Lucie Baudouin
email: lucie.baudouin@laas.fr

Masahiro Yamamoto

Inverse problem on a tree-shaped network: unified approach for uniqueness

Keywords: networks, inverse problem, Carleman estimate AMS subject classifications: 35R30, 93C20, 34B45

In this article, we prove uniqueness results for coefficient inverse problems regarding wave, heat or Schrödinger equation on a tree-shaped network, as well as the corresponding stability result of the inverse problem for the wave equation. The objective is the determination of the potential on each edge of the network from the additional measurement of the solution at all but one external end points. Several results have already been obtained in this precise setting or in similar cases, and our main goal is to propose a unified and simpler method of proof of some of these results. The idea which we will develop for proving the uniqueness is to use a more traditional approach in coefficient inverse problems by Carleman estimates. Afterwards, using an observability estimate on the whole network, we apply a compactness-uniqueness argument and prove the stability for the wave inverse problem.

Introduction and main results

Systems known as multi-link structures have large applicative relevance and their mathematical study attracts a growing interest. The modeling of the evolution of these structures is generally quite complex and can be often described with systems of partial differential equations on networks or graphs. As large as the literature on this subject can be (see e.g. [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] and the references therein), from a mathematical and scientific point of view, there are still many necessary works needed to complete the theory. Regarding more specific topics such as the control and inverse problems on networks, a full theory cannot be developed by simply superposing the existing results for partial differential equations on one-dimensional domains. Indeed, the interaction between the different components of a multi-link structure may generate unexpected phenomena, as [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] pointed out mainly for vibrations on networks.

Concerning other operators on metric graphs, they can arise in many applications, e.g., chemistry or engineering, and we refer to high-temperature granular superconductors [START_REF] Alexander | Superconductivity of networks. a percolation approach to the effects of disorder[END_REF] as one example. One can look deeper in the review papers [START_REF] Exner | A general approximation of quantum graph vertex couplings by scaled schrödinger operators on thin branched manifolds[END_REF], [START_REF] Kuchment | Graph models for waves in thin structures[END_REF] (or even in the introduction of [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF]) for more information on this topic.

Besides, inverse problems have been studied only recently on networks, and we refer to the articles [START_REF] Belishev | Boundary spectral inverse problem on a class of graphs (trees) by the BC method[END_REF] and [START_REF] Avdonin | On an inverse problem for tree-like networks of elastic strings[END_REF] for uniqueness results with the boundary control approach, or in [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] and [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] for uniqueness and stability results with Carleman estimates.

Here we discuss the same type of inverse problems for the wave and the Schrödinger equations as in [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] and [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF], while for the heat equation, our inverse problem concerns the initial-boundary value problem and is different from the case of the heat equation treated in [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] where initial values are also unknown in the inverse problem. Our purpose is to propose a unified and simpler method to prove some of these results by using previously known results for the inverse problem of determining a coefficient.

Let us now give the detailed mathematical framework within which we will work. Let Λ be a tree-shaped network composed of N + 1 open segments (ej)j=0,1,...,N of length ℓj, linked by N1 internal node points belonging to the set Π1 and let us denote by Π2 the set of N2 exterior end points where only one segment starts. By a "tree-shaped network", we mean that Λ does not contain any closed loops, and and one can see for instance the network pictured in Figure 1. Moreover we assume that at each interior node point, at least three segments ej meet.

We define any function on the network f : Λ → R by f = (f0, ..., fN ), where fj = f |e j denotes the restriction of f to the edge ej. Moreover, for any internal node P ∈ Π1 where nP segments, say e1, ..., en P , meet, we set We consider on this plane 1-d tree-shaped network Λ either wave or heat or even Schrödinger equations, with a different potential term x → pj(x) in L ∞ (ej) on each segment.

Our first system of interest is the following 1-d wave equation on the network Λ:

   ∂ 2 t uj -∂ 2
x uj + pj(x)uj = 0 ∀j ∈ {0, 1, ..., N }, (x, t) ∈ ej × (0, T ), u(Q, t) = h(t), ∀Q ∈ Π2, t ∈ (0, T ), u(x, 0) = u 0 (x), ∂tu(x, 0) = u 1 (x),

x ∈ Λ, [START_REF] Alexander | Superconductivity of networks. a percolation approach to the effects of disorder[END_REF] assuming some compatibility condition between the boundary and initial data. Moreover we assume the continuity and what is called the Kirchhoff law at any internal node P ∈ Π1, which are given by uj (P, t) = u k (P, t) =: u(P, t), ∀j, k ∈ {1, ..., nP } , 0 < t < T,

[ux(t)] P := n P j=1 ∂n e uj(P, t) = 0, 0 < t < T.

Here and henceforth we choose an orientation of Λ such that to two endpoints of each segment e, correspond an initial node I(e) and a terminal node T (e), and further define the outward normal derivative ∂n e uj at a node P of ej by ∂n e uj (P, t) = -∂xuj (P, T ), if P ∈ I(ej), ∂xuj (P, T ), if P ∈ T (ej).

Since one can prove the unique existence of solution to (1) -(3) in a suitable function space (from Lions and Magenes [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF] or more explicitly in [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF]Chapter 3], and see also Lemma 1 below), we denote the solution by u[p](x, t), and we set u[p] = (u[p]0, ..., u[p]N ).

Moreover we consider the following heat system on the same network Λ

   ∂tuj -∂ 2 x uj + pj(x)uj = 0 ∀j ∈ {0, 1, ..., N }, ∀(x, t) ∈ ej × (0, T ), ∂xu(Q, t) = 0, ∀Q ∈ Π2, ∀t ∈ (0, T ), u(x, 0) = u 0 (x), ∀x ∈ Λ, (4)
and the Schrödinger system on the network Λ

   i∂tuj -∂ 2 x uj + pj(x)uj = 0 ∀j ∈ {0, 1, ..., N }, ∀(x, t) ∈ ej × (0, T ), u(Q, t) = h(t), ∀Q ∈ Π2, ∀t ∈ (0, T ), u(x, 0) = u 0 (x), ∀x ∈ Λ, (5)
both under the same node conditions (2) and (3). Here and henceforth we set i = √ -1. If there is no possible confusion, by the same notation u[p] we denote the solution to (4) or ( 5) under (2) and (3). Again the uniqueness, existence and regularity of solutions to these systems can be deduced by a standard method (see e.g. [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]).

This article focuses on results on networks concerning the following question.

Inverse Problem: Is it possible to retrieve the potential p everywhere in the whole network Λ from measurements at all external nodes except one? As it will be clear when reading the proof, we cannot expect the uniqueness by data ∂xu[p](Q, t) at external nodes Q except more than one. This inverse problem is nonlinear and we will give here the proof of the uniqueness of the solution with an argument which do not use a global Carleman estimate. Very recent papers on coefficient inverse problems on networks, as Baudouin, Crépeau and Valein [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] for the wave equation, and Ignat, Pazoto and Rosier [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] for the heat and the Schrödinger equations, give indeed the stability and therefore the uniqueness in the determination of the potential from boundary measurements using appropriate global Carleman estimates.

For the wave, the heat and the Schrödinger equations, our method yields the same uniqueness results as in the existing works, and we basically want to give a technically different approach of the same questions. However, our method is widely applicable to other equations such as a system of beam equations on networks and can yields novel uniqueness results. The question of the proof of the Lipschitz stability in the case of the wave equation will be addressed afterwards, using a compacteness-uniqueness argument, and relies on the observability estimate on the whole network which was already proved in the literature in several situations. However, this Lipschitz stability result given in Theorem 4 will require more regularity of the solutions than [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF], as a consequence of the use of the compactness-uniqueness argument.

Concerning the precise topic we are considering, the bibliography lies in two different domains, namely coefficient inverse problems for partial differential equation on the one hand and control and stabilization in networks on the other hand.

Actually, as the first answer to the uniqueness for a coefficient inverse problem with a single measurement, we refer to Bukhgeim and Klibanov [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF]. Later on, the works of Klibanov [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF] and Yamamoto [START_REF] Yamamoto | Uniqueness and stability in multidimensional hyperbolic inverse problems[END_REF] are also important references related to our topic. Here we do not intend to give an exhaustive list of references but still, we can mention a recent comprehensive survey [START_REF] Klibanov | Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems[END_REF]. After the proof of uniqueness using the basic 1-d result on the basis of local Carleman estimates, the idea beneath this article is to take advantage of an observability estimate to obtain the Lipschitz stability of the inverse problem with a compactness-uniqueness argument (see [START_REF] Yamamoto | Uniqueness and stability in multidimensional hyperbolic inverse problems[END_REF]). We can refer to Isakov [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] which addresses other techniques linked to the study of inverse problems for several partial differential equations.

Nowadays, many results on the stability of inverse problems are derived directly from global Carleman estimates (e.g., [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] and [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF]) for networks, or also in [START_REF] Baudouin | A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem[END_REF] for the related case of a hyperbolic transmission equation. We should underline that this "global" method goes back to [START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF] and [START_REF] Imanuvilov | Global uniqueness and stability in determining coefficients of wave equations[END_REF] for the wave equation. One should also know that studies on inverse problems and controllability of partial differential equations share some technical materials such as Carleman estimates and observability inequalities. In the particular network setting, we would like to make use of classical results such as well-known 1-d local Carleman estimates and observability estimates on the network, borrowed from control studies, in order to obtain first uniqueness and then stability results, using a compactness-uniqueness argument (e.g. [START_REF] Yamamoto | Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns[END_REF], [START_REF] Puel | Generic well-posedness in a multidimensional hyperbolic inverse problem[END_REF]).

