Mathias Rousset

AND Yushun Xu

Pierre-Andr É Zitt

A WEAK OVERDAMPED LIMIT THEOREM FOR LANGEVIN PROCESSES

Keywords: . 1991 Mathematics Subject Classification. 35P25, 35Q55. Key words and phrases. Langevin dynamics, overdamped asymptotics, perturbed test function

In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.

Introduction

This paper focuses on the overdamped asymptotics of Langevin dynamics. The Langevin Stochastic Differential Equation (SDE) describes the dynamics of a classical mechanical system perturbed by a stochastic thermostat. The system state at time t ≥ 0 is encoded by its position Q t and its momentum P t . More formally, the equation reads:

dQ t = P t dt, dP t = -∇V (Q t )dt -P t dt + 2β -1 dW t ,
where in the above, Q t takes values in the d-dimensional torus T d , P t takes values in ×R d , the function V : T d → R is the particles' potential energy, β > 0 the inverse temperature, and t → W t ∈ R d is a standard d-dimensional Brownian motion. The term 2β -1 dW t is a fluctuation term bringing energy into the system, while this energy is dissipated through the friction term -P t dt; the sum of these two terms forming the so-called thermostat part. The remaining terms are simply Newton's equation of motion. For more details on this equation, we refer to [LRS10, Section 2.2].

The case we consider here is the so-called overdamped asymptotics, where the time scale of the large damping due to friction is much smaller than the time scale of the Hamiltonian dynamics, so that the momentum becomes a fast variable compared to the slow position variable. We introduce a parameter ε for the ratio of the time scales, and consider

dQ ε t = 1 ε P ε t dt, dP ε t = -1 ε ∇V ε (Q ε t )dt -1 ε 2 P ε t dt + 1 ε 2β -1 dW t .
(1.1)

Note that we allow the potential V ε ∈ C 1 (T d ) to depend on ε and will only suppose that it converges to a limit V ; see below for a precise statement. The Markov generator L ε associated with (1.1) is given by

L ε f (q, p) := 1 ε 2 1 β ∆ p f -p • ∇ p f + 1 ε (p • ∇ q f -∇ q V ε • ∇ p f ) , (1.2)
where f denotes any smooth test function of the variables (q, p) ∈ T d × R d .

Overdamped processes are stochastic dynamics on the system position (Q t ) t≥0 only. The overdamped Langevin SDE is given by:

(1.3)

dQ t = -∇V (Q t )dt + 2β -1 dB t ,
where V : T d → R d is a potential energy, limit of V ε when ε → 0 in some appropriate sense, and t → B t ∈ R d is a standard d-dimensional Wiener process. The Markov generator L associated with (1.3) acts on smooth test functions f of the variable q as follows:

Lf (q) := -∇ q V • ∇ q f + 1 β ∆ q f.
Our main result is the proof of the convergence in distribution of the Langevin position process (Q ε t ) t≥0 towards its overdamped counterpart (Q t ) t≥0 , assuming the uniform convergence of the gradient potential as well as a control of moments of the initial kinetic energy.

Theorem 1.1 (Overdamped limit of the Langevin dynamics). For any ε > 0, suppose that (Q ε t , P ε t ) t≥0 ∈ T d × R d is a weak solution to the SDE (1.1). Assume that the following conditions hold:

(1) V ε is C 1 (T d ), and converges to V in the sense that ∇V ε -∇V ∞ ---→ ε→0 0,

(2) The following moment bound holds true:

lim ε→0 εE(|P ε 0 | 3 ) = 0 (3)
The initial position distribution is converging to some limit: Law

(Q ε 0 ) ---→ ε→0 Law (Q 0 ).
Then, when ε → 0, the process (Q ε t ) t≥0 ∈ C(R + → T d ) converges in distribution to the unique weak solution of the overdamped SDE (1.3).

Remark 1.2. In Theorem 1.1, the space of trajectories C(R + → T d ) is endowed with uniform convergence on compact sets; making it Polish (metrizable for a separable and complete metric).

The literature on diffusion approximations is very rich; we refer for instance to Stuart-Pavliotis in [START_REF] Pavliotis | Multiscale methods[END_REF] for a recent pedagogical overview of related issues. Historically, a possible chain of seminal references is given by Stratonovich in [START_REF] Stratonovich | I: General theory of random processes. Nonlinear transformations of signals and noise[END_REF], Khas'minskii in [START_REF] Khas | minskii, A limit theorem for the solutions of differential equations with random right-hand sides[END_REF], Papanicolaou-Varadhan in [START_REF] Papanicolaou | A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations[END_REF], as well as Papanicolaou-Kohler in [START_REF] Papanicolaou | Asymptotic theory of mixing stochastic ordinary differential equations[END_REF]; complemented with the more modern viewpoint of Ethier-Kurtz in [START_REF] Ethier | Markov processes[END_REF], Chapter 12 "Random evolutions".

In the present case, the momentum variable is averaged out with the diffusion approximation, so that the problem may be labeled as "diffusion approximation with averaging". Broadly speaking, the problem can be approached using strong or weak convergence techniques. For an example of the strong convergence approach, the results in [START_REF] Sancho | Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients[END_REF] rely on estimating the dynamics of Q ε t and its limit using a Gronwall argument; this approach requires the Lipschitz continuity of ∇V ε uniformly in ε. Similar strong convergence results for more advanced models (infinite dimensional, inhomogeneous in space) can be found for instance in [START_REF] Cerrai | On the smoluchowski-kramers approximation for a system with an infinite number of degrees of freedom[END_REF][START_REF] Hottovy | The smoluchowski-kramers limit of stochastic differential equations with arbitrary state-dependent friction[END_REF].

On the other hand, weak convergence results rely on the so-called "perturbed" test function or "corrector" approach, that have been developed since Panicolaou-Stroock-Varadhan in [START_REF] Papanicolaou | Martingale approach to some limit theorems[END_REF]. The case of the overdamped limit (1.1) is not directly covered by these results. Indeed, the correctors are not bounded in the present case, due to the fact that the state space of the momentum variable is not compact.

In a series of papers [PV01, PV03, PV05], Pardoux-Veretennikov extend the classical diffusion approximation with averaging to the non-compact state space case. In the latter setting however, the slow variable has a dynamics independent of the fast one, which is not the case in the Langevin case (1.1).

We now give a physically motivated example that satisfies our assumptions but was not covered by previous works.

Example 1.3. Let V ε (q) = V (q) + α ε χ(k ε q),
where χ ∈ C ∞ (T d ), and the scaling coefficients

k ε ∈ N and α ε ∈ R satisfy k ε → ∞, α ε k ε → 0.
Physically, the potential α ε χ(k ε q) may model the interaction between a particle with unit energy and a periodic crystal of small period k -1 ε , and small energy range of order α ε . When k ε → +∞ but α ε k ε = 1 and ε is kept constant, the effective action of the periodic crystal on the particle can not be neglected, especially for grazing velocities co-linear to the principal directions of the crystal. Indeed, in the latter case, on times of order 1, the crystal exerts on the particle a total force also of order 1, making it deviating from its trajectory.

Our result shows that the physically necessary condition α ε k ε → 0 is in fact sufficient for neglecting the crystal effect in the overdamped regime. Note that if

α ε k 2 ε → +∞, when ε → 0, then ∇V ε -∇V ∞ ε→0 ---→ 0, but still ∇ 2 V ε ∞ ∼ α ε k 2 ε ∇ 2 χ ∞ ε→0 ---→ +∞,
preventing ∇V ε from being Lipschitz uniformly in ε; and hence forbidding results based on strong convergence.

In order to prove Theorem 1.1, we will establish a more general weak convergence result. We consider a sequence (indexed by a small parameter ε > 0) of Markov processes of the form t → (Q ε t , P ε t ) ∈ T d × R d taking value in the Skorokhod path space D T d ×R d . Our general convergence result, namely Theorem 3.5, gives general conditions under which (Q ε t ) t≥0 converges in distribution to the unique solution of a particular martingale problem. The proof follows the usual pattern: first we prove tightness for the family of distributions of (Q ε t ), and then characterize the limit through martingale problems. For both steps, we use the perturbed test function method. The key sufficient criteria yielding the results of both steps is given in Assumption 3.4, which states that to any smooth f : T d → R, we can associate a perturbed test function f ε :

T d × R d → R such that for all T > 0, lim ε→0 E sup t≤T |f (Q ε t ) -f ε (Q ε t , P ε t )| = 0 and lim ε→0 E T 0 |Lf (Q ε t ) -L ε f ε (Q ε t , P ε t )| dt = 0.
Remark 1.4 (On the choice of the state space). Theorem 3.5 can be useful for càd-làg processes, which explains the fact that we work in Skorokhod space. We have chosen to work in T d × R d for notational simplicity, but Theorem 3.5 could be extended to more general product spaces of the type E × F , where E and F are Polish spaces. If E is compact, the extension is straightforward. If E is locally compact, then one can work with E ∪ {∞}, the one point compactification of E at infinity (see [START_REF] Ethier | Markov processes[END_REF]Chapter 4]). If E is not locally compact, then one needs to use Theorem 9.1 in [EK86, Chapter 3] instead of Theorem 2.12 below which is a corollary of the former. In the latter case: (i) the a priori compact containment condition (9.1) of Theorem 9.1 in [EK86, Chapter 3] has to be proven; and (ii) one has to show the tightness of Law (f (Q ε t )) t≥0 ε≥0 for all f in a space of functions dense in C b (E) for the topology of uniform convergence on compacts. Such extensions to infinite dimensional spaces are left for future work.

The paper is organized as follows. Section 2 starts with some notation and preliminaries. In Section 3, we state and prove the general convergence result Theorem 3.5. This general method is then applied in Section 4 to the overdamped Langevin limit, proving Theorem 1.1.

