3 :

 3 Figure 3: Effect of survey-related variability on body size-temperature relationship for the 10 considered species. The lines represent predictions based on linear mixed models relating body size to temperature and latitude while considering the season as a random effect variable including a random intercept and a random slope of temperature.

Figure 4 :

 4 Figure 4: Response surface graph of linear regressions showing the predicted mean body sizes as a function of temperature and depth.

Figure 5 :

 5 Figure 5: Patterns of body size variation in relation to temperature and depth. The colour gradient indicates the observed mean body sizes (in mm) for each combination of temperature and depth values.

Figure 6 :

 6 Figure 6: Effect of sub-sampling on body size-temperature relationships. Violin plots indicate the body size distributions in each bin of 1 °C. Numbers indicate the sample size (number of trawls) per temperature bin. The histograms represent the distributions of slopes and the R² of the 999 linear models fitted using the re-sampling procedure.

Table 2 :

 2 Parameter estimates for linear models relating mean body size to temperature and depth. 1

	Species	Scientific names		Fixed effect			Random effect
			Predictors (Intercept) Latitude Temperature	Residual variance (mm 2 )	Month variance (mm 2 )	Temperature (slope)	N	Marginal / Conditional R 2
			Estimates 206.45	-1.25	-2.07
	European anchovy	Engraulis encrasicolus	p	<0.001 <0.001	<0.001	227.38	62.90	0.07	1303	0.171/ 0.202
			df	1296.00 1296.00 1296.00
			Estimates 232.55	-1.67	-1.84
	Sardine	Sardina pilchardus	p	<0.001 <0.001	<0.001	456.22	291.13	0.64	1228	0.101 / 0.166
			df	1221.00 1221.00 1221.00
			Estimates 440.68	-4.42	-3.43
	Atlantic chub mackerel	Scomber colias	p	<0.001	0.001	0.005	3663.37	0.00	1.35	427	0.075 / 0.170
			df	420.00 420.00	420.00
			Estimates 113.05	-0.16	-0.40
	Sprat	Sprattus sprattus	p	<0.001	0.768	0.532	132.41	446.75	1.08	423	0.005 / 0.313
			df	416.00 416.00	416.00
			Estimates 191.08	-1.42	0.59
	Atlantic horse mackerel	Trachurus trachurus	p	<0.001	0.061	0.782	2058.11 5117.44	15.38	628	0.007 / 0.123
			df	621.00 621.00	621.00
			Estimates 234.86	-2.27	1.05
	Round sardinella	Sardinella aurita	p	<0.001	0.049	0.479	1830.55 656.15	4.56	235	0.020 / 0.160
			df	228.00 228.00	228.00
			Estimates 243.80	-0.98	-1.18
	Mediterranean horse mackerel	Trachurus mediterraneus	p	<0.001	0.279	0.209	2865.76 164.66	0.78	542	0.009 / 0.037
			df	535.00 535.00	535.00
			Estimates 310.15	-5.36	4.87
	Atlantic mackerel	Scomber scombrus	p	<0.001 <0.001	0.017	2095.35 4744.51	11.12	482	0.079 / 0.284
			df	475.00 475.00	475.00

Acknowledgments

This study was carried out in the framework of the MEDIAS within the DCF programs of EU and the Directorates of Fishery of the National Ministries. In Croatia it was supported by the Croatian Ministries of Agriculture and of Science and Education (Grant No. 001-0013077-0532). The authors are grateful to all the participants involved in the MEDIAS surveys and to the Captains and crews of

differentiation, Journal of theoretical biology, 183(3), 329-340. Walters, R. J., and Hassall, M., 2006. The temperature-size rule in ectotherms: may a general explanation exist after all?, The American Naturalist ,167(4),510511512513514515516517518519520521522523 .