Besides, the control, observation and stabilization problems of networks have been objects of recent and intensive researches such as e.g., Dáger and Zuazua [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF], Lagnese, Leugering and Schmidt [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF], Zuazua [START_REF] Zuazua | Control and stabilization of waves on 1-d networks[END_REF]. More specifically, the control being only applied at one single end of the network, the articles Dáger [START_REF] Dáger | Observation and control of vibrations in tree-shaped networks of strings[END_REF], Dáger and Zuazua [START_REF] Dáger | Controllability of star-shaped networks of strings[END_REF][START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] prove controllability results for the wave equation on networks, using observability inequalities under assumptions about the irrationality properties of the ratios of the lengths of the strings. We can also underline that many results of controllability on networks concern only the wave equation without lower order terms (see [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF] and [START_REF] Schmidt | On the modelling and exact controllability of networks of vibrating strings[END_REF] for instance). However, it is difficult to consider such measurements at a more limited number of nodes for the inverse problem and we do not consider the measurements at less external nodes than Π2 \ {QN 2 }.

In the sequel, we shall use the following notations:

L γ (Λ) = {f ; fj ∈ L γ (ej), ∀j ∈ {0, 1, ..., N }} , γ ≥ 1, H 1 0 (Λ) = f ; fj ∈ H 1 (ej), ∀j ∈ {0, 1, ..., N }, fj (P ) = f k (P )
if ej and e k meet at P , ∀P ∈ Π1, and f (Q) = 0, ∀Q ∈ Π2 .

For shortness, for f ∈ L 1 (Λ), we often write,

Λ f dx = N j=0 e j fj (x)dx,
where the integral on ej is oriented from I(ej) to T (ej). Then the norms of the Hilbert spaces L 2 (Λ) and H 1 0 (Λ) are defined by

f 2 L 2 (Λ) = Λ |f | 2 dx and f 2 H 1 0 (Λ) = Λ |∂xf | 2 dx.
We are ready to state our first main result:

Theorem 1 (Uniqueness) Let r > 0 be an arbitrary constant. Assume that p, q ∈ L ∞ (Λ) and the initial value u 0 satisfies |u 0 (x)| ≥ r > 0, a.e. in Λ.

Assume further that the solutions u[p], u[q] of (1)-( 2)-(3) belong to

C 1 ([0, T ]; H 1 (Λ)) ∩ C 2 ([0, T ]; L 2 (Λ)).
Then there exists sufficiently large T0 > 0 such that for all T ≥ T0, if

∂xu[p](Q, t) = ∂xu[q](Q, t) for each t ∈ (0, T ) and Q ∈ Π2 \ {QN 2 },
then we have p = q in Λ.

The proof of this result in Section 2 relies on the uniqueness for the determination of potential in the one-dimensional wave equation and a "removing" argument.

Our argument gives the uniqueness for the inverse problems of determination of potentials on tree-shaped networks also for the heat and the Schrödinger equations using only measurements at N2 -1 exterior end points. In fact, our arguments in proving the uniqueness for the wave and the Schrödinger equations are essentially the same and are based on local Carleman estimates, while the uniqueness for the inverse heat problem is reduced to the uniqueness for the corresponding inverse wave problem (in a sense to be detailed later).

Theorem 2 (Uniqueness for the heat inverse problem) Assume that p, q ∈ L ∞ (Λ), the initial value u 0 satisfies |u 0 (x)| ≥ r > 0, a.e. in Λ for some constant r, and the solutions u[p] and u[q] to (4)-( 2)-(3) satisfy

∂ j t u[p], ∂ j t u[q] ∈ H 1 (0, T ; L 2 (Λ)) ∩ L 2 (0, T ; H 2 (Λ)), j = 0, 1.
Then there exists T > 0 such that if

u[p](Q, t) = u[q](Q, t) for each t ∈ (0, T ) and Q ∈ Π2 \ {QN 2 },
then we have p = q in Λ.

We note that our inverse heat problem is an inverse problem for the initial-boundary value problem and is different from Ignat, Pazoto and Rosier [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] where an initial value is known but value at positive time is given. In general dimensions, the uniqueness for the inverse problem of determining coefficients in a parabolic equation by overdetermining boundary data on arbitrary subboundary is an open problem if we are not given u(x, t0) over the whole domain at t0 > 0. In other words, even the uniqueness the inverse heat problem for the classical initial-boundary value problem is not solved except for the measurement subboundary is a sufficienly large part of the whole boundary, because one cannot have relevant Carleman estimates for the heat equation for t ≥ 0.

Theorem 3 (Uniqueness for the Schrödinger inverse problem) Assume that p, q ∈ L ∞ (Λ), the initial value u 0 satisfies |u 0 (x)| ≥ r > 0, a.e. in Λ for some constant r, and the solutions u[p] and u[q] to (5)-( 2)-( 3) satisfy

∂ j t u[p], ∂ j t u[q] ∈ H 1 (0, T ; L 2 (Λ)) ∩ L 2 (0, T ; H 2 (Λ)), j = 0, 1.
Then there exists T > 0 such that

∂xu[p](Q, t) = ∂xu[q](Q, t) for each t ∈ (0, T ) and Q ∈ Π2 \ {QN 2 },
then we have p = q in Λ.

In Theorems 2 and 3, we may be able to relax the regularity condition for u[p] and u[q], but we do not want to go into further details in this article.

One can refer to [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] for the same inverse problem in the wave equation on a network where the proof is detailed in a star-shaped network but is actually generalizable to tree-shaped networks. The paper [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF] treats the Schrödinger case in a star-shaped network and needs measurements at all external nodes. We do not know any uniqueness result for non-tree graphs, which are graphs containing a closed cycle.

For M ≥ 0, we introduce the set

L ∞ M (Λ) = q = (q0, ..., qN ); qj ∈ L ∞ (ej), ∀j ∈ {0, 1, ..., N } such that q L ∞ (Λ) ≤ M .
For the inverse problem in the wave equation case, we state Theorem 4 (Stability) Let M > 0, r > 0 and K > 0 be arbitrarily fixed constants. Assume that p ∈ L ∞ M (Λ) and the solutions u[p] and u

[q] to (1)-(2)-(3) satisfy u[p], u[q] ∈ C 1 ([0, T ]; H 1 (Λ)) ∩ C 2 ([0, T ]; L ∞ (Λ)) with u C 1 ([0,T ];H 1 (Λ))∩C 2 ([0,T ];L ∞ (Λ)) ≤ K. ( 6
)
Assume also that the initial data u 0 satisfies

|u 0 (x)| ≥ r > 0, a.e. in Λ.
Then there exists sufficiently large T0 > 0 such that for all T ≥ T0, there exists a constant

C = C(T, M, r, K, Λ) > 0 such that ||q -p|| L 2 (Λ) ≤ C N 2 -1 j=1 ∂xu[p]j (Qj) -∂xu[q]j (Qj) H 1 (0,T ) . (7)
One can notice that for our proof based on a compactness-uniqueness argument, we assume a little more regularity than in Theorem 1.

This paper is composed of five sections. The proof of uniqueness in the inverse problem in the wave equation case (Theorem 1) is presented in Section 2. Section 3 is devoted to the proofs of Theorems 2 and 3. Theorem 4 is finally proven in Section 5 by a compactness-uniqueness argument and an observability estimate on the whole network.

We conclude this section with a classical result on the existence and regularity of solutions of the wave system and provide the corresponding energy estimates for the solution which we will need later.

Lemma 1 Let Λ be a tree-shaped network and assume that p

∈ L ∞ M (Λ), g ∈ L 1 (0, T ; L 2 (Λ)), u 0 ∈ H 1 0 (Λ) and u 1 ∈ L 2 (Λ).
We consider the 1-d wave equation on the network with the conditions (2) and (3):

           ∂ 2 t u -∂ 2 x u + p(x)u = g(x, t), in Λ × (0, T ), u(Q, t) = 0, in (0, T ), Q ∈ Π2, uj(P, t) = u k (P, t), in (0, T ), P ∈ Π1, j, k ∈ {1, ..., nP }, [∂xu(t)] P = 0, in (0, T ), P ∈ Π1, u(0) = u 0 , ∂tu(0) = u 1 , in Λ. ( 8
)
The Cauchy problem is well-posed and equation (8) admits a unique weak solution

u ∈ C([0, T ]; H 1 0 (Λ)) ∩ C 1 ([0, T ]; L 2 (Λ)).
Moreover there exists a constant C = C(Λ, T, M ) > 0 such that for all t ∈ (0, T ), the energy

E(t) = ||∂tu(t)|| 2 L 2 (Λ) + ||∂xu(t)|| 2 L 2 (Λ) , 0 ≤ t ≤ T of the system (8) satisfies E(t) ≤ C ||u 0 || 2 H 1 0 (Λ) + ||u 1 || 2 L 2 (Λ) + g 2 L 1 (0,T,L 2 (Λ)) , 0 ≤ t ≤ T (9)
and we also have the following trace estimate

N 2 j=1 ∂xuj (Qj) 2 L 2 (0,T ) ≤ C ||u 0 || 2 H 1 0 (Λ) + ||u 1 || 2 L 2 (Λ) + ||g|| 2 L 1 (0,T,L 2 (Λ)) . (10)
The proof of the unique existence of solution to equation ( 8) can be deduced from [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF], vol.1, Chapter 3 or [23, Chapter 3].

Estimate ( 9) is a classical result which can be formally obtained by multiplying the main equation in ( 8) by ∂tuj , summing up for j ∈ {0, ..., N } the integral of this equality on (0, T ) × ej and using some integrations by parts. Estimate ( 10) is a hidden regularity result which can be obtained by multipliers technique (we refer to [25, Chapter 1]). Formally, for the particular case of a star-shaped network of vertex P = 0 for example, it comes from the multiplication of (8) by m(x)∂xuj, where m ∈ C 1 ( Λ) with m(0) = 0 and mj (lj) = 1, summing up the integrals of this equality on (0, T ) × (0, lj) over j ∈ {0, ..., N } and using integrations by parts. Here we omit the details and we concentrate on the inverse problems.

Uniqueness of the inverse problem -wave network case

As already evoked in the introduction, the proof of Theorem 1 will use a well-known uniqueness result for the inverse problem by a local Carleman estimate. We state it in the following lemma.