Notation and Preliminaries

In what follows, we introduce notation and recall some known results.

2.1. General notation. Let (E, d) be a Polish space, that is, a topological space which is metric, complete and separable. Denote C(E) the Banach space of all continuous functions and C b (E) the Banach space of all bounded continuous functions. We denote by P(E) the space of probability measures on the Borel σ-field B(E). The notation F X t means the natural filtration of càd-làg processes (X t ) t≥0 , that is F X t = σ(X s , 0 ≤ s ≤ t). For any (s, t) ∈ R × R, we denote by s ∧ t the minimum of s and t.

The Skorokhod space.

A càd-làg function (from the French "continu à droite, limité à gauche", also called RCLL for "right continuous with left limits") is a function defined on R + that is everywhere right-continuous and has left limits everywhere. The collection of càd-làg functions on a given domain is known as the Skorokhod space. We denote D E the space of càd-làg functions with values in a Polish space E. We recall that this path space D E may be equipped with the Skorokhod topology (see Section 5 of [EK86, Chapter 3]): a family of trajectories (q ε s ) s≥0 indexed by ε converges to a limit trajectory (q 0 s ) s≥0 if there exists a sequence (λ ε ) ε≥0 in the space of strictly increasing continuous bijections of [0, ∞[, such that for each T > 0: lim ε→0 sup t≤T |λ ε (t) -t| = 0 and lim ε→0 sup t≤T d q ε t , q 0 λε(t) = 0. The following result will be useful in the proof of Theorem 3.5. Lemma 2.1. Integration with respect to time is continuous with respect to the Skorokhod topology: if (q ε t ) t≥0 converges to (q 0 t ) t≥0 in D E , and ψ : E → R is bounded and continuous, then for each T > 0,

T 0 ψ(q ε t )dt ---→ ε→0 T 0 ψ(q 0 t )dt.
Proof. Let us denote by J T := t ∈ [0, T ], q 0 t -= q 0 t the countable set of jump times in [0, T ] of q 0 . By definition of convergence in the Skorokhod space,

lim ε→0 q ε s = q 0 s ∀s ∈ [0, T ] \ J T .
Since J T has Lebesgue measure 0 and ψ is continuous and bounded, dominated convergence yields the result.

Martingale problems.

Let us first recall some basics on martingales and stochastic calculus. Let (Ω, F, P, (F t ) t≥0 ) a filtered probability space. A càd-làg real-valued process (X t ) t≥0 is said to be adapted if X t is F t -measurable for all t ≥ 0, and is called a

(F t ) t≥0 - martingale if E(|X t | |F s ) < +∞ and E(X t |F s ) = X s for any 0 ≤ s ≤ t.
We will often need the technical tool of localization by stopping times, to deal with the unboundedness of the momentum variable. We follow here the presentation of [EK86, Chapter 4].

Definition 2.2 (Local martingale).

A càd-làg real-valued process (X t ) t≥0 defined on (Ω, F, P, (F t ) t≥0 ) is called a local martingale with respect to (F t ) t≥0 if there exists a nondecreasing sequence (τ n ) n∈N of (F t ) t≥0 -stopping times such that τ n → ∞ P-almost surely, and for every n ∈ N, X t∧τn t≥0 is an (F t ) t≥0 -martingale.

Let us now state precisely what it means for a process to solve a martingale problem.

Definition 2.3 (Martingale problem)

. Let E be a Polish space. Let L be a linear operator mapping a given space D ⊂ C b (E) into bounded measurable functions. Let µ be a probability distribution on E. A càd-làg process (X t ) t≥0 with values in E solves the martingale problem for the generator L on the space D with initial measure µ -in short, X solves MP(L, D(L), µ) -if Law (X 0 ) = µ and if, for any ϕ ∈ D,

(2.1) t → M t (ϕ) := ϕ(X t ) -ϕ(X 0 ) - t 0 Lϕ(X s )ds
is a martingale with respect to the natural filtration

F X t = σ (X s , 0 ≤ s ≤ t) t≥0 .
Moreover, the martingale problem MP(L, D, µ) is said to be well-posed if:

• There exists a probability space and a càd-làg process defined on it that solves the martingale problem (existence); • whenever two processes solve MP(L, D, µ), then they have the same distribution on D E (uniqueness). (2.2)

dX t = b(X t )dt + σ(X t )dW t ,
with an initial condition Law (X 0 ) = µ 0 . Let L be the formal generator

L := d i=1 b i ∂ i + 1 2 d i,j=1 a ij ∂ i ∂ j , (2.3)
where a = σσ T .

Definition 2.4 (Weak solution of the SDE).

A continuous process (X t ) t≥0 is a weak solution of (2.2) if there exists a filtered probability space (Ω, F, P, (F t ) t≥0 ) such that:

• t → W t is a (F t ) t≥0 -Brownian motion, that is, an (F) t≥0 -adapted process such that Law(W t+h -W t |F t ) = N (0, h).
• X is a continuous, (F t ) t≥0 -adapted process and satisfies the stochastic integral equation

X t = X 0 + t 0 b(X s )ds + t 0 σ(X s )dW s a.s.
We now quote two results from [START_REF] Ethier | Markov processes[END_REF] concerning existence and uniqueness of solutions to SDEs and martingale problems. The first is an existence result, and can be found in [EK86, Section 5.3] (Corollary 3.4 and Theorem 3.10).

Theorem 2.5. Assume that b, σ are continuous. If there exists a constant K such that for any t ≥ 0, x ∈ R d :

|σ| 2 ≤ K(1 + |x| 2 ); (2.4) x • b(x) ≤ K(1 + |x| 2 ), (2.5)
then there exists a weak solution of the stochastic differential equation (2.2) corresponding to (σ, b, µ), which is also solution of the martingale problem

MP(L, C ∞ c (R d ), µ), C ∞ c (R d
) being the set of smooth functions with compact support. Remark 2.6. For the Langevin equation (1.1)) we first remark that the latter can be set in

R d × R d using the Z d -periodic extension of V ε . Then b(q, p) = 1 ε p, -1 ε ∇V ε (q) -1 ε 2 p and σ = (0, 1 ε 2β -1 Id R d ) are continuous since V ε ∈ C 1 (R d ). Moreover, |σ| 2 = σσ ⊤ = (0, 2 βε 2 Id R d )
, and on the other hand

(q, p) • b(q, p) = 1 ε pq - 1 ε p∇V ε (q) - 1 ε 2 p 2 ≤ 1 2ε (1 + ∇V ε ∞ )(1 + |p| 2 + |q| 2 ),
which implies the existence of weak solution of (1.1) in R d . One then obtains existence of a weak solution in T d of the original (1.1) using the canonical continuous mapping

R d → T d := R d /Z d .
The next result follows from [EK86] (Theorem 1.7 in Section 8.1) and [SV07] (Theorem 10.2.2 and the discussion following their Corollary 10.1.2) .

Theorem 2.7. Assume that the bounds (2.4) and (2.5) hold. Suppose that a := σσ ⊤ is continuous and uniformly elliptic:

∃C a > 0, ∀ξ ∈ R d , ∀x ∈ R d , ξ ⊤ a(x)ξ ≥ C a |ξ| 2 .
Then for any initial condition µ, there is a unique weak solution of the stochastic differential equation (2.2). This solution is also the unique solution of the martingale problem µ). Remark 2.8. For the overdamped Langevin equation (1.3), we remark again that the latter can be set in R d using the Z d -periodic extension of V ε . One then obtains well-posedness of the martingale problem

MP(L, C ∞ c (R d ),
MP(L, C ∞ c (R d ), µ) in R d since ∇V
U k := (x 1 , . . . , x d ) ∈ R d | |x i -k i /8| ≤ 1/4 ∀i = 1 . . . d
where k ∈ Z d and then remark that by partition of unity for smooth functions, any ϕ ∈ C ∞ c (R d ) can be written as a finite sum of smooth functions with compact support in each given U k , k ∈ Z d .

Convergence in distribution.

As we said before, we are interested here in proving convergence in distribution for processes. Let us briefly recall several key results that will be used later.

For completeness, we start by recalling the very classical Prohorov theorem, characterizing relative compactness by tightness (see for example Section 2 in[EK86, Chapter 3]).

Theorem 2.9 (Prohorov theorem). Let (µ ε ) ε be a family of probability measures on a Polish space E. Then the following are equivalent:

(1) (µ ε ) ε is relatively compact for the topology of convergence in distribution.

(2) (µ ε ) ε is tight, that is to say, for any δ > 0, there is a compact set K δ such that

inf ε µ ε (K δ ) ≥ 1 -δ.
Over the years several relative compactness criteria in Skorokhod space have been developed. We will use the following one [EK86, Theorem 8.6, Chapter 4].

Theorem 2.10 (Kurtz-Aldous tightness criterion). Consider a family of stochastic processes ((X

ε t ) t≥0 ) ε in D R . Assume that Law(X ε 0 ) ε is tight. ∀δ ∈ (0, 1
) and T > 0, there exists a family of nonnegative random variable Γ ε,δ , such that:

∀ 0 ≤ t ≤ t + h ≤ t + δ ≤ T E |X ε t+h -X ε t | 2 |F X ε t ≤ E Γ ε,δ |F X ε t ; (2.6) with lim δ→0 sup ε E(Γ ε,δ ) = 0. (2.7)
Then the family of distributions (Law ((X ε t ) t≥0 )) ε is tight.

Remark 2.11 (On using sequences). If ε > 0 is a real number and that instead of (2.7), one considers the condition lim δ→0 lim sup ε→0 + E(Γ ε,δ ) = 0, then the conclusion becomes the following: (Law ((X εn t ) t≥0 )) εn is tight for any (ε n ) n≥1 -sequence such that ε n > 0 and lim n→+∞ ε n = 0. This version will be the one used in the present paper.