Lemma 2 Let r > 0, p ∈ L ∞ (0, ℓ) and T > ℓ. Let f ∈ L 2 (0, ℓ), R ∈ H 1 (0, T ; L ∞ (0, ℓ)) and ∂tR ∈ L ∞ ((0, ℓ) × (0, T )) and assume that |R(x, 0)| ≥ r > 0 a.e. in (0, ℓ). Let y satisfy ∂ 2 t y -∂ 2 x y + p(x)y = f (x)R(x, t), (x, t) ∈ (0, ℓ) × (0, T ), y(x, 0) = 0, ∂ty(x, 0) = 0, x ∈ (0, ℓ), (11)
Assume further that

y ∈ C 1 ([0, T ]; H 1 (0, ℓ)) ∩ C 2 ([0, T ]; L 2 (0, ℓ)). (12)
Therefore, if y(0, t) = ∂xy(0, t) = 0 for all t ∈ (0, T ), then we have f ≡ 0 in (0, ℓ). Moreover if 0 < T1 < Tℓ, then we obtain y(x, t) = 0 for 0 < x < ℓ and 0 < t < T1.

This lemma is a classical uniqueness result for the inverse source problem in a wave equation and the proof can be done by the method in [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] on the basis of a Carleman estimate and the even extension of y to negative times t. We further refer to Imanuvilov and Yamamoto [START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF], [START_REF] Imanuvilov | Global uniqueness and stability in determining coefficients of wave equations[END_REF], Klibanov [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF], Klibanov and Timonov [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF] for example, and for completeness, we prove it in Appendix A.

Proof of Theorem 1. We define the following operation of "removing" segments from the tree-shaped network Λ, starting from all the external nodes except for one where we make measurements. We divide the proof into three steps.

Step 1. From Lemma 2, we can easily prove that if ej is a segment of Λ which ends at an external node Qj ∈ Π2, and if the solutions u[p] and u[q] to (1) satisfy ∂xu[p](Qj , t) = ∂xu[q](Qj, t) for all t ∈ (0, T ), then p = q on the segment ej and u[p](x, t) = u[q](x, t) for all x ∈ ej and all t ∈ (0, T1), for T1 ∈ (0, Tℓ).

Indeed, if we set y = u[p]j -u[q]j , then    ∂ 2 t y -∂ 2 x y + pj(x)y = (qj -pj)(x)u[q]j(x, t) (x, t) ∈ (0, ℓ) × (0, T ), y(Qj, t) = 0, t ∈ (0, T ), y(x, 0) = 0, ∂ty(x, 0) = 0, x ∈ (0, ℓ), (13)
and noting that T > 0 is sufficiently large, we can apply Lemma 2 on ej since ∂xy(Qj, t) = 0 for all t ∈ (0, T ). Moreover we note that u

[q]j ∈ C 1 ([0, T ]; H 1 (ej)) ⊂ H 1 (0, T ; L ∞ (ej)) and ∂tu[q]j ∈ C([0, T ]; H 1 (ej )) ⊂ L ∞ (ej × (0, T )
) by the Sobolev embdedding, and we also note |u 0 j (x)| ≥ r > 0 on ej, so that R := u[q]j satisfies the required assumptions. Thus, Lemma 2 allows to obtain pj ≡ qj on ej and consequently u[p]j (x, t) = u[q]j (x, t) in ej × (0, T1).

Therefore, for any segment e with the end points P and Q such that Q ∈ Π2 \ {QN 2 }, we see that p = q on e and (u[p]|e)(P, t) = (u[q]|e)(P, t), (∂xu[p]|e)(P, t) = (∂xu[q]|e)(P, t) for 0 < t < T1.

Let Π 2 1 be all the interior node points P of segments of Λ having their other end point in Π2 \ {QN 2 }. We note that Π 2 1 ⊂ Π1. Applying the above argument to all the exterior end points except for QN 2 , we have u[p]j (P, t) = u[q]j (P, t), ∂xu[p]j (P, t) = ∂xu[q]j (P, t) for each P ∈ Π 2 1 , 0 < t < T1 and j ∈ {1, ..., N3}. Here by e1, ..., eN 3 , we enumerate the segments connecting a point in Π 2 1 and a point in Π2 \ {QN 2 }.

Step 2. Let P ∈ Π1 be a given node such that nP segments, say, e1, ..., en P meet at P and e1, ..., en P -1 connect P with exterior end points, say, Q1, ..., Qn P -1 ∈ Π2 and

u[p]j (P, t) = u[q]j (P, t), ∂xu[p]j (P, t) = ∂xu[q]j (P, t), j ∈ {1, ..., nP -1}, 0 < t < T1. ( 14
)
Using the continuity (2) and the Kirchhoff law (3) at node P , we can deduce that u[p]n P (P, t) = u[q]n P (P, t), ∂xu[p]n P (P, t) = ∂xu[q]n P (P, t), 0 < t < T1.

Step 3. Let Λ 2 be the graph generated from Λ by removing e1, ..., eN 3 . Since T0 > 0 is sufficiently large by assumption in Theorem 1, when applying the same argument as in Step 1 to the graph Λ 2 , the corresponding time T1 > 0 just adds up to the largest one induced by Step 2. Thus we can repeat this operation to obtain the sets Λ 3 , and then Λ 4 ,..., Λ n . Hence, let L k be the set of all the open segments of Λ k , Π k 1 the set of the interior node points of Λ k , Π k 2 the set of external endpoints of Λ k . Setting Λ 1 = Λ, we note that

L 1 = {e0, ..., eN }, Π 1 1 = {P1, ..., PN 1 }, Π 1 2 = {Q1, ..., QN 2 }.
By ( 2) and ( 3), we see that

Π k-1 1 ⊃ Π k 1 , ∀k ∈ N and Λ k = L k ∪ Π k 1 ∪ Π k 2 , L k ∩ Π k 1 = L k ∩ Π k 2 = Π k 1 ∩ Π k 2 = ∅, ∀k ∈ N.
In order to complete the proof, it is sufficient to prove there exists n ∈ N such that Λn = ∅.

(

) 15
Assume contrarily that Λn = ∅ for all n ∈ N. Since every segment with exterior end point in Π2 \ {QN 2 }, can be removed (meaning that u[p] = u[q] on the segment) by the above operation, we obtain that there exists n0 ∈ N such that Λn 0 = L n 0 ∪ Π n 0 1 , i.e., Π n 0 2 = ∅. Then Λn 0 must be a closed loop since it possesses no external end points. By the assumption, there exist no closed loops in a tree-shape network. This is a contradiction and thus the proof of [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF], and therefore, the one of Theorem 1 is completed.

3 Uniqueness for the inverse problem -Schrödinger and heat network cases 3.1 Proof of Theorem 2 -Heat case.

We apply an argument similar to the proof of Theorem 4.7 in [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF] which is based on the reduction of the inverse heat problem to an inverse wave problem by a kind of Laplace transform called the Reznitzkaya transform (e.g., [START_REF] Isakov | Inverse problems for partial differential equations[END_REF], [START_REF] Lavrent'ev | skiȋ. Ill-posed problems of mathematical physics and analysis[END_REF], [START_REF] Romanov | Inverse problems of mathematical physics[END_REF]).

First we define an operator ∆Λ in L 2 (Λ) by ∆Λu = ∂ 2 x uj in ej , for all j ∈ {0, 1, ...., N } with

D(∆Λ) = u = (u0, ..., uN ); uj ∈ H 2 (ej), ∂xu(Q) = 0 for Q ∈ Π2, uj satisfying (2) and (3) .
Here, ej is oriented from I(ej) to T (ej) when defining ∂ 2

x . Then, similarly to [START_REF] Ignat | Inverse problem for the heat equation and the Schrödinger equation on a tree[END_REF], we can prove that ∆Λ is self-adjoint and (∆Λu, u) L 2 (Λ) := N j=0 (∂ 2 x uj , uj ) L 2 (e j ) ≤ 0. Therefore ∆Λ generates an analytic semigroup e t∆ Λ , t > 0 (e.g., Pazy [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], Tanabe [START_REF] Tanabe | Equations of evolution[END_REF]).

Since p ∈ L ∞ (Λ), the perturbed operator ∆Λp generates an analytic semigroup (e.g., Theorem 2.1 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], p.80). Therefore by the semigroup theory (e.g., [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], [START_REF] Tanabe | Equations of evolution[END_REF]), we know that the solutions u[p](x, t) and u[q](x, t) of equation ( 4) are analytic in t for any fixed x ∈ Λ. More precisely, u[p], u[q] : (0, ∞) -→ H 2 (Λ) are analytic in t > 0.

By u H [p] we denote the solution of the heat system (4) and by u H [q] the corresponding solution when the potential is q. By the analyticity in t and the assumption in the theorem, we have

u H [p](Q, t) = u H [q](Q, t), ∀Q ∈ Π2 \ {QN 2 }, ∀t > 0. ( 16
)
On the other hand, we denote by u[p] the solution of the wave system

   ∂ 2 t uj -∂ 2 x uj + pj(x)uj = 0, ∀j ∈ {0, 1, ..., N }, ∀(x, t) ∈ ej × (0, ∞), ∂xu[p](Q, t) = 0, ∀Q ∈ Π2, ∀t ∈ (0, ∞), u[p](x, 0) = 0, ∂tu(x, 0) = u 0 (x), ∀x ∈ Λ (17)
and by u[q] the corresponding solution when the potential is q. Then we obtain (e.g., [24, pp.251

- 252]) that 1 2 √ πt 3 ∞ 0 τ e -τ 2 4t u[p](x, τ )dτ
satisfies [START_REF] Baudouin | Global Carleman estimates for waves and applications[END_REF]. The uniqueness of solution to equation ( 4) implies

u H [p](x, t) = 1 2 √ πt 3 ∞ 0 τ e -τ 2 4t u[p](x, τ )dτ, ∀x ∈ Λ, ∀t > 0
and the same equality with q. By assumption [START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF], we obtain

1 2 √ πt 3 ∞ 0 τ e -τ 2 4t ( u[p] -u[q])(Q, τ )dτ = 0, ∀Q ∈ Π2 \ {QN 2 }, ∀t > 0.
By the change of variables s = 1 4t and τ 2 = η, we obtain

∞ 0 e -sη ( u[p] -u[q])(Q, √ η)dη = 0, ∀Q ∈ Π2 \ {QN 2 }, ∀s > 0
and the injectivity of the Laplace transform yields

( u[p] -u[q])(Q, √ η) = 0, ∀Q ∈ Π2 \ {QN 2 }, ∀η > 0. ( 18
)
Applying the same argument as in Section 2 for the wave system, we prove p = q in Λ. Thus the proof of Theorem 2 is completed.

Proof of Theorem 3 -Schrödinger case.

It is sufficient to prove the following lemma.

Lemma 3 Let r > 0 and p ∈ L ∞ (0, ℓ), f ∈ L 2 (0, ℓ) be real-valued, and T > 0 be arbitrarily fixed. We consider a 1-d Schrödinger equation:

   i∂ty -∂ 2 x y + p(x)y = f (x)R(x, t), ∀(x, t) ∈ (0, ℓ) × (0, T ), y(0, t) = 0, ∀t ∈ (0, T ), y(x, 0) = 0, ∀x ∈ (0, ℓ),
where R ∈ H 1 (0, T ; L ∞ (0, ℓ)), ∂tR ∈ L ∞ ((0, ℓ) × (0, T )) and |R(x, 0)| ≥ r > 0 a.e. in (0, ℓ).

If ∂xy(0, t) = 0 for all t ∈ (0, T ), then we have f = 0 in (0, ℓ) and y = 0 in (0, ℓ) × (0, T ).

Using the same method as the one for the proof of Lemma 2, this lemma is proved by means of the following Carleman estimate: Lemma 4 For x0 ∈ [0, ℓ] and β > 0 arbitrarily fixed, we set

Sv = i∂tv -∂ 2 x v, ϕ(x, t) = e λ(|x-x 0 | 2 -βt 2 ) , (x, t) ∈ (0, ℓ) × (0, T ),
where λ > 0 is chosen sufficiently large. Then, there exists s0 > 0 and a constant C > 0 such that

T 0 ℓ 0 (s|∂xv| 2 + s 3 |v| 2 )e 2sϕ dxdt ≤ C T 0 ℓ 0 |Sv| 2 e 2sϕ dxdt
for all s > s0 and all v ∈ L 2 (0, T ; H 2 0 (0, ℓ)) ∩ H 1 0 (0, T ; 2 (0, ℓ)). This is a Carleman estimate with regular weight function λ(|x -x0| 2βt 2 ) and for the proof, we refer to e.g., [START_REF] Yuan | Lipschitz stability in inverse problems for a Kirchhoff plate equation[END_REF]Lemma 2.1], [START_REF] Yuan | Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality[END_REF]. Concerning a Carleman estimate for the Schrödinger equation in a bounded domain Ω ⊂ R n with singular weight function ϕ, we can refer for example to [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF].

On the basis of this lemma, the proof of Lemma 3 is done by a usual method by Bukhgeim and Klibanov [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] by using the extension of y to -T < t < 0 by y(•, t) = y(•, -t) and a cut-off argument and the proof is similar to the proof of Lemma 2 in Appendix. We omit the details.

Observability in the wave network

The proof of the stability result will rely strongly on the classical result of observability that we are now presenting and proving. One should specifically mention a survey [START_REF] Zuazua | Control and stabilization of waves on 1-d networks[END_REF] and books [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF], [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF], where the question of observability in networks of strings (or wave equations) is widely explored in different cases.

We concentrate here on the case where the available observation comes from all but one external nodes, in a setting with a system of wave equations with potential. Since most of the literature on string networks focus only on the wave equation without lower order terms (see [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF] or [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] for instance), we detail here how to obtain the observability result for the wave equation with potential. In some other cases, we can prove the observability inequality directly by a global Carleman estimate (e.g., [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF]).

Theorem 5 (Observability inequality) On the tree-shaped network Λ, assuming p ∈ L ∞ (Λ), let us consider the system of 1-d wave equations under the continuity and Kirchhoff law assumptions (2) and (3):

           ∂ 2 t u -∂ 2 x u + p(x)u = 0, in Λ × (0, T ), u(Q, t) = 0, in (0, T ), ∀Q ∈ Π2, uj (P, t) = u k (P, t), in (0, T ), ∀P ∈ Π1, ∀j, k ∈ {1, ..., nP }, [∂xu(t)] P = 0, in (0, T ), ∀P ∈ Π1, u(x, 0) = 0, ∂tu(x, 0) = a(x), in Λ, (19)
Then there exists a constant T such that for all T > T , the following observability estimate holds for u solution of (19):

Λ |a(x)| 2 dx ≤ C N 2 -1 j=1 T 0 |∂xuj (Qj, t)| 2 dt. ( 20
)
Proof of Theorem 5. Let v be the solution of the system

   ∂ 2 t v -∂ 2 x v = -pu ∀(x, t) ∈ Λ × (0, T ), v(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), vj (x, 0) = 0, ∂tvj(x, 0) = 0, ∀j ∈ {0, 1, ..., N }, x ∈ ej,
under conditions (2) and (3). Then [START_REF] Dáger | Observation and control of vibrations in tree-shaped networks of strings[END_REF] in Lemma 1 and p ∈ L ∞ (Λ) yields

N 2 j=1 T 0 |∂xvj (Qj, t)| 2 dt ≤ C T 0 Λ |pu| 2 dxdt ≤ C T 0 Λ |u| 2 dxdt. (21)
Setting w = uv, we still have ( 2), (3) and the following equation

   ∂ 2 t w -∂ 2 x w = 0 ∀(x, t) ∈ Λ × (0, T ), w(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), wj (x, 0) = 0, ∂twj(x, 0) = a(x), ∀x ∈ Λ.
Therefore, using the existing observability inequality in the case where p = 0 (see e.g., [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF][START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF]), we have

Λ |a(x)| 2 dx ≤ C N 2 -1 j=1 T 0 |∂xwj (Qj, t)| 2 dt.
Hence, by ( 21), we have

Λ |a(x)| 2 dx ≤ C N 2 -1 j=1 T 0 |∂xuj(Qj, t)| 2 dt + C N 2 -1 j=1 T 0 |∂xvj (Qj, t)| 2 dt ≤ C N 2 -1 j=1 T 0 |∂xuj (Qj, t)| 2 dt + C T 0 Λ |u| 2 dxdt. (22)
Therefore a usual compactness-uniqueness argument yields the observability inequality [START_REF] Klibanov | Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems[END_REF]. Indeed, if [START_REF] Klibanov | Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems[END_REF] is not satisfied, then we can assume that there exists a n ∈ L 2 (Λ), n ∈ N such that

a n L 2 (Λ) = 1, ∀n ∈ N and lim n→+∞ N 2 -1 j=1 T 0 |∂xu n j (Qj, t)| 2 dt = 0. ( 23
)
Here u n j is the solution to [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF] with initial value a n . Using the energy estimate (9) of Lemma 1 to u n , we obtain

||u n (t)|| 2 H 1 0 (Λ) + ∂tu n (t) 2 L 2 (Λ) ≤ C||a n || 2 L 2 (Λ) ≤ C. Since the embedding C([0, T ]; H 1 0 (Λ)) ∩ C 1 ([0, T ]; L 2 (Λ)) ⊂ L 2 (Λ × (0, T )
) is compact, we can extract a subsequence, denoted again by the same notation, such that (u n ) n∈N * is convergent in L 2 (Λ). Therefore, using [START_REF] Kuchment | Graph models for waves in thin structures[END_REF], we obtain

Λ |a n -a m | 2 dx ≤ C N 2 -1 j=1 T 0 |∂xu n j (Qj, t)| 2 dt + C N 2 -1 j=1 T 0 |∂xu m j (Qj, t)| 2 dt + C T 0 Λ |u n -u m | 2 dxdt
so that (23) and lim n,m→∞

u n -u m L 2 (Λ×(0,T )) = 0 imply lim n,m→∞
||a n -a m || 2 L 2 (Λ) = 0. Consequently, there exists a limit a0 such that lim n→+∞ a n = a0 in L 2 (Λ) and from [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF], we have a0 L 2 (Λ) = 1 and the solution u[a0] of system [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF] with initial data a0 that satisfies

∂xuj [a0](Q, t) = 0, ∀t ∈ (0, T ), ∀Q ∈ Π2.
The goal is now to deduce u[a0] ≡ 0 in Λ × (0, T ), and for it we will need a classical unique continuation result for a 1-d wave equation and a "removing" argument similar to the one in the proof of Theorem 1.

Lemma 5 We choose T > ℓ. Let y ∈ C([0, T ]; H 1 (0, ℓ)) ∩ C 1 ([0, T ]; L 2 (0, ℓ)) satisfy    ∂ 2 t y(x, t) -∂ 2
x y(x, t) + p(x)y = 0, (x, t) ∈ (0, ℓ) × (0, T ), y(0, t) = 0, t ∈ (0, T ), y(x, 0) = 0,

x ∈ (0, ℓ).

If ∂xy(0, •) ∈ L 2 (0, T ) and ∂xy(0, t) = 0 for 0 < t < T , then for 0 < T1 < T -ℓ we have y(x, t) = 0 for 0 < x < ℓ and 0 < t < T1.

This is a type of unique continuation and for completeness, the proof is given in Appendix B. On the basis of Lemma 5, with (0, ℓ) replaced by ej for j ∈ {1, . . . , N2 -1}, we follow the ideas of Steps 1 -3 of the proof of Theorem 1 to obtain that u[a0] ≡ 0 everywhere in Λ × (0, T ), which implies ∂tu[a0](x, 0) = a0(x) = 0 for all x ∈ Λ. It finally yields a contradiction with a0 L 2 (Λ) = 1, so that this ends the proof of Theorem 5.

Proof of the stability for the wave network inverse problem

This section is devoted to the proof of Theorem 4. The proof relies on a compactness-uniqueness argument and the observability estimate (Theorem 5) on the whole network.