If the processes, say (Q ε t ) t≥0 , is defined in a general state space E, it is natural to consider the image processes (f (Q ε t )) t≥0 for various observables, or test functions, f . The following result enables us to recover the tightness for the original process from the tightness of the observed processes (Corollary 9.3 Chapter 3 in [START_REF] Ethier | Markov processes[END_REF]).

Theorem 2.12 (Tightness from observables). Let E be a compact Polish space and ((Q ε t ) t≥0 ) ε>0 be a family of stochastic processes in D E . Assume that there is an algebra of test functions D ⊂ C b (E), dense for the uniform convergence, such that for any

f ∈ D, ((f (Q ε t )) t≥0 ) ε>0 is tight in D R . Then (Law(Q ε t ) t≥0 ) ε>0 is tight in D E .
Remark 2.13. Again, the above theorem will be used for families indexed by sequences (ε n ) n≥1 such that ε n > 0 and lim n→+∞ ε n = 0.

Finally, the following two lemmas will be useful when we considering martingale problems. The first one states that the distribution of jumps of càd-làg processes have atoms in a countable set (see Lemma 7.7 Chapter 3 in [START_REF] Ethier | Markov processes[END_REF]).

Lemma 2.14. Let (X t ) t≥0 be a random process in the Skorokhod path space D E . The set of instants where no jump occurs almost surely:

C Law(X) := {t ∈ R + |P(X t -= X t ) = 1}, has countable complement in R + . In particular, it is a dense set.
The second one is a very useful way to check whether a process is a martingale or not (see page 174 in Ethier-Kurtz [START_REF] Ethier | Markov processes[END_REF]).

Lemma 2.15 (Martingale equivalent condition). Let (M t ) t≥0 and (X t ) t≥0 be two càd-làg proceses and let C be an arbitrary dense subset of R + . Then (M t ) t≥0 is F X t -martingale if and only if

E (M t k+1 -M t k )ϕ k (X t k )...ϕ 1 (X t 1 ) = 0, for any time ladder t 1 ≤ ... ≤ t k+1 ∈ C ⊂ R + , k ≥ 1, and ϕ 1 , ..., ϕ k ∈ C b (E).

A general perturbed test function method

In this section, we consider a sequence of stochastic processes, indexed by a small parameter ε > 0, of the form

t → (Q ε t , P ε t ) ∈ T d × R d ,
taking value in the Skorokhod path space D T d ×R d associated with the (Polish) product state space T d × R d . Our goal is to describe a general framework to prove the convergence of the (slow) variables Q towards a well-identified dynamics. We use standard tightness arguments and characterization through martingale problems, emphasizing the technical role of perturbed test functions.

Notation and Assumptions.

For each ε, we consider a càd-lag process t → (Q ε t , P ε t ) ∈ T d × R d . The natural filtration of the full process and the process (Q ε t ) t≥0 are denoted respectively by

F Q ε , P ε t := σ ((Q ε s , P ε s ), 0 ≤ s ≤ t), and F Q ε t := σ (Q ε s , 0 ≤ s ≤ t).
We now state the key assumptions that will imply convergence in distribution of the process (Q ε t ) t≥0 towards the solution of a martingale problem. Assumption 3.1 (Generator of the process (Q ε t , P ε t ) ). There exists a linear operator

L ε acting on C ∞ (T d × R d ) which is the extended Markov generator of (Q ε t , P ε t ) t≥0 in the sense that, for all f ∈ C ∞ (T d × R d ), L ε f is locally bounded and t → M ε t (f ) := f (Q ε t , P ε t ) -f (Q ε 0 , P ε 0 ) - t 0 L ε f (Q ε s , P ε s )ds is a (F Q ε , P ε t ) t≥0 -local martingale.
Assumption 3.2 (The limit process). There exists a linear operator L mapping C ∞ (T d ) to C(T d ) such that the martingale problem MP(L, C ∞ (T d ), µ) is well-posed for any initial condition µ. Assumption 3.3 (Initial condition). The initial condition (Law(Q ε 0 )) ε>0 converge to a limit µ 0 , when ε → 0. Assumption 3.4 (Existence of perturbed test functions). For all f ∈ C ∞ (T d ), there exists a perturbed test function

f ε ∈ C ∞ (T d × R d ),
such that for all T , the rest terms

R ε 1,t (f ) := |f (Q ε t ) -f ε (Q ε t , P ε t )| and R ε 2,t (f ) := |Lf (Q ε t ) -L ε f ε (Q ε t , P ε t )| satisfy the following bounds: lim ε→0 E sup 0≤t≤T R ε 1,t (f ) = 0, (3.1) lim ε→0 E T 0 R ε 2,t (f )dt = 0. (3.2)
3.2. The general convergence theorem. We are now in position to state our main abstract result. Theorem 3.5. Under the Assumptions 3.1, 3.2, 3.3, and 3.4, the family Law(Q ε t ) t≥0 ε>0 converges when ε → 0 to the unique solution of martingale problem MP(L, C ∞ (T d ), µ).

The proof follows the classical pattern, in two steps: we first prove that the processes Q ε t are relatively compact in D T d ; then we show that any possible limit must solve the martingale problem MP(L, C ∞ (T d ), µ).

3.2.1.

Step one: The proof of tightness. We want to prove that for each sequence (ε n ) n≥1 satisfying lim n ε n = 0, (Law(Q εn t )) n≥1 is tight. By Theorem 2.12, it is enough to prove the tightness of (Law (f (Q εn t ))) n≥1 for all f ∈ C ∞ (T d ). The latter fact will follow from Theorem 2.10, if we are able to construct, for any function f ∈ C ∞ (T d ) and any ε, δ > 0 and any T > 0, a random variable Γ ε,δ (f ) such that for all 0

≤ t ≤ t + h ≤ t + δ ≤ T , one has E f (Q ε t+h ) -f (Q ε t ) 2 F Q ε t ≤ E Γ ε,δ (f ) F Q ε t , (3.3) where lim δ→0 lim sup ε≥0 E [Γ ε,δ (f )] = 0. (3.4)
We claim that the following variant: Lemma 3.6. For any g ∈ C ∞ (T d ), and any δ, ε, T > 0, there exists a random variable

Γ ′ ε,δ (g) such that for all 0 ≤ t ≤ t + h ≤ t + δ ≤ T , E g(Q ε t+h ) -g(Q ε t ) F Q ε t ≤ E Γ ′ ε,δ (g) F Q ε t , (3.5)
where lim

δ→0 lim sup ε≥0 E Γ ′ ε,δ (g) = 0. (3.6)
is a sufficient condition. Indeed, the required estimates (3.3), (3.4) will follow easily from the basic decomposition

f (Q ε t ) -f (Q ε t+h ) 2 = f (Q ε t+h ) 2 -(f (Q ε t )) 2 -2f (Q ε t ) f (Q ε t+h ) -f (Q ε t ) . since we get E f (Q ε t+h ) -f (Q ε t ) 2 F Q ε t ≤ E Γ ′ ε,δ (f 2 ) F Q ε t + 2 f ∞ E Γ ′ ε,δ (f ) F Q ε t , (3.7) and it is enough to let Γ ε,δ (f ) = Γ ′ ε,δ (f 2 ) + 2 f ∞ Γ ′ ε,δ (f ) to conclude.
Let us now prove the Lemma 3.6. Let g be an arbitrary smooth function, and let g ε be the perturbed test function given by Assumption 3.4. An elementary rewriting leads to

g(Q ε t+h ) -g(Q ε t ) = g(Q ε t+h ) -g ε (Q ε t+h , P ε t+h ) -(g(Q ε t ) -g ε (Q ε t , P ε t )) - t+h t (Lg(Q ε s ) -L ε g ε (Q ε s , P ε s )) ds + t+h t Lg(Q ε s )ds -M ε t (g ε ) + M ε t+h (g ε ), (3.8)
where (M ε t (g ε )) t≥0 is a local F Q ε , P ε -martingale by Assumption 3.1. Let τ n be an associated localizing sequence of stopping times. Applying (3.8) at times t ∧ τ n and (t + h) ∧ τ n , we get

g(Q ε (t+h)∧τn ) -g(Q ε t∧τn ) = g(Q ε (t+h)∧τn ) -g ε Q ε (t+h)∧τn , P ε (t+h)∧τn -g(Q ε t∧τn ) -g ε (Q ε t∧τn , P ε t∧τn ) - t+h t (Lg(Q ε s ) -L ε g ε (Q ε s , P ε s )) 1 s≤τn ds + t+h t Lg(Q ε s )1 s≤τn ds -M ε t∧τn (g ε ) + M ε (t+h)∧τn (g ε ).
Taking the conditional expectation with respect to F Q ε t , the martingale terms cancel out, and we get:

E g(Q ε (t+h)∧τn ) -g(Q ε t∧τn ) F Q ε t ≤ E g(Q ε (t+h)∧τn ) -g ε Q ε (t+h)∧τn , P ε (t+h)∧τn F Q ε t + E g(Q ε t∧τn ) -g ε (Q ε t∧τn , P ε t∧τn ) F Q ε t + t+h t E Lg(Q ε s ) -L ε g ε (Q ε s , P ε s )|F Q ε t ds + h sup q∈T d |Lg(q)| ≤ E R ε 1,(t+h)∧τn + R ε 1,t∧τn F Q ε t + t+h t E R ε 2,s |F Q ε t ds + δ sup q∈T d |Lg(q)| ≤ 2E sup s∈[0,T ] R ε 1,s F Q ε t + T 0 E R ε 2,s |F Q ε t ds + δ sup q∈T d |Lg(q)| .
The right hand side does not depend on n any longer. On the left hand side, we apply dominated convergence for n → ∞ to get

E g(Q ε (t+h) ) -g(Q ε t ) F Q ε t ≤ E Γ ′ ε,δ (g) F Q ε t for Γ ′ ε,δ (g) = 2 sup [0,T ] R ε 1,t + T 0 R ε 2,s ds + δ Lg ∞ .
The controls on the rest terms given by Assumption 3.4, and the continuity of Lg (Assumption 3.2) ensure that

lim δ→0 lim sup ε→0 Γ ′ ε,δ (g) = 0,
and the proof of tightness is concluded.