Let us denote by u[p] the solution of (1) under the assumptions ( 2) and ( 3). Henceforth we always assume the conditions ( 2) and ( 3). We consider

y = ∂t (u[p] -u[q]) that satisfy    ∂ 2 t y -∂ 2 x y + q(x)y = (q -p)∂tu[p] ∀(x, t) ∈ Λ × (0, T ), y(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), y(x, 0) = 0, ∂ty(x, 0) = (q -p)u 0 (x), ∀x ∈ Λ, (24)
We set f = qp. We define ψ and φ as the solutions of

   ∂ 2 t ψ -∂ 2 x ψ + q(x)ψ = f ∂tu[p] ∀(x, t) ∈ Λ × (0, T ), ψ(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), ψ(x, 0) = 0, ∂tψ(x, 0) = 0, ∀x ∈ Λ, (25)
and

   ∂ 2 t φ -∂ 2 x φ + q(x)φ = 0 ∀(x, t) ∈ Λ × (0, T ), φ(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), φ(x, 0) = 0, ∂tφ(x, 0) = f u 0 (x), ∀x ∈ Λ. ( 26
)
Then y = ψ + φ. We can apply Theorem 5 to equation ( 26) so that

Λ |f u 0 | 2 dx ≤ C N 2 -1 j=1 T 0 |∂xφj(Qj, t)| 2 dt. ( 27
)
On the other hand, taking the time derivative of equation ( 25) and applying [START_REF] Dáger | Observation and control of vibrations in tree-shaped networks of strings[END_REF] from Lemma 1, by ( 6) we have

N 2 j=1 ∂xψj(Qj) 2 H 1 (0,T ) ≤ C ||f u 1 || 2 L 2 (Λ) + ||f ∂tu[p]|| 2 W 1,1 (0,T,L 2 (Λ)) ≤ 2CK 2 ||f || 2 L 2 (Λ) , (28)
where we used that u

[p] ∈ C 1 ([0, T ], H 1 (Λ)) ∩ C 2 ([0, T ], L ∞ (Λ)) yields ∂tu[p] ∈ C([0, T ]; L ∞ (Λ)) so that u 1 ∈ L ∞ (Λ) and (6) gives u 1 L ∞ (Λ) ≤ K. By the compact embedding H 1 (0, T ) ⊂ L 2 (0, T ), we see that the operator Ψ : L 2 (Λ) → L 2 (0, T ) defined by (Ψf )(t) = N 2 j=1 ∂xψj (Qj, t), 0 < t < T is compact.
Therefore, since we have |u 0 (x)| ≥ r > 0 almost everywhere in Λ, by ( 27) and ( 28), we obtain

||f || 2 L 2 (Λ) ≤ C Λ |f u 0 | 2 dx ≤ C N 2 -1 j=1 T 0 |∂xφj(Qj, t)| 2 dt ≤ C N 2 -1 j=1 T 0 |∂xyj(Qj, t)| 2 dt + C N 2 j=1 T 0 |∂xψj (Qj, t)| 2 dt ≤ C N 2 -1 j=1 T 0 |∂xyj(Qj, t)| 2 dt + C||Ψf || 2 L 2 (0,T ) (29)
≤ C N 2 -1 j=1 ∂xu[p]j (Qj) -∂xu[q]j (Qj) 2 H 1 (0,T ) + C||Ψf || 2 L 2 (0,T ) .
We aim at proving that we can get rid of the second term on the right-hand side of the last estimate in order to obtain [START_REF] Belishev | Boundary spectral inverse problem on a class of graphs (trees) by the BC method[END_REF]. Again, a compactness-uniqueness argument will be the key and it relies here on the compactness of Ψ and the uniqueness result of Theorem 1.

In fact, by contradiction, we assume that

||f || L 2 (Λ) ≤ C N 2 -1 j=1 ∂xyj(Qj) L 2 (0,T ) ,
which is equivalent to [START_REF] Belishev | Boundary spectral inverse problem on a class of graphs (trees) by the BC method[END_REF], does not hold. Then one can assume that there exists

f n ∈ L 2 (Λ), n ∈ N such that f n L 2 (Λ) = 1, ∀n ∈ N and lim n→+∞ N 2 -1 j=1 ∂xy n j (Qj) L 2 (0,T ) = 0. (30)
Here

y n = (y n 0 , ..., y n N ) is the solution to    ∂ 2 t y n -∂ 2 x y n + q(x)y n = f n ∂tu[p] ∀(x, t) ∈ Λ × (0, T ), y n (Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), y n (x, 0) = 0, ∂ty n (x, 0) = (f n u 0 )(x), ∀x ∈ Λ.
First, since the sequence (f n ) n∈N is bounded in L 2 (Λ), we can extract a subsequence denoted again by (f n ) n∈N such that it converges towards some f 0 ∈ L 2 (Λ) weakly in L 2 (Λ). Since Ψ is a compact operator, we obtain therefore the strong convergence result lim n,m→∞

Ψf n -Ψf m L 2 (0,T ) = 0. (31)
Then, from [START_REF] Puel | Generic well-posedness in a multidimensional hyperbolic inverse problem[END_REF] we can write

||f n -f m || 2 L 2 (Λ) ≤ C N 2 -1 j=1 ∂xy n j (Qj) 2 L 2 (0,T ) + C N 2 -1 j=1 ∂xy m j (Qj) 2 L 2 (0,T ) + C||Ψf n -Ψf m || 2 L 2 (Λ)
and deduce from ( 30) and ( 31) that lim n,m→∞

f n -f m L 2 (Λ) = 0, so that there exists f 0 ∈ L 2 (Λ) such that lim n→∞ f n -f 0 L 2 (Λ) = 0 with f 0 L 2 (Λ) = 1. (32)
By Lemma 1, there exists a solution y 0 ∈ C([0, T ]; H 1 0 (Λ)) ∩ C 1 ([0, T ]; L 2 (Λ)) to the system (24) with initial data (0, f 0 u 0 ) and source term f 0 ∂tu[p]. Moreover, using the trace estimate (10) of Lemma 1 for the solution y n of system [START_REF] Lavrent'ev | skiȋ. Ill-posed problems of mathematical physics and analysis[END_REF] with initial data f n u 0 and source term f n ∂tu[p], we obtain (in the same way we proved (28))

N 2 -1 j=1 ∂xy n j (Qj) 2 L 2 (0,T ) ≤ C ||f n u 0 || 2 L 2 (Λ) + ||f n ∂tu[p]|| 2 L 1 (0,T,L 2 (Λ)) ≤ 2CK 2 f n 2 L 2 (Λ) .
Thus we can write

lim n→∞ N 2 -1 j=1 ∂xy n j (Qj) -∂xy 0 j (Qj) 2 L 2 (0,T ) ≤ 2CK 2 lim n→∞ f n -f 0 2 L 2 (Λ) = 0,
which, combined with (30), gives

∂xy 0 j (Qj, t) = 0, ∀Qj ∈ Π2 \ {QN 2 }, ∀t ∈ (0, T ). Therefore we have    ∂ 2 t y 0 -∂ 2 x y 0 + q(x)y 0 = f 0 ∂tu[p] ∀(x, t) ∈ Λ × (0, T ), y 0 (Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), y 0 (x, 0) = 0, ∂ty 0 (x, 0) = (f 0 u 0 )(x),
∀x ∈ Λ and ∂xy 0 (Q, t) = 0, for all Q ∈ Π2 \ {QN 2 } and t ∈ (0, T ). The last objective is now to infer that f0 ≡ 0 in Λ, and in order to be able to apply the unique continuation given by Lemma 2 and the "removing" argument of the proof of Theorem 1, we need to work on a primitive of y 0 .

Therefore, we set y(x, t) = t 0 y 0 (x, ξ)dξ for (x, t) ∈ Λ × (0, T ). In view of the initial data y 0 (x, 0) = 0 and ∂ty 0 (x, 0) = (f 0 u 0 )(x) for all x ∈ Λ, we can verify that y ∈ C 1 ([0, T ];

H 1 0 (Λ)) ∩ C 2 ([0, T ]; L 2 (Λ)) and y satisfies       ∂ 2 t y -∂ 2 x y + q(x) y = f 0 u[p] ∀(x, t) ∈ Λ × (0, T ), y(Q, t) = 0, ∀Q ∈ Π2, t ∈ (0, T ), ∂x y(Q, t) = 0, ∀Q ∈ Π2 \ {QN 2 }, t ∈ (0, T ), y(x, 0) = ∂t y(x, 0) = 0, x ∈ Λ.
Since T > T0, noting that |u[p](x, 0)| = |u 0 (x)| > 0 for x ∈ Λ and using Lemma 2, we repeat the arguments of the proof of Theorem 1 to conclude that f 0 = 0 everywhere in Λ. This contradicts [START_REF] Tanabe | Equations of evolution[END_REF] and thus the proof of Theorem 4 is complete.

A Proof of Lemma 2

We begin this appendix by recalling the following known Carleman estimate:

Lemma 6 We assume that p0 ∈ L ∞ ((0, ℓ) × (-T, T )), x0 > ℓ, β ∈ (0, 1), and we set

ψ(x, t) = |x -x0| 2 -βt 2 , ϕ(x, t) = e λψ(x,t) ,
where we fix λ > 0 sufficiently large. Then there exist contants s0 > 0 and C > 0 independent of s, such that

T -T ℓ 0 (s|∂xv| 2 + s|∂tv| 2 + s 3 v 2 )e 2sϕ dxdt ≤ C T -T ℓ 0 ∂ 2 t v -∂ 2 x v + p0v 2 e 2sϕ dxdt (33)
for all s ≥ s0 and v ∈ L 2 (-T, T ;

H 1 0 (Ω)) satisfying ∂ 2 t v -∂ 2 x v ∈ L 2 ((0, ℓ) × (-T, T )), v(•, ±T ) = ∂tv(•, ±T ) = 0, in (0, ℓ), and ∂xv(0, •) = ∂xu(ℓ, •) = 0 in (-T, T ).
Proof. This result is not new and the proof of this Lemma can be done in several ways. For convenience of the reader, we refer to [START_REF] Baudouin | Global Carleman estimates for waves and applications[END_REF] where the detailed proof of a more general Carleman estimate, given in Theorem 2.1, can be read.

Lemma 6 is a particular case of [4, Theorem 2.1]. On the one hand, by choosing large s0 > 0, a Carleman estimate for the operator ∂ 2 t -∂ 2 x + p0 with p0 ∈ L ∞ ((0, ℓ) × (-T, T )) is very easily deduced from a Carleman estimate in the case of p0 = 0 which is given by Theorem 2.1. Moreover we notice that the weight function used in Theorem 2.1 is ψ0(x, t) = |x -x0| 2βt 2 + C0 with some constant C0, but, since λ is fixed large enough in our lemma, we do not need C0 as we can write 2sϕ = 2se λψ = 2se λψ 0 e -λC 0 and set s := se -λC 0 .

The Carleman estimate for a hyperbolic equation is well-known for functions with compact supports, i.e. u ∈ H 2 0 ((0, ℓ) × (0, T )), and we can refer to several works for the proof , e.g., Hörmander [START_REF] Hörmander | Linear partial differential operator[END_REF] for a general theory, Klibanov and Timonov [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF], and Chapter 1 in Bellassoued and Yamamoto [START_REF] Bellassoued | Carleman estimates and applications to inverse problems for hyperbolic systems[END_REF] where a direct proof is given. In view of such a classical Carleman estimates, Lemma 6 can also be proved by approximating the right-hand side by suitable smooth functions.

Proof of Lemma 2.

Our proof is based on Bukhgeim and Klibanov [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] but here we use the argument from Imanuvilov and Yamamoto [START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF].

Step 1. First we extend y in (0, T ) to (-T, T ) by setting y(x, t) = y(x, -t) for t < 0. We also set R(x, t) = R(x, -t) for t < 0. By the initial conditions in [START_REF] Dáger | Controllability of star-shaped networks of strings[END_REF], we can directly verify that

y ∈ C 1 ([-T, T ]; H 1 0 (0, ℓ)) ∩ C 2 ([-T, T ]; L 2 (0, ℓ)) and    ∂ 2 t y -∂ 2 x y + p(x)y = f (x)R(x, t) (x, t) ∈ (0, ℓ) × (-T, T
), y(0, t) = 0, ∂xy(0, t) = 0, t ∈ (-T, T ), y(x, 0) = 0, ∂ty(x, 0) = 0,

x ∈ (0, ℓ).

In order to apply Lemma 6, we need to design and use a cut-off function χ because y does not satisfy all the expected zero boundary conditions. Let 0 < δ0 < δ and set x0 = ℓ + δ0. Since T > ℓ, we can choose δ > 0 small enough such that ℓ 2 + 2ℓδ < T 2 . Therefore we can choose 0 < β < 1 such that ℓ 2 +2ℓδ

T 2 < β < 1, i.e., x 2 0 -δ 2 β < T. ( 34
) ℓ T 0 ψ = 0 ψ = δ 2 ψ = 4 δ 2 ψ = 9 δ 2 x 0 Ω
Figure 2: Isovalues of the weight function ψ(x, t) = |x-x 0 | 2 -βt 2 , β ∈ (0, 1), for Ω = (0, ℓ), x 0 = ℓ+δ 0 , 0 < δ 0 < δ and T = x 0 / √ β.

Indeed, we easily have:

x 2 0 -δ 2 = (ℓ + δ0) 2 -δ 2 = ℓ 2 + 2ℓδ0 + δ 2 0 -δ 2 < ℓ 2 + 2ℓδ < βT 2 . With this β and x0, we set ψ(x, t) = |x -x0| 2 -βt 2 , ϕ(x, t) = e λψ(x,t)
with large fixed λ > 0 (see Figure 2). We further set

Qµ = {(x, t); 0 < x < ℓ, ψ(x, t) > µ 2 }.
Then we can easily verify Q δ ⊂ (0, x0δ) × -

x 2 0 -δ 2 β , x 2 0 -δ 2 β
so that, by [START_REF] Yamamoto | Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns[END_REF], we have

Q δ ⊂ (0, ℓ) × (-T, T ).
Thus, we define a cut-off function χ ∈ C ∞ (R+ × R) such that 0 ≤ χ ≤ 1 and

χ(x, t) = 1, (x, t) ∈ Q 3δ , 0, (x, t) ∈ ((0, ℓ) × (-T, T )) \ Q 2δ (35)
and we set z = χ(∂ty)e sϕ .

Then, in the sense of distribution in (0, ℓ) × (-T, T ), we easily have ∂tz = (∂tχ)(∂ty)e sϕ + χ(∂

where Q1(s) denotes a linear combination of ∂ j t y with j = 0, 1, 2 whose coefficients include the time derivatives of χ and the power s k with k = 0, 1 as factors, and Q2(s) denotes a linear combination of ∂ j

x ∂ty with j = 0, 1 whose coefficients include the space derivatives of χ and the power s k with k = 0, 1 as factors. Therefore we obtain

∂ 2 t z -∂ 2 x z + pz = χ∂t(∂ 2 t y -∂ 2 x y + py)e sϕ + sχ((∂tϕ)∂ 2 t y -(∂xϕ)(∂x∂ty))e sϕ + s(∂ 2 t ϕ -∂ 2 x ϕ)z + s((∂tϕ)∂tz -(∂xϕ)∂xz) + (Q1(s) -Q2(s))e sϕ = χf (x)(∂tR)e sϕ + sχ((∂tϕ)∂ 2 t y -(∂xϕ)(∂x∂ty))e sϕ + s(∂ 2 t ϕ -∂ 2 x ϕ)z + s((∂tϕ)∂tz -(∂xϕ)∂xz) + (Q1(s) -Q2(s))e sϕ . ( 37
)
Now if we set w = χ∂ty (such that z = e sφ w), then, by [START_REF] Zuazua | Control and stabilization of waves on 1-d networks[END_REF] with s = 0 and the regularity assumptions of y and R, we have

∂ 2 t w -∂ 2 x w + pw = χf ∂tR + (Q1(0) -Q2(0)) in L 2 ((0, ℓ) × (-T, T )).
Since y(0, t) = ∂xy(0, t) = 0, -T < t < T , we see by [START_REF] Yuan | Lipschitz stability in inverse problems for a Kirchhoff plate equation[END_REF] that w ∈ C([-T, T ]; H 1 0 (0, ℓ)) ∩ C 1 ([-T, T ]; L 2 (0, ℓ)), ∂ j t w(x, ±T ) = 0 for 0 < x < ℓ and j = 0, 1, ∂xw(0, t) = ∂xw(ℓ, t) = 0 for -T < t < T and ∂ 2 t w -∂ 2 x w = χf ∂tR + Q1(0) -Q2(0)pw ∈ L 2 ((0, ℓ) × (-T, T )). Hence we can apply the Carleman estimate of Lemma 6 to w:

Q 2δ (s|χ∂x∂ty| 2 + s|χ∂ 2 t y| 2 + s 3 |χ∂ty| 2 )e 2sϕ dxdt ≤ 2 Q 2δ (s|∂xw| 2 + s|∂tw| 2 + s 3 w 2 )e 2sϕ dxdt + 2 Q 2δ (s|∂ty| 2 |∂xχ| 2 + s|∂ty| 2 |∂tχ| 2 )e 2sϕ dxdt.
Therefore, from the (35) of the cut-off function, and using (38), the definition of Q 3δ and the regularity assumptions ( 12) on y, we obtain

Q 2δ (s|χ∂x∂ty| 2 + s|χ∂ 2 t w| 2 + s 3 |χ∂ty| 2 )e 2sϕ dxdt ≤ C Q 2δ |f | 2 e 2sϕ dxdt+C Q 2δ \Q 3δ |Q1(0)-Q2(0)| 2 e 2sϕ dxdt+Cs Q 2δ (|∂xχ| 2 +|∂tχ| 2 )|∂ty| 2 e 2sϕ dxdt ≤ C Q 2δ |f | 2 e 2sϕ dxdt + Cs 2 e 2se 9λδ 2 (39)
since derivatives of χ, as well as Q1(0) -Q2(0), are vanishing everywhere outside Q 2δ \ Q 3δ .

Step 2. Next we set

Q + δ = {(x, t) ∈ Q δ ; t > 0}. Then Q + δ = (x, t); 0 < x < x0 -δ, 0 < t < |x -x0| 2 -δ 2 β .
Multiplying (37) by ∂tz and integrating over Q + δ , we have

Q + δ (∂ 2 t z -∂ 2 x z + pz)∂tz dxdt = 4 k=1 I k := Q + δ χf (∂tR)e sϕ (∂tz) dxdt + s Q + δ χ((∂tϕ)∂ 2 t y -(∂xϕ)(∂x∂ty))(∂tz)e sϕ dxdt + s Q + δ {(∂ 2 t ϕ -∂ 2 x ϕ)z + ((∂tϕ)∂tz -(∂xϕ)∂xz)}∂tz dxdt + Q + δ (Q1(s) -Q2(s))(∂tz)e sϕ dxdt.
On the one hand, by integration by parts and using z = ∂xz = ∂tz = 0 on ∂Q + δ \ {t = 0} and z = ∂xz = 0 on ∂Q + δ ∩ {t = 0}, we have

Q + δ (∂ 2 t z -∂ 2 x z + pz)∂tz dxdt = 1 2 x 0 -δ 0 |∂tz(x, 0)| 2 dx.
Moreover, by ∂ty(x, 0) = 0 for 0 < x < ℓ and z = χ(∂ty)e sϕ , we have ∂tz(x, 0) = χ(x, 0)∂ 2 t y(x, 0)e sϕ(x,0) . Therefore, using [START_REF] Dáger | Controllability of star-shaped networks of strings[END_REF] and |R(x, 0)| ≥ r on [0, ℓ], we obtain