3.2.2.

Step two: identification of the limit. In this step, we suppose that a sequence Q n t = Q εn t converges in distribution to a limit Q 0 t , and we prove that necessarily, Q 0 solves the martingale problem for the generator L.

Let f ∈ C ∞ (T d ), we have to check that

M t (Q 0 t ) := f (Q 0 t ) -f (Q 0 0 ) - t 0 Lf (Q 0 s )ds (3.9) is a martingale with respect to F Q 0 t = σ(Q 0 s , 0 ≤ s ≤ t). Consider a time sequence 0 ≤ t 1 ≤ • • • ≤ t p ≤ t p+1 for p ≥ 1,
taken in the continuity set C Law (Q) given by Lemma 2.14. Recall that C Law (Q) is dense in R. Let ϕ 1 , ..., ϕ p ∈ C b (T d ) be p test functions. By Lemma 2.15, it is enough to prove that

I 0 := E f (Q 0 t p+1 ) -f (Q 0 tp ) - t p+1 tp Lf (Q 0 s )ds ϕ 1 (Q 0 t 1 ) • • • ϕ p (Q 0 tp ) = 0.
Let I ε be the corresponding quantity for ε > 0, that is,

I ε := E f (Q ε t p+1 ) -f (Q ε tp ) - t p+1 tp Lf (Q ε s )ds ϕ 1 (Q ε t 1 ) • • • ϕ p (Q ε tp ) .
Let us first show that I ε converges to 0. We first condition on F Q ε tp to get:

|I ε | ≤ E E f (Q ε t p+1 ) -f (Q ε tp ) - t p+1 tp Lf (Q ε s )ds F Q ε tp ϕ 1 (Q ε t 1 ) • • • ϕ p (Q ε tp ) ≤ E E f (Q ε t p+1 ) -f (Q ε tp ) - t p+1 tp Lf (Q ε s )ds F Q ε tp ϕ 1 ∞ • • • ϕ p ∞ .
Using again the perturbed test function f ε and the decomposition (3.8), we get by the same localization argument as in Step 1 that

|I ε | ≤ E R ε 1,t p+1 (f ) + R ε 1,tp (f ) + t p+1 tp R ε 2,s (f ) ϕ 1 ∞ ... ϕ p ∞ .
The estimates on the rest term from Assumption 3.4 then imply that I ε → 0.

Let us now prove that I ε converges to I 0 . Let Φ :

D T d → R be the functional Φ : (q t ) t≥0 → f (q t p+1 ) -f (q tp ) - t p+1 tp Lf (q s )ds ϕ 1 (q t 1 ) • • • ϕ p (q tp ) so that I ε = E [Φ((Q ε t ) t≥0 )] and I 0 = E Φ((Q 0 t ) t≥0 ) . Let us first check that, if q 0 ∈ D T d satisfies q 0 t - k = q 0 t k for each 1 ≤ k ≤ p + 1,
then the functional Φ is continuous at the trajectory q 0 . Indeed, since Lf is continuous and bounded by Assumption 3.2, Lemma 2.1 shows that the map (q t ) t≥0 → t p+1 tp Lf (q s )ds is continuous with respect to Skorokhod topology; moreover, by assumption, q 0 is continuous at the time t k for each 1 ≤ k ≤ p + 1, so the map (q t ) t≥0 → ϕ k (q t k ) is continuous at q 0 ∈ D T d .

Let now (ε n ) n≥1 be any sequence such that ε n → 0 and (Q εn t ) t≥0 converges in distribution to (Q 0 t ) t≥0 . The Skorokhod representation theorem (Theorem 1.8 in [EK86, Chapter 3]) ensures that one can construct a probability space where the distribution of (Q εn t ) t≥0 for each n is unchanged but for which lim n→+∞ Q εn = Q 0 almost surely in D T d . Since t k ∈ C Law (Q 0 ) for each k = 1 . . . p+1, Ψ is almost surely continuous at Q 0 and we can apply the dominated convergence theorem to obtain lim n→+∞ I εn = I 0 . Since the choice of the vanishing sequence (ε n ) n≥1 is arbitrary, we conclude that lim ε→0 I ε = I 0 . The limit process thus solves the martingale problem MP(L, C ∞ (T d ), µ).

Conclusion.

For each sequence (ε n ) n≥1 satisfying lim n ε n = 0, we have proven that (Law(Q εn t )) n≥1 is tight and that any converging subsequence is solution to the martingale problem MP(L, C ∞ (T d ), µ). By uniqueness of the latter according to Assumption 3.2, this identifies the limit, showing that (Law(Q εn t )) n≥1 converges to the solution of MP(L, C ∞ (T d ), µ). Since the sequence (ε n ) n≥1 is arbitrary and convergence in distribution is metrizable, (Law(Q ε t )) ε>0 also converges to the solution of MP(L, C ∞ (T d ), µ), proving Theorem 3.5.

Overdamped limit of the Langevin dynamics

In the section, we will use the perturbed test function method presented in last section to prove Theorem 1.1. We will first state the key estimates on (|P ε t |) t≥0 . These estimates are then used to check the assumptions of our general Theorem 3.5 in the specific case of Langevin processes. In a last section we will detail the proof of the key estimates.

4.1. Some moments estimates for Langevin processes. We start by giving a few facts about the solution to the Langevin SDE (1.1). We first check that the operator

L ε acting on C ∞ (T d , R d ) by L ε f (q, p) := 1 ε 2 1 β ∆ p f -p • ∇ p f + 1 ε (p • ∇ q f -∇ q V ε • ∇ p f )
is the generator the process, in the sense that Assumption 3.1 holds.

Proposition 4.1. If (Q ε t , P ε t ) t≥0 is a weak solution of the Langevin SDE (1.1), then for any smooth function f :

T d × R d → R d , the process t → M ε t (f ) = f (Q ε t , P ε t ) -f (Q ε 0 , P ε 0 ) - t 0 L ε f (Q ε s , P ε s )ds, is a (F Q ε ,P ε t ) t≥0 -local martingale.
Proof. This is a very classical result. By Itô calculus we write

df ε (Q ε t , P ε t ) = L ε f ε (Q ε t , P ε t )dt + 1 ε 2β -1 ∇ p f ε (Q ε t , P ε t )dW t .
Defining the sequence of

F Q ε ,P ε t t≥0
-stopping time

τ n = inf{t ≥ 0, |P ε t | ≥ n}, (4.1)
which converge almost surely to infinity, we obtain that

M ε,n t (f ε ) := 1 ε 2β -1 t 0 ∇ p f ε (Q ε s , P ε s )1 s≤τn dW s is a F Q ε ,P ε t t≥0
-martingale for any n ≥ 0, which is the definition of a local martingale.

We now state several bounds on the momentum variable P ε t , which are the key technical estimates needed later to control the rest terms appearing in the perturbed test function method. For any continuous V : T d → R we denote by osc(V ) the oscillation defined by osc(V ) = max Vmin V. Lemma 4.2 (Propagation of moments). For any γ ≥ 1, any M > 0 and any β > 0, there is a numerical constant C(γ, M, β) such that for any ε > 0, if osc(V ε ) ≤ M , then

(4.2) sup t≥0 E |P ε t | 2γ ≤ C(γ, M, β) E |P ε 0 | 2γ + 1 .
Lemma 4.3 (Moment of suprema). For any M > 0, any β > 0 and any T > 0, there is a numerical constant C(M, β, T ) such that for any ε ∈ (0, 1), if osc(V ε ) ≤ M , then

E sup 0≤t≤T |P ε t | 2 ≤ E |P ε 0 | 2 + 1 ε C(M, β, T ) E |P ε 0 | 2 + 1 1/2 . (4.3) In particular, if lim ε→0 ε 2 E |P ε 0 | 2 = 0, then lim ε→0 ε 2 E sup 0≤t≤T |P ε t | 2 = 0.
The proofs of these estimates use classical techniques of stochastic calculus and are postponed to Section 4.3. 4.2. The perturbed test functions in the Langevin case. In this section we apply the general method described in Section 3 to the specific Langevin case, in order to prove Theorem 1.1.