Q + δ (∂ 2 t z -∂ 2 x z + pz)∂tz dxdt = 1 2 x 0 -δ 0 |χ(x, 0)| 2 |∂ 2 t y(x, 0)| 2 e 2sϕ(x,0) dx = 1 2 x 0 -δ 0 |χ(x, 0)| 2 |f (x)| 2 |R(x, 0)| 2 e 2sϕ(x,0) dx ≥ C x 0 -δ 0 |χ(x, 0)| 2 |f (x)| 2 e 2sϕ(x,0) dx. ( 40
)
On the other hand, by ∂tR ∈ L ∞ ((0, ℓ) × (0, T )) and the Cauchy-Schwarz inequality, we have Therefore, since |∂tχ| 2 and |∂xχ| 2 vanish everywhere outside Q 2δ \ Q 3δ and χ = 0 outside of Q 2δ , using the definition of Q 3δ and the regularity assumptions [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] on y, we have

I1 ≤ C Q + δ χ 2 |f (x)| 2 e 2sϕ dxdt + C Q + δ |∂tz| 2 dxdt, I2 ≤ Cs Q + δ (χ|∂ 2 t y| + χ|∂t∂xy|)|∂tz|e sϕ dxdt ≤ Cs Q δ (|χ∂ 2 t y| 2 + |χ∂t∂xy| 2 )e 2sϕ dxdt + C Q δ |∂tz| 2 dxdt, I3 ≤ Cs Q + δ (|z| + |∂tz| + |∂xz|)|∂tz|dxdt ≤ Cs Q δ (z 2 + |∂tz| 2 + |∂xz| 2 ) I4 ≤ Cs 2 e 2se 9λδ 2 + C Q δ |∂tz| 2 dxdt
Q δ (s|∂tz| 2 + s|∂xz| 2 + s 3 z 2 )dxdt ≤ C Q 2δ
(s|χ∂ 2 t y| 2 + s|χ∂x∂ty| 2 + s 3 |χ∂ty| 2 )e 2sϕ dxdt + Cs 2 e 2se 9λδ 2 .

Hence, using also (39), we finally obtain e 2s(ϕ(x,t)-ϕ(x,0)) dt dx. By µ4 -µ3 > 0, letting s → ∞, we see that the right-hand side tends to 0, and f (x) = 0 for 0 ≤ x ≤ ℓ + δ0 -4δ. We can choose δ, δ0 > 0 arbitrarily small, provided that 0 < δ0 < δ, so that f = 0 in (0, ℓ).

Step 4. It remains to prove the latter part of the Lemma 2. Since f = 0 in (0, ℓ), estimate (39) yields for all s ≥ s0. Letting s → ∞, we obtain Q 4δ |∂ty| 2 dxdt = 0. One can directly verify that we have the inclusion

Q 2δ s 3 |χ(x,
(0, x0 -5δ) × - δ √ β , δ √ β δ ⊂ Q 4δ , (43)
so that ∂ty(x, t) = 0 in (0, x0 -5δ) × -δ √ β , δ √ β .

We fix δ > 0 arbitrarily small and choose the weight function ψ θ (x, t) = |x -x0| 2β(tθ) 2 for each θ ∈ (-T1, T1). Noting that |T -θ| > ℓ for |θ| < T1, we can apply the above arguments from (39) and obtain ∂ty(x, t) = 0 in (0, ℓ + δ0 -5δ) × (-T1, T1). Since y(x, 0) = 0 for 0 < x < ℓ, we have y(x, t) = 0 in (0, ℓ + δ0 -5δ) × (-T1, T1). Since δ0, δ > 0 can be chosen arbitrarily small, we can obtain y = 0 in (0, ℓ) × (-T1, T1). Thus the proof of Lemma 2 is completed.

B Proof of Lemma 5

We choose the same x0, δ0, δ, β and use the same notations as in the proof of Lemma 2. We extend y(x, t) to t < 0 by setting y(x, -t) = -y(x, t) for t > 0. Then by y(•, 0) = 0 in (0, ℓ), we can verify that y ∈ C([-T, T ]; H 1 (0, ℓ)) ∩ C 1 ([-T, T ]; L 2 (0, ℓ)) and y satisfies by assumption ∂ 2 t y -∂ 2 x y + p(x)y = 0, (x, t) ∈ (0, ℓ) × (-T, T ), y(0, t) = 0, ∂xy(0, t) = 0, t ∈ (-T, T ).

Defining the cut-off function χ by [START_REF] Yuan | Lipschitz stability in inverse problems for a Kirchhoff plate equation[END_REF], we apply Lemma 6 to y1 := χy. Similar but simpler arguments to the proof of Lemma yields y = 0 in Q 4δ . We also repeat the proof of the final part of Lemma 2, in view of (43), so that the proof of Lemma 5 is completed.

Figure 1 :

 1 Figure 1: A star-shaped network with 10 edges (N = 9, N 1 = 4, N 2 = 7).

(s|∂xw| 2 +χ 2

 22 s|∂tw| 2 + s 3 w 2 )e 2sϕ dxdt |f | 2 |∂tR| 2 e 2sϕ dxdt + C -Q2(0)| 2 e 2sϕ dxdt. (38) Now, since χ∂x∂ty = ∂xw -(∂ty)∂xχ and χ∂ 2 t y = ∂tw -(∂ty)∂tχ, we have

t y| 2 +s|χ∂x∂ty| 2 +s 3 |χ∂ty| 2 )e 2sϕ dxdt+Cs 2 e 2se 9λδ 2 ≤ C Q 2δ |f | 2 e

 222 2sϕ dxdt + Cs 2 e 2se 9λδ 2 . (41)Step 3. From (40) and (41), we obtainx 0 -δ 0 |χ(x, 0)| 2 |f (x)| 2 e 2sϕ(x,0) dx ≤ C Q 2δ |f | 2 e 2sϕ dxdt + Cs 2 e 2se 9λδ 2.Noting that (0, x0mδ) × {0} ⊂ Q mδ and Q mδ ⊂ Q 2δ for m ≥ 3 and χ = 1 in Q 3δ , setting µm = e m 2 λδ 2 , we havex 0 -3δ 0 |f (x)| 2 e 2sϕ(x,0) dx ≤ C Q 3δ |f | 2 e 2sϕ dxdt + C Q 2δ \Q 3δ|f | 2 e 2sϕ dxdt + Cs 2 e 2sµ 3 .

4 x 0 -4δ 0 |f | 2

 402 Here ϕ(x, t)ϕ(x, 0) = e λ(|x-x 0 | 2 -βt 2 )e λ|x-x 0 | 2 ≤ min 0≤x≤ℓ e λ|x-x 0 | 2 (e -λβt 2 -1) < 0 for t = 0 and 0 ≤ x ≤ ℓ, and the Lebesgue theorem yieldsmax 0≤x≤ℓ T -T e 2s(ϕ(x,t)-ϕ(x,0)) dt ≤ C T -T exp(2sC(e -λβt 2 -1))dt = o(1)as s → ∞. Consequently,Q 3δ |f | 2 e 2sϕ dxdt = o(1) x 0 -3δ 0 |f (x)| 2 e 2sϕ(x,0) dx. Hence (1o(1)) x 0 -4δ 0 |f (x)| 2 e 2sϕ(x,0) dx ≤ (1o(1)) x 0 -3δ 0 |f (x)| 2 e 2sϕ(x,0) dx ≤ C Q 2δ \Q 3δ |f | 2 e 2sϕ dxdt + Cs 2 e 2sµ 3 .Therefore, since (0, x0 -4δ) × {0} ⊂ Q 4δ implies e 2sϕ(x,0) ≥ e 2sµ 4 for 0 < x < x0 -4δ and|e 2sϕ | ≤ e 2sµ 3 in Q 2δ \ Q 3δ , we have (1o(1))e 2sµ dx ≤ Ce 2sµ 3 f 2 L 2 (0,ℓ) + Cs 2 e 2sµ 3 .Choosing s > 0 large and dividing by e 2sµ 4 , we obtainℓ+δ 0 -4δ 0 |f (x)| 2 dx ≤ C(s 2 + f 2 L 2 (0,ℓ) )e -2s(µ 4 -µ 3 )

  .dxdt, using also that Q1(s) -Q2(s) vanishes everywhere outsideQ 2δ \ Q 3δ .Now, analyzing the terms in z on the right-hand side of these estimates, from (36), we have that in Q δ ,|∂tz| 2 + |∂xz| 2 ≤ C(|χ∂2 t y| 2 + |χ∂t∂xy| 2 )e 2sϕ + Cs 2 |χ∂ty| 2 e 2sϕ + (|∂tχ| 2 + |∂xχ| 2 )|∂ty| 2 e 2sϕ .

  t)| 2 |∂ty| 2 e 2sϕ dxdt ≤ Cs 2 e 2sµ 3 (42) for all s ≥ s0. Since Q 4δ s 3 |χ(x, t)| 2 |∂ty| 2 e 2sϕ dxdt ≤ Q 2δ s 3 |χ(x, t)| 2 |∂ty| 2 e 2sϕ dxdt,e 2sϕ ≥ e 2sµ 4 in Q 4δ and χ = 1 in Q 4δ , from (42) we obtain

	Q 4δ	|∂ty| 2 dxdt ≤	C s	e -2s(µ 4 -µ 3 )

Acknowledgements. The authors thank anonymous referees for valuable comments which were very useful for the improvements of the manuscript.


Nizar Demni
email: nizar.demni@univ-rennes1.fr

Alain Rouault
email: alain.rouault@uvsq.fr

Marguerite Zani
email: marguerite.zani@univ-orleans.fr

In Memoriam

Marc Yor

Large deviations for clocks of self-similar processes

.

Introduction

This problem is an extension of a question raised by Marc Yor during the defense of the thesis of Marguerite, under the supervision of Alain, long time ago in 2000. The last part of this thesis was dedicated to the study of large deviations principles (LDP) for Maximum Likelihood Estimates of diffusion coefficients (for squared-radial Ornstein-Uhlenbeck processes, squared Bessel processes and Jacobi processes). The main tool there was a convenient Girsanov change of probability. This method allowed to convert the computation of Laplace transform of some additive functionals into the computation of Laplace transform of a single variable.

it "reduction method". In those times, Marc was interested in exponential functionals of Brownian motion, Lamperti transform and Asian options and he guessed that this LDP could be applied to the Bessel clock and solved effectively the problem with Marguerite in [START_REF] Yor | Large deviations for the Bessel clock[END_REF] a couple of weeks later. Marc suggested to extend it to the Cauchy clock and gave a sketch of proof, but a technical difficulty stopped the project. Meanwhile, Marguerite and Nizar published a paper [START_REF] Demni | Large deviations for statistics of the Jacobi process[END_REF] on Jacobi diffusions where the reduction method is again crucial. Recently, the three of us felt the need to revisit the problem of Cauchy clock with the hope of new ideas.