We will use the following standard notation for multidimensional derivatives:

∇ k f (p 1 , . . . , p k ) := d i 1 ,...,i k =1 ∂ i 1 . . . ∂ i k f × p i 1 1 × . . . × p i k k
where in the above p 1 , . . . , p k ∈ R d . Note that as usual ∆f = Tr ∇ 2 f . We first construct explicitly, for any

f ∈ C ∞ (T d ), a perturbed test function f ε ∈ C ∞ (T d × R d ).
Let us look for f ε in the following form (see [START_REF] Papanicolaou | Martingale approach to some limit theorems[END_REF])

f ε (q, p) = f (q) + εg 1 (q, p) + ε 2 g 2 (q, p). (4.4)
Applying the generator L ε , using the fact that f does not depend on p, and grouping terms with respect to powers of ε, we get

L ε f ε (q, p) = 1 ε p • ∇ q [f (q) + εg 1 (q, p) + ε 2 g 2 (q, p)] - 1 ε ∇ q V (q) • ∇ p [εg 1 (q, p) + ε 2 g 2 (q, p)] - 1 ε 2 p • ∇ p [εg 1 (q, p) + ε 2 g 2 (q, p)] + 1 ε 2 β ∆ p [εg 1 (q, p) + ε 2 g 2 (q, p)] = 1 ε p • ∇ q f -p • ∇ p g 1 + 1 β ∆ p g 1 + p • ∇ q g 1 -∇ q V ε • ∇ p g 1 -p • ∇ p g 2 + 1 β ∆ p g 2 + ε (p • ∇ q g 2 -∇ p g 2 • ∇ q V ε ) . (4.5)
In order for L ε f ε to converge to Lf , the ε -1 -order terms should vanish, and the ε 0 -order terms should converge at least formally to L(f ). As a consequence g 1 and g 2 should solve the following equations:

0 = p • ∇ q f -p • ∇ p g 1 + 1 β ∆ p g 1 , (4.6) Lf (q) = p • ∇ q g 1 -∇ q V • ∇ p g 1 -p • ∇ p g 2 + 1 β ∆ p g 2 .. (4.7)
The function g 1 (q, p) = p • ∇ q f (q) clearly solves (4.6). With this choice, (4.7) becomes

Lf (q) = ∇ 2 q f (p, p) -∇ q V • ∇ q f -p • ∇ p g 2 + 1 β ∆ p g 2 .
Since Lf (q) = 1 β ∆ q f -∇ q V • ∇ q f , it is easy to check that g 2 (q, p) = 1 2 ∇ 2 q f (p, p) solves the equation.

Therefore, in view of Eq. (4.4), we defined the perturbed test function by : (4.8)

f ε (q, p) = f (q) + εp • ∇ q f + 1 2 ε 2 ∇ 2 q f (p, p).
With this choice, we get using previous calculations and the last line of (4.5)

L ε f ε (q, p) -Lf (q) = (∇ q V -∇ q V ε ) • ∇ q f + ε (p • ∇ q g 2 -∇ p g 2 • ∇ q V ε ) = (∇ q V -∇ q V ε ) • ∇ q f + 1 2 ε ∇ 3 q f (q)(p, p, p) -∇ 2 q f (p, ∇ q V ε ) . (4.9)
We now need to show that Assumption 3.4 holds for this choice of a perturbed test function, that is, we want to show that the differences f εf and L ε f ε -Lf are small in the following appropriate sense. Recalling the notation

R ε 1,t (f ) = |f (Q ε t ) -f ε (Q ε t , P ε t )| , R ε 2,t (f ) = |Lf (Q ε t ) -L ε f ε (Q ε t , P ε t )| , we need to prove that lim ε→0 E sup 0≤t≤T R ε 1,t (f ) = 0, (4.10) lim ε→0 E T 0 R ε 2,t (f )dt = 0. (4.11) Since f ∈ C ∞ (T d ), there exists a C f = max ∇f ∞ , ∇ 2 f ∞ such
that for all (q, p) and all δ ∈ (0, 1/2)

|f ε (q, p) -f (q)| = ε |p • ∇ q f (q)| + 1 2 ε 2 ∇ 2 q f (q) • (p, p) ≤ C f (ε |p| + ε 2 |p| 2 ) ≤ δC f + 1 δ C f ε 2 |p| 2 ,
where we have used that for any δ > 0,

ε |p| ≤ 1 2 δ + 1 2 ε 2 |p| 2 /δ. Therefore E sup t∈[0,T ] R ε 1,t (f ) ≤ δC f + 1 δ C f ε 2 E sup t∈[0,T ] |P ε t | 2 .
By assumption, lim ε→0 εE

|P ε 0 | 3 = 0, so ε 2 E |P ε 0 | 2 ≤ ε 4/3 (εE |P ε 0 | 3 ) 2/3
also goes to zero by Jensen's inequality. By the key Lemma 4.3 this entails that the last term in the previous display disappears in the limit and we get lim sup

ε→0 E sup t∈[0,T ] R ε 1,t (f ) ≤ δC f ,
which proves (4.10) since δ is arbitrary.

We now turn to the proof of (4.11), that is, we want to compare L ε f ε and Lf . By the expression (4.9), we have for some constant

C f = max ∇f ∞ , ∇ 2 f ∞ , ∇ 3 f ∞ |L ε f ε (q, p) -Lf (q)| ≤ C f ∇ q V -∇ q V ε ∞ + C f ε |p| 3 + ∇ q V ε ∞ |p| .
We get rid of the product term with Young's inequality ab ≤ a 3 /3 + 2 3 b 3/2 ≤ a 3 + b 3/2 and get

E R ε 2,t ≤ C f ∇ q V -∇ q V ε ∞ + εC f E 2 |P ε t | 3 + ∇V ε 3/2 ∞ .
We integrate in t to obtain

T 0 E R ε 2,t dt ≤ C f ∇ q V -∇ q V ε ∞ T + εC f T sup t∈[0,T ] E 2 |P ε t | 3 + ∇V ε 3/2 ∞ .
By assumption, lim ε→0 εE |P ε 0 | 3 = 0, and by the uniform convergence of ∇V ε to ∇V we can find a uniform bound M such that osc(V ε ) ≤ M for all ε, so we may apply Lemma 4.2 with γ = 3/2 and get lim

ε→0 ε sup t∈[0,T ] E |P ε t | 3 = 0,
for any T ≥ 0. Together with the convergence of ∇V ε to ∇V this yields

lim ε→0 T 0 E R ε 2,t dt = 0.
from which (4.11) follows.

4.3. Proofs of the moment bounds. We now come back to the proofs of the moment bounds (Lemmas 4.2 and 4.3). It will prove useful to work with the Hamiltonian of the system rather than directly with P ε t . For convenience's sake we assume without loss of generality that 0 ≤ V ε (q) ≤ osc(V ε ). Definition 4.4 (Hamiltonian). We denote by H ε the Hamiltonian of the system:

H ε (q, p) = 1 2 |p| 2 + V ε (q).
We will also write

H ε t := H ε (Q ε t , P ε t ). By Itô's formula, dH ε t = P ε t dP ε t + ∇ q V ε (Q ε t )dQ ε t + 1 2 d i,j=1 d (P ε ) i , (P ε ) j t = - 1 ε 2 |P ε t | 2 + 1 ε 2 β dt + 1 ε 2β -1 P ε t dW t (4.12) = - 2 ε 2 H ε t + 2 ε 2 V ε (Q ε t ) + 1 ε 2 β dt + 2β -1 ε P ε t dW t . (4.13)
Again, by Itô's formula, we thus get for any smooth function (t, h) → φ(t, h)

(4.14) dφ(t, H ε t ) = ∂ t φ(t, H ε t )dt + ∂ h φ(t, H ε t )dH ε t + 1 ε 2 β ∂ 2 h φ(t, H ε t ) |P ε t | 2 dt.
Proof of Lemma 4.2. Let γ ≥ 1. We apply (4.14) to φ(t, x) = e αt h γ and plug in (4.13) to get:

d(e αt (H ε t ) γ ) = γ(H ε t ) γ-1 α γ H ε t - 2 ε 2 H ε t + 2 ε 2 V ε (Q ε t ) + 1 ε 2 β e αt dt + 2β -1 ε γ(H ε t ) γ-1 P ε t e αt dW t + γ(γ -1) ε 2 β (H ε t ) γ-2 |P ε t | 2 e αt dt.
The choice α = 2γ/ε 2 cancels the higher order term in the first bracket. We integrate in time, multiply by e -αt and regroup the finite variation terms to get:

(H ε t ) γ = (H ε 0 ) γ + t 0 γ(H ε s ) γ-1 2 ε 2 V ε (Q ε s ) + 1 ε 2 β + γ(γ -1) ε 2 β (H ε s ) γ-2 |P ε s | 2 e -α(t-s) ds + 2β -1 ε t 0 γ(H ε s ) γ-1 P ε s e -α(t-s) dW s . Since (1/2) |P ε s | 2 ≤ H ε s ≤ (1/2) |P ε s | 2 + osc(V ε ), (4.15) (H ε t ) γ ≤ (H ε 0 ) γ + 2γ ε 2 osc(V ε ) + γ β t 0 (H ε s ) γ-1 e -α(t-s) ds + 2β -1 ε t 0 (H ε s ) γ-1 P ε s e -α(t-s) dW s .
To deal with the unboundedness of the momentum P , we define the following stopping times:

(4.16)

τ n := inf{t : |P t | = n}. When s ≤ τ n , we have |P ε s | ≤ n and H ε s ≤ (osc(V ε ) + n 2 2
). This entails that t → t∧τn 0 (H ε s ) γ-1 P ε s dW s is martingale. Writing (4.15) at time t ∧ τ n and taking expectations, the martingale part disappears; recalling that α = 2γ/ε 2 we get

E (H ε t∧τn ) γ ≤ E [(H ε 0 ) γ ] + osc(V ε ) + γ β αE t∧τn 0 (H ε s ) γ-1 e -α(t-s) ds ≤ E [(H ε 0 ) γ ] + osc(V ε ) + γ β sup s≤t E (H ε s ) γ-1 ds.
Sending n to infinity, we apply Fatou's lemma to get

E [(H ε t ) γ ] ≤ E [(H ε 0 ) γ ] + osc(V ε ) + γ β sup s≤t E (H ε s ) γ-1 ,
and thus

(4.17) sup t≥0 E [(H ε t ) γ ] ≤ E [(H ε 0 ) γ ] + osc(V ε ) + γ β sup t≥0 E (H ε s ) γ-1 .
We are now ready to conclude. Say that γ is good if there exists a C(γ, M, β) such that for all ε,

sup t E [(H ε t ) γ ] ≤ C(γ, M, β)(1 + E [(H ε 0 ) γ ] ,
whenever osc(V ε ) ≤ M . The bound (4.17) immediately shows that γ = 1 is good. If γ is good and γ ≤ γ ′ ≤ γ + 1, using the elementary inequality x a ≤ 1 + x b valid for any x > 0 and any 1 ≤ a < b, we get

sup t≥0 E (H ε t ) γ ′ ≤ E (H ε 0 ) γ ′ + osc(V ε ) + γ ′ β sup t≥0 E (H ε s ) γ ′ -1 ≤ E (H ε 0 ) γ ′ + M + γ ′ β 1 + sup t≥0 E [(H ε s ) γ ] ≤ E (H ε 0 ) γ ′ + M + γ ′ β (1 + C(γ, M, β)E [(H ε 0 ) γ ]) ≤ E (H ε 0 ) γ ′ + M + γ ′ β 1 + C(γ, M, β) 1 + E (H ε 0 ) γ ′
showing that γ ′ is itself good. Therefore all γ ≥ 1 are good. Using the bounds (1/2)p 2 ≤ H ε (q, p) ≤ (1/2)p 2 + M it is easy to translate this into bounds on E |P ε t | 2γ , concluding the proof of Lemma 4.2.

Proof of Lemma 4.3. Let us fix an arbitrary T > 0, and prove (4.3), that is, prove the existence of a numerical constant C(β, M, T ) such for any ε ∈ (0, 1),

E sup 0≤t≤T |P ε t | 2 ≤ E |P ε 0 | 2 + 1 ε C(β, M, T ) E |P ε 0 | 2 +
H t ≤ E [H ε 0 ] + osc(V ε ) + 1 β + 8 T β -1 ε 2 osc(V ε ) + E [H ε 0 ] + 1 β 1/2
, concluding the proof of (4.18).

Appendix A. Stopped martingale problem

Let E be a Polish space. Let L be a linear operator mapping a given space D ⊂ C b (E) into bounded measurable functions. Let µ be a probability distribution on E. Let U ⊂ E be an open set. A càd-làg process (X t ) t≥0 with values in E solves the stopped martingale problem for the generator L on the space D with initial measure µ and domain U -in short, X solves sMP(L, D(L), µ, U ) -if, denoting τ U := inf {t ≥ 0|X t / ∈ U or X t -/ ∈ U } , (i) Law (X 0 ) = µ; (ii) X t = X t∧τ U ; and (iii) if for any ϕ ∈ D, t → M t (ϕ) := ϕ(X t )ϕ(X 0 ) - • There exists a probability space and a càd-làg process defined on it that solves the stopped martingale problem (existence); • whenever two processes solve sMP(L, D, µ, U ), then they have the same distribution on D E (uniqueness). The following theorem is a synthesis of the localization technique of Theorem 6.1 and 6.

2. 4 .

 4 Weak solutions of SDEs. Let b : R d → R d and σ : R d → R d×n be locally bounded. Consider a stochastic differential equation in R d of the form:

  is bounded and continuous by assumption. This solution obviously solves MP(L, C ∞ (T d ), µ) in T d . The fact that uniqueness of MP(L, C ∞ c (R d ), µ) implies uniqueness of MP(L, C ∞ c (T d ), µ) is technically less obvious. It can be treated using the localization technique of Theorem A.1 stated in appendix. More precisely, using the notation of Theorem A.1, one can defines the covering of R d by the open sets

t∧τ U 0

 0 Lϕ(X s )ds is a martingale with respect to the natural filtrationF X t = σ (X s , 0 ≤ s ≤ t) t≥0 .Moreover, the stopped martingale problem sMP(L, D, µ, U ) is said to be well-posed if:

  2 of [EK86, Chapter 4]. It gives a simple criteria ensuring equivalence of uniqueness between (i) a global martingale problem, and (ii) local stopped martingale problems. Theorem A.1. Let (U k ) k∈K be a countable family of open subsets of E such that k∈K U k = E. Assume that for any initial ν, there exists a solution to MP(L, D, µ). Then uniqueness of MP(L, D, µ) for all µ is equivalent to uniqueness of sMP(L, D, µ, U k ) for all µ and all k ∈ K.

  1 ) ≤ M . As before, since 2H ε t -2M ≤ (P ε t ) 2 ≤ 2H ε t , it is enough to prove the statement with H ε t instead of |P ε t | 2 . We start by recalling (4.15) for γ = 1 and α = 2/ε 2 : In order to control the expectation of the supremum, we must control the stochastic integral. Define M t = t 0 P ε s dW s and integrate by parts: By Doob's martingale maximal inequality, Itô's isometry and the bound (4.20) we get

	whenever osc(V H ε t ≤ H ε 0 + osc(V ε ) + (4.19)	1 β	+	2β -1 ε	0	t	e -α(t-s) P ε s dW s .
	Recall that this led by a localization argument to the following bound (4.17):
	(4.20)				sup t≥0	E [H ε t ] ≤ E [H ε 0 ] + M +	1 β	.
	0	t	e -α(t-s) P ε s dW s =	0	t	e -α(t-s) dM s = M t -α	0	t	e -α(t-s) M s ds
					≤ |M t | + sup	|M s |
										s∈[0,t]
					≤ 2 sup	|M s | .
							s∈[0,T ]
	Plugging this in (4.19) yields
	(4.21)			sup t∈[0,T ]	H t ≤ H ε 0 + osc(V ε ) +	1 β	+	2β -1 ε	t∈[0,T ] 2 sup	|M t | .
	E sup 0≤t≤T	|M ε t | 2 ≤ 4E |M ε T | 2 = 4E	 	0	T	P ε s dW s	2	  = 4E	0	T	(P ε s ) 2 ds
					≤ 8T osc(V ε ) + sup t∈[0,T ]	E [H ε t ]
					≤ 8T 2 osc(V ε ) + E [H ε 0 ] +	1 β	.
	Injecting this in (4.21) and applying Cauchy-Schwarz inequality yields
	(4.22) E sup
	t∈[0,T ]
	(4.18)											1/2

ε

This work is partially supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement number 614492.


D Di

Senior Member, IEEE, I. Planes S Girard

Student Member, IEEE A Cebollada

G Li

A Alessi

Student Member, IEEE I Reghioua

C Cangialosi

Student Member, IEEE, S. Rizzolo A Ladaci

Student Member, IEEE V Lecoeuche

A Boukenter

A Champavère

Y Ouerdane

I Planes

S Rizzolo

Radiation Hardened Architecture of a Single-Ended Raman-Based Distributed Temperature Sensor

Keywords: Differential attenuation, distributed temperature sensor, irradiation, optical fiber, radiation hardened by design, raman, RHD-DTS. I

Raman-based Distributed Temperature Sensors (RDTS) allow performing spatially resolved (1 m) reliable temperature measurements over several km long Optical Fibers (OFs). These systems are based on the temperature dependence of the intensities of both the Stokes and anti-Stokes components of the Raman back-scattered signal. One of the specific issues associated with RDTS technology in radiation environments is the differential Radiation Induced Attenuation (RIA) between the two components that induces huge errors in the temperature evaluation. Such problem is particularly evident for commercially available single-ended DTS using one laser source. Doubleended configuration could be used to correct for the differential attenuation but are limited by RIA in terms of sensing range. In the present work, we show how a Radiation-Hardened-by-Design DTS (RHD-DTS) overcomes the observed radiation issues keeping the single-ended interrogation scheme. In the tested RHD-DTS two infrared excitation laser sources (∼1550 nm and ∼1650 nm) are employed: the wavelength of the Stokes component due to the first excitation source coincides with the wavelength of the second excitation; vice versa, the wavelength of the anti-Stokes component due to the second excitation source coincides with the wavelength of the first excitation. The overall result is that the two signal intensities are automatically corrected for the differential RIA all along the OF sensor length and the temperature measurements becomes robust against radiation effects. This study demonstrates the potential of such a sensor by reporting preliminary experimental results obtained with a prototype developed by Viavi Solutions exploiting radiationsensitive or radiation-hardened optical fibers.

nuclear power plants, research facilities. Indeed, OFs combine several remarkable properties: electromagnetic immunity, light weight, small sizes, good radiation tolerance and the possibility to be employed as several kilometers long distributed sensors. Raman-based Distributed Temperature Sensor (RDTS) fall within the latter category [START_REF] Dakin | Distributed optical fibre Raman temperature sensor using a semiconductor light source and detector[END_REF], [START_REF] Culshaw | Fiber-optic sensing: A historical perspective[END_REF]. This type of optoelectronic sensors combines the Optical Time Domain Reflectometry (OTDR) measurement technique to the detection of Stokes and anti-Stokes components of the Raman back-scattered light. The theory of Raman scattering relates the intensities of these two components [START_REF] Ferraro | Introductory Raman Spectroscopy[END_REF]. Once a calibration procedure is applied, in order to correct for the detection sensitivities, excitation laser power and initial differential attenuation, a silica-based OF can efficiently be employed as a distributed temperature sensor. However, when the sensing OF is exposed to radiation new optical losses are induced [START_REF] Girard | Radiation effects on silica-based optical fibers: Recent advances and future challenges[END_REF]. This well-known phenomenon is called Radiation Induced Attenuation (RIA) and is generally wavelength-dependent. As a consequence, the backscattered Stokes and anti-Stokes components generated by the same probe laser are not equally affected in the flight back to the detector, which causes the initial calibration to fail and the temperature measurement to be wrong [START_REF] Cangialosi | Development of a temperature distributed monitoring system based on Raman scattering in harsh environment[END_REF]. In order to mitigate such effect several approaches were adopted: employment of radiation resistant OFs [START_REF] Cangialosi | Development of a temperature distributed monitoring system based on Raman scattering in harsh environment[END_REF], applying an H 2 -loading pre-treatment to the OF sensor [START_REF] Cangialosi | Effects of radiation and hydrogen-loading on the performances of Raman-distributed temperature fiber sensors[END_REF], [START_REF] Girard | On-site regeneration technique for hole-assisted optical fibers used in nuclear facilities[END_REF], using a double-ended configuration system [START_REF] Fernandez | Radiation-tolerant Raman distributed temperature monitoring system for large nuclear infrastructures[END_REF], [START_REF] Toccafondo | Raman distributed temperature sensing at CERN[END_REF]. Although the latter system successfully manages to retrieve correct temperature measurements, in order to maintain the performances of a single-ended system, it requires the access to both ends of the OF, a twice longer OF length and, therefore, a twice larger optical budget.