We planned to discuss with Marc, and promised him we would keep him informed of our progress. We did not have the time...

In this paper, we extend the methods and results of [START_REF] Yor | Large deviations for the Bessel clock[END_REF] to a large class of clocks issued from positive self-similar Markov processes. In Section 2 we recall some basic results about the Lamperti correspondence between these processes and Lévy processes and we give the definition of the clocks. We also define a generalized Ornstein-Uhlenbeck process which will be useful in the sequel. In Section 3 and Section 4, we present the main results: Law of Large Numbers and Large Deviations for the clocks. In Section 5 we show some examples illustrating our main theorems.

Positive self-similar Markov processes and Lamperti transformation

In [START_REF] Lamperti | Semi-stable Markov processes[END_REF] Lamperti defined (positive) semi-stable process which are nowadays called positive self-similar Markov process.

Definition 1. For α > 0, a positive self-similar Markov process (pssMp) of index α, is a [0, ∞)-valued strong Markov process (X, Q a ), a > 0 with càdlàg paths, fulfilling the scaling property

({bX b -α t ,t ≥ 0}, Q a ) (law) = ({X t ,t ≥ 0}, Q ba ) (1)
for every a, b > 0.

Lamperti [START_REF] Lamperti | Semi-stable Markov processes[END_REF] has shown that these processes can be connected to Lévy processes by a one-to-one correspondence, that we develop below. We refer to Kyprianou [START_REF] Kyprianou | Fluctuations of Lévy processes with applications[END_REF] especially Chapter 13 for properties of Lévy processes and pssMp. One can also see [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF] for the Lamperti's correspondence. One can notice that there is a little confusion in the notion of index of these processes. In [START_REF] Lamperti | Semi-stable Markov processes[END_REF], [START_REF] Bertoin | The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes[END_REF] and [START_REF] Bertoin | Entrance from 0+ for increasing semi-stable markov processes[END_REF], the index is 1/α, and in [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF], [START_REF] Caballero | Explicit identities for Lévy processes associated to symmetric stable processes[END_REF] and [START_REF] Kyprianou | Fluctuations of Lévy processes with applications[END_REF], the index is α. We take this latter convention. These processes have a natural application in the theory of self-similar fragmentations : see [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF], references therein, and [START_REF] Caballero | On the asymptotic behaviour of increasing self-similar Markov processes[END_REF]. For other areas of application, such as diffusions in random environments, see Section 6 of [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF].

From X to ξ

Any pssMp X which never reaches the boundary state 0 may be expressed as the exponential of a Lévy process not drifting to -∞, time changed by the inverse of its exponential functional. More formally, if (X, (Q a ) a>0 ) is a pssMp of index α which never reaches 0, set

T (X) (t) = t 0 ds X α s , (t ≥ 0) (2)
and let A (X) be its inverse, defined by

A (X) (t) = inf{u ≥ 0 : T (X) (u) ≥ t} , (3)
and let ξ be the process defined by

ξ t = log X A (X) (t) -log X 0 , (t ≥ 0) . (4)
Then, for every a > 0, the distribution of (ξ t ,t ≥ 0) under Q a does not depend on a and is the distribution of a Lévy process starting from 0. Moreover, if we set

A (ξ ) (t) = t 0 e αξ s ds (5)
and τ (ξ ) its inverse defined by

τ (ξ ) (t) = inf{u ≥ 0 : A (ξ ) (u) ≥ t} (6)
we have

τ (ξ ) (t) = T (X) (tX α 0 ) . (7)
Let us remark that the self-similarity property (1) leads to the following relation:

the law of T (X) (. b -α ) under Q a is the law of T (X) (.) under Q ba . (8)
2.2 From ξ to X Let ξ t be a Lévy process starting from 0 and let P and E denote the underlying probability and expectation, respectively. Fix α > 0. Let A (ξ ) be its exponential functional defined by [START_REF] Caballero | Explicit identities for Lévy processes associated to symmetric stable processes[END_REF]. When ξ drifts to -∞, this functional is very popular in mathematical finance (see [START_REF] Yor | Exponential functionals of Brownian motion and related processes[END_REF]), with important properties of the perpetuity A (ξ ) (∞).

Here, we rather assume that ξ does not drift to -∞ i.e. satisfies, lim sup t↑∞ ξ t = ∞. We define the inverse process τ (ξ ) of A (ξ ) by ( 6).

For every a > 0, let Q a be the law under P of the time-changed process

X t = a exp ξ τ (ξ ) (ta -α ) , (t ≥ 0) , (9)
then (X, (Q a ) a>0 ) is a pssMp of index α which never reaches 0 and we have the fundamental relation [START_REF] Ph | On exponential functionals of certain levy processes[END_REF].

Index and starting point

If (X, (Q a ) a>0 ) is an pssMa of index α, then the process Y = (X α , (Q a α ) a>0 ), is a pssMp of index 1. Conversely if (Y, (Q a ) a>0 ) is a pssMp if index 1 then, for any α > 0, the process (X = Y 1/α , (Q a 1/α ) a>0
) is a pssMp of index α. Hence we may and will assume without loss of generality, that α = 1, except some examples in Section 5.

Let us stress also the fact that the distribution of T (X) is considered under Q a for a > 0, and the distribution of τ (ξ ) is considered under P.

Ornstein-Uhlenbeck process

Having defined the above pair of associated processes (ξ , X) it is usual to consider a third process, very useful in the sequel. To introduce it, we need additional properties on the pssMp X. In Theorem 1 of [START_REF] Bertoin | The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes[END_REF], Bertoin and Yor have shown that whenever the support of ξ is not arithmetic and E(ξ 1 ) > 0, then as a ↓ 0, the probability measures Q a (defined in section 2.2) converge in the sense of finitely dimensional marginals to a probability measure denoted by Q 0 . This latter measure is an entrance law for the semigroup Q t f (x) = E x f (X t ) and satisfies

Q 0 ( f (X t )) = 1 Eξ 1 E[I -1 ∞ f (t/I ∞ )] (10)
where

I ∞ = ∞ 0 e -ξ s ds .
Now we define the associated generalized Ornstein-Uhlenbeck (OU) process by

U(t) := e -t X(e t ) .
This process has been studied in [START_REF] Ph | On the distribution and asymptotic results for exponential functionals of lévy processes[END_REF], and further on by [START_REF] Rivero | A law of iterated logarithm for increasing self-similar markov processes[END_REF] and [START_REF] Caballero | On the asymptotic behaviour of increasing self-similar Markov processes[END_REF]. It is strictly stationary, Markovian and ergodic under E 0 , and its invariant measure is the law of X 1 under Q 0 , i.e.

µ( f ) = 1 Eξ 1 E[I -1 ∞ f (I ∞ )] .
The purpose of this paper is the study of the asymptotic behavior of the process T (X) , called the clock of the pssMp X, as t → ∞ under Q a for any a > 0, or equivalently the asymptotic behavior of the process τ (ξ ) , as t → ∞ under P. The two points of view are complementary: the clock as a functional of the pssMp or the clock as a functional of the Lévy process. We will restrict us to the case of Eξ 1 ∈ (0, ∞), which ensures a Law of Large Numbers (LLN) and a Large Deviation Principle (LDP).

Law of Large Numbers

The first step of our study is the a.s. convergence of our functionals.

Theorem 1. If Eξ 1 is finite positive (hence = ψ ′ (0) > 0), then for every a > 0, we have T (X) (t) logt → (Eξ 1 ) -1 , Q a -a.s. ( 11
)
or equivalently

τ (ξ ) (t) logt → (Eξ 1 ) -1 , P -a.s. ( 12
)
Proof. From [START_REF] Ph | On exponential functionals of certain levy processes[END_REF] it is enough to prove [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. Let us now repeat verbatim the trick from [START_REF] Caballero | On the asymptotic behaviour of increasing self-similar Markov processes[END_REF] Section 2. By the ergodic theorem, we have that for f ∈ L 1 (µ):

1 t t 0 f (U s )ds → µ( f ) , Q 0 -a.s.
or, by scaling 1 logt

t 1 f (u -1 X u ) du u → µ( f ) , Q 0 -a.s. (13)
For f (x) = x -1 this yields

1 log(1 + t) 1+t 1 du X u → (Eξ 1 ) -1 , Q 0 -a.s.
If H denotes the set where the convergence holds, the Markov property yields:

E 0 Q X 1 T (X) (t) log(1 + t) → (Eξ 1 ) -1 = P 0 (H ) = 1 .
If we remember that µ is the law of X 1 under Q 0 , we have

Q x T (X) (t) log(1 + t) → (Eξ 1 ) -1 = 1 , µ -a.e. .
Fixing such an x and using the scaling property (8), we get [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. ⊓ ⊔

It is then natural to look for a Large Deviation Principle (LDP) to characterize the speed of this convergence.

LDP

We present a LDP for the clock T (X) (t) (Theorem 2), whose proof relies on the reduction method (Lemma 1). The remaining of the section is devoted to some illustrative remarks and the proof of a technical lemma. In view of [START_REF] Ph | On exponential functionals of certain levy processes[END_REF], the following statements are valid when T (X) and Q a are replaced by τ (ξ ) and P, respectively.

Main result

Theorem 2. Assume 1 that dom ψ = (m -, m + ) ∋ 0 and that ψ ′ (0) > 0. Set m 0 = inf{θ : ψ ′ (θ ) > 0} , τ + = 1 ψ ′ (m + ) , τ 0 = 1 ψ ′ (m 0 ) and