In the present work we explore the potentialities of a singleended Raman-based DTS with two probe laser sources instead of one [START_REF] Bibby | Temperature measurement[END_REF]- [START_REF] Suh | Auto-correction method for differential attenuation in a fiber-optic distributed-temperature sensor[END_REF], which, to the best of our knowledge, is the first experimental evidence of the feasibility and efficiency of a Radiation-Hardened-by-Design DTS (RHD-DTS) based on such technology. In the RHD-DTS the wavelengths of the two excitation lasers are carefully chosen in such a way that their energy separation is close to the spectral shift of the main Raman band of silica (about 450 cm -1 ). The Stokes component excited by the shorter wavelength source has the same wavelength of the other source. Conversely, the anti-Stokes component generated by the higher wavelength source is located in the spectral domain of the shorter wavelength source. Therefore, the intensity traces of the Stokes and anti-Stokes components are automatically corrected for any possible differential RIA all along the OF length, as long as the RIA variation is negligible during the measurement time operation (typically a few minutes). It is worth mentioning that the same system also corrects for the losses due to bending, hydrogen diffusion inside the OF and all the factors that might alter the attenuation along the sensing OF. In the present work, we report a first study regarding irradiation tests during which the RHD-DTS was used in combination with a phosphorous doped single-mode OF (radiation sensitive) as well as a fluorine doped single-mode OF (radiation resistant). P-doped OF are among the most radiation sensitive OFs in the infrared domain [START_REF] Regnier | Low-dose radiation-induced attenuation at infrared wavelengths for P-doped, Ge-doped and pure silica-core optical fibres[END_REF]- [START_REF] Francesca | Radiation response of ce-codoped germanosilicate and phosphosilicate optical fibers[END_REF], therefore the success of our preliminary tests strongly supports the employment of the RHD-DTS in radiation environments for temperature or dose measurements.

II. EXPERIMENTAL DETAILS

A. Tested Optical Fiber

The tested singlemode OFs were produced by iXBlue Photonics Division through the Modified Chemical Vapor Deposition (MCVD) method [16]- [START_REF] Richard | Coupled theoretical and experimental studies for the radiation hardening of silica-based optical fibers[END_REF]. The samples, which are named iXSM-P and iXSM-F, are step-index single-mode OFs with an external cladding diameter of 125 μm. The iXSM-P has a pure silica cladding and a P-doped silica core, whereas the iXSM-F has low level of F-doping in the core and a higher one in the cladding.

B. The RHD-DTS System

The RHD-DTS system used for this study is a prototype produced by VIAVI Solutions France SAS [20]. A synoptic scheme of the system is shown in Fig. 1.

The RHD-DTS is equipped with two pulsed excitation lasers, at ∼1550 nm and ∼1650 nm. The 1550 nm laser is used for the detection of the Stokes traces, whereas the 1650 nm is used for the detection of the anti-Stokes ones. The system employs single-mode OFs as sensing element. The back-scattered Raman signals are generated by the laser pulses injected in the sensing OF. As shown in Fig. 1, the system uses adequate filters to isolate the Raman signal of interest. Therefore, Stokes and anti-Stokes measurements are performed in sequence. The RHD-DTS also authorizes performing the usual OTDR traces at both laser excitation wavelengths.

The ratio between the Stokes and anti-Stokes intensities is related to the absolute temperature via the following formula:

I Stokes I anti Stokes = ν 0 -ν m ν 0 + ν m 4 e hcνm kT ( 1
)
where ν 0 and ν m are the wavenumbers of both the excitation probe laser and the specific Raman band (main band of silica, for instance) respectively, his Planck's constant, cis the velocity of light, kis Boltzmann's constant, and T is the absolute temperature. In Raman-based DTS, eq. 1 is exploited for the determination of the temperature. In order to calibrate the instrument response, two references are provided to the RHD-DTS by placing two sections of the tested OF at two different controlled temperatures. In order to assure the stability of these two references, these parts of the OF sensor were coiled up and immerged in water inside isolating containers with well controlled temperatures. The remaining part of the OF can then be employed for the actual temperature measurement. The temperature is calculated by the following formula.

T (x) = A (x) + B ( 2
)
where T (x) is the temperature at position x along the OF, (x) is the difference between the Stokes and anti-Stokes traces (measured in dB) at position x. A and B are simply calibration constants determined on the basis of the two external temperature references. Assuming that no errors affect the A and B constants in relation (2), a relative error of the order of 1% affecting the variable would not cause relative errors exceeding 1% in the estimation of the absolute temperature. Moreover, for temperatures higher than room temperature, such error would be less than 0.5%.

Concerning the constant A and B, if they are affected by small errors, of 1% for example, they would produce certainly systematic errors in the measurement of the absolute temperature, which, however, would remain of the order of 1%.

Each temperature measurement lasted ∼1 min whereas the OTDR measurements could be performed in about 10 s with a good signal to noise ratio. The duration of the used laser pulses was fixed to 30 ns allowing a spatial resolution of about 3 m.

C. Experimental Procedure

The irradiations were performed with the Moperix X-ray machine of the Laboratoire Hubert Curien in St. Etienne (France). The X-ray tube was operated at 100 kV with generations of photons of ∼50 keV average energy.

As shown in Fig. 2 the RHD-DTS is connected to an OF which first goes inside the two (cold and hot) controlled external temperature systems. Afterwards, the fiber under test goes inside the irradiation chamber, where it is kept at room temperature (the temperature is monitored via thermocouples). Each of the three coils in Fig. 2 has a length of about 65 m and they are interconnected by splices. The temperature of the cold reference was ∼20 • C, whereas the hot one was ∼55 • C. Irradiations and RHD-TDS measurements were carried out in sequence in the case of the radiation sensitive P-doped OF and simultaneously in the case of the radiation resistant F-doped OF (online measurements). For the P-doped sample at each step, the deposited radiation dose was increased by a multiple of 2 in order to obtain an exponential increase of the radiation dose and span a large range of RIA losses (up to ∼125 dB/km). The irradiation dose-rate was set to ∼220 mGy(SiO 2 )/s. For the F-doped OF the RHD-TDS measurements were performed online during the irradiation, with a maximum deposited dose of about 240 kGy(SiO 2 ) and a dose-rate of 760 mGy(SiO 2 )/s.

III. EXPERIMENTAL RESULTS

A. OTDR Measurement and RIA in the P-Doped OF

In Fig. 3 we report the OTDR traces recorded during the experiment as the radiation dose is increased.

The first 15 m of the traces correspond to an internal OF located inside the instrument. The two following plateaus correspond to the two coils of OF that are kept at controlled temperatures and that served as references. The last segment is the irradiated part of the P-doped OF.

It is evident that the irradiation induces a strong variation of the slope of the traces at both 1550 and 1650 nm, this slope being directly representative of linear optical losses and the RIA at these two wavelengths.

In Fig. 4 we report the corresponding values as a function of the radiation dose as well as their differential attenuation.

As already mentioned, the P-doped OF employed in this investigation is very sensitive to irradiation, and a total radiation dose of only 35 Gy(SiO 2 ) is high enough to induce losses of the order of 100 dB/km in the spectral domain of interest. For the purpose of this investigation, it is also important to notice that the differential attenuation at 1550 and 1650 nm increases as a function of the dose as well.

B. RHD-DTS Measurements of the P-Doped OF

The Stokes and anti-Stokes trace evolutions with the radiation dose are reported in Fig. 5.

Similarly to Fig. 3, we can notice two plateaus corresponding to the two coils used as temperature references. In this case it is easy to recognize the Stokes and anti-Stokes components thanks to the inverted step occurring at the boundary between the cold and hot reference. Moreover, as in Fig. 3, the effect of irradiation on the third coil is very strong.

The temperature curves can be evaluated from the difference of the Stokes and anti-Stokes using eq. 2 (A and B constants were automatically calculated by the software of the RHD-DTS).

In Fig. 6 we finally report the RHD-DTS traces recorded for all the irradiation doses. All the traces overlap very well and we can consider that the RHD-DTS measurement is reliable up to radiation induced optical losses of ∼120 dB/km and differential attenuation of ∼5 dB/km.

C. RHD-DTS Measurements of the F-Doped OF

In contrast with the iXSM-P sample, in the case of the iXSM-F fiber a script was prepared that allowed the continuous acquisition of the RHD-DTS traces during the irradiation tests. In such experiment only the Stokes and anti-Stokes components were recorded, which are sufficient for the distributed temperature measurements. Fig. 7 shows the Stokes and anti-Stokes traces as a function of the radiation dose. Once again, these traces are composed by three sections: the first two are associated to the controlled temperature references; the third one corresponds to the irradiated part of the sensing OF. As expected, we observe an absolute increase of the slope of the traces in correspondence of the irradiated section, which is clearly due to the radiation induced attenuation. In the inset of the same figure we report the evolution of the RIA as a function of the dose, evaluated from the recorded traces. This represents the average RIA value at the two probing wavelengths, since both of them are employed in a single measurement either as excitation light or back-scattered Raman signal. In the inset we also added a point associated to the RIA at 240 kGy(SiO 2 ), the highest radiation dose we reached, which has been used in the postirradiation study (reported below).

In Fig. 8 we show the temperature traces as a function of the radiation dose.

The temperature measured at the irradiated sample increased with the dose, i.e. with time, during the irradiation. Such effect is due to an actual increase of temperature of the irradiation chamber. To prove it, in Fig. 9 we show the comparison between the temperature recorded by a thermocouple placed in proximity of the irradiated iXSM-F fiber and by the RHD-DTS. Clearly, the two temperature sensors measured the same temperature increased during the experiment. The temperature statistical error affecting the RHD-DTS curve is about ±0.2 • C, which may be sufficient for some applications. Although the thermocouple curve is significantly less noisy, we should take in consideration that the experiment was performed with a prototype instrument which will be further improved and optimized. The same effect was not observed during the tests with the iXSM-P sample because the deposited radiation dose was much lower with much shorter irradiation periods. Therefore, no temperature variations were induced inside the radiation chamber. Although the online experiment lasted up to the dose of 60 kGy(SiO 2 ), the actual irradiation went on until a total radiation dose of 240 kGy(SiO 2 ) was reached. A post-mortem study was carried out on the sample irradiated at the highest dose. Comparison between the temperature measurements inside the radiation chamber recorded by the RHD-DTS system and by a thermocouple placed in proximity of the iXSM-F fiber. In Fig. 10 we report the post-irradiation temperature trace of the irradiated iXSM-F connected to the two temperature references. In the inset of the same figure we highlight the zone associated to the OF under test. As expected, the temperature trace is constant along the irradiated fiber sample in agreement with the ambient conditions at the moment of the measurements.

The OTDR traces at 1550 and 1650 nm were also measured and are reported in Fig. 11. Performing the OTDR measurement at the two excitation wavelengths used by the RHD-DTS system allows to appreciate the significant differential RIA. In particular, after a total radiation dose of 240 kGy(SiO 2 ) the RIA values are 16.3 and 22.7 dB/km at 1550 nm and 1650 nm respectively. Therefore, the total differential RIA in the iXSM-F sample is 6.4 dB/km. RIA in the iXSM-P sample X-rays irradiated at a dose of 35 Gy(SiO2). induced defect which absorbs light at about 1.6 μm [START_REF] Regnier | Low-dose radiation-induced attenuation at infrared wavelengths for P-doped, Ge-doped and pure silica-core optical fibres[END_REF], [START_REF] Griscom | Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorus-doped silica glass and optical fibers[END_REF]. Such defect is called P1 and corresponds to an unpaired electron localized on a threefold coordinated P atom [START_REF] Griscom | Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorus-doped silica glass and optical fibers[END_REF]. An example of the infrared RIA spectrum of the iXSM-P OF is reported in Fig. 12. In this case we carried out an irradiation up to 35 Gy(SiO 2 ), in order to record a RIA spectrum as described in ref. [START_REF] Francesca | X-ray irradiation effects on fluorine-doped germanosilicate optical fibers[END_REF] that was induced during the testing of the RHD-TDS. Fig. 12 shows clearly the absorption band due to the P1 defect in the center of the spectral domain of interest. We notice that the RIA value at 1550 nm is higher than at 1650 nm and the values are in agreement with the result reported in Fig. 4. We should also mention that the position of the maximum of the absorption band due to the P1 is blueshifted compared to the true absorption band of this defect because of the guiding properties of the OF [START_REF] Regnier | Low-dose radiation-induced attenuation at infrared wavelengths for P-doped, Ge-doped and pure silica-core optical fibres[END_REF].

The choice of using a P-doped OF permitted to reach high RIA levels at fairly low radiation doses and to perform the whole experiment in a limited time. Indeed, at a radiation dose of only 35 Gy(SiO 2 ) we measured optical attenuations of about 120 dB/km. At the same time, we notice that a differential attenuation of 5 dB/km did not impact the measurement of temperature performed by the RHD-DTS. Previous studies showed that such a small differential attenuation is sufficient to produce large errors in the temperature measurement that increases with the probed distance if a classic single-ended Raman-based DTS is employed (one excitation wavelength), up to 30 • C on a 100 m long sensor [START_REF] Cangialosi | Development of a temperature distributed monitoring system based on Raman scattering in harsh environment[END_REF].

Finally, concerning the P-doped sample it is interesting to notice that the RIA in Fig. 4 is linearly related to the radiation dose under the investigated conditions (∼220 mGy(SiO 2 )/s and room temperature). As we suggested previously [START_REF] Girard | Feasibility of radiation dosimetry with phosphorus-doped optical fibers in the ultraviolet and visible domain[END_REF], [START_REF] Francesca | Radiation response of ce-codoped germanosilicate and phosphosilicate optical fibers[END_REF], such circumstance could be exploited to perform dose measurements. However, in the case of the RHD-DTS system, this would mean the possibility to perform simultaneous and online distributed dose and temperature measurements once an appropriate sensing OF is chosen (such as the P-doped one here investigated). It is worth to mention that such aspect is of major relevance for the dosimetry application, since it was recently proven that the radiation response of silica-based OF depends strongly on temperature of irradiation [START_REF] Girard | Combined high dose and temperature radiation effects on multimode silica-based optical fibers[END_REF] and that the RHD-DTS system would allow correcting for the irradiation temperature at each point of the OF sensor.

In order to really appreciate the potentiality of the RHD-TDS we should now consider that, by employing radiation resistance OFs, RIA values of ∼120 dB/km would only be reached after total radiation doses largely exceeding 1 MGy(SiO 2 ).

In Fig. 13 we report the infrared optical attenuation spectrum of the iXSM-F sample recorded after a total dose of 240 kGy(SiO 2 ). This is a post-irradiation spectrum which was measured by means of the usual cut-back technique. The inset of Fig. 13 highlights the spectral region of interest. The optical attenuation values recorded at 1550 nm and 1650 nm match very well the ones measured via the OTDR as shown in Fig. 11.

In contrast with the P-doped sample, no clear absorption band can be detected in the F-doped OF, with the exception of the well-known OH signal at 1.38 μm, which however does not interfere with the RHD-DTS measurements. Since the differential RIA is caused by the tail of the infrared attenuation, in the iXSM-F sample the attenuation value is higher at 1650 nm than 1550 nm, in contrast with the previous example of the iXSM-P OF. As shown in Fig. 9, the online measurements carried out by means of the RHD-DTS are in excellent agreement with the ones reported by a standard thermocouple. On one hand our preliminary study shows that RHD-DTS overcome the difficulties associated with classical RDTS systems, which are affected by the differential attenuation, as well as the double-ended ones, reduced sensor length. On the other end, since the instrument we employed was a prototype we can imagine to further optimize it in order to achieve higher accuracies than 0.2 • C.

As differential attenuation does not represent a limitation for the RHD-DTS, we can easily imagine that the system developed by Viavi Solutions will be well suited for a large number of applications in nuclear environments. In addition to its intrinsic radiation resistance, the radiation hardened architecture is robust against radiation effects as well as to bending losses that may occur during fiber deployment in large facilities.

V. CONCLUSION

We have validated the feasibility of a Radiation-Hardenedby-Design DTS by experimentally testing the proposed sensor architecture under radiation. The RHD-DTS works in a singleended interrogation configuration and employs two suitably chosen excitation probe lasers. In particular, we have shown that the usual differential attenuation induced by radiation does not represent an issue for the RHD-DTS since the measured Stokes and anti-Stokes components are automatically corrected for such effect. By employing a radiation resistant OF, we have also highlighted the potentiality of this system to be employed in temperature monitoring applications having to operate in harsh environment at MGy(SiO 2 ) dose levels. Furthermore, the results reported in this paper on a radiation sensitive OF also suggest that such sensor can be used to perform simultaneous online distributed dose and temperature measurements on a unique optical fiber.

Fig. 1 .

 1 Fig. 1. Synoptic scheme of the RHD-DTS system.

Fig. 2 .

 2 Fig. 2. Schematic representation of the experimental setup that was used for the irradiation test.

Fig. 3 .

 3 Fig. 3. Evolution of the OTDR traces at 1550 nm (left panel) and 1650 nm (right panel) with the radiation dose.

Fig. 4 .

 4 Fig. 4. Growth of the RIA at 1550 and 1650 nm with the irradiation dose. The differential RIA (RIA1550-RIA1650) is also reported.

Fig. 5 .

 5 Fig. 5. Evolution with the radiation dose of the measured Stokes and anti-Stokes components in the P-doped OF.

Fig. 6 .

 6 Fig. 6. RHD-DTS temperature traces for all the radiation doses.

Fig. 7 .

 7 Fig. 7. Evolution with the radiation dose of the measured Stokes and anti-Stokes components in the iXSM-F sample. In the inset the average RIA is plotted as a function of the dose.

Fig. 8 .

 8 Fig. 8. Temperature traces in the iXSM-F sample as a function of the radiation dose.

Fig. 9 .

 9 Fig. 9.

Fig. 10 .

 10 Fig. 10. Post-mortem RHD-DTS temperature trace recorded on the iXSM-F irradiated at 240 kGy(SiO2).

  IV. DISCUSSION P-doped silica-based OFs are among the most sensitive OFs in the infrared domain because of a well-known radiation

Fig. 11 .

 11 Fig. 11.Post-mortem OTDR traces of the iXSM-F OF irradiated at 240 kGy(SiO2).

Fig. 12 .

 12 Fig. 12.RIA in the iXSM-P sample X-rays irradiated at a dose of 35 Gy(SiO2).

Fig. 13 .

 13 Fig. 13. Post-mortem optical attenuation measurement of the iXSM-F OF irradiated at a dose of 240 kGy(SiO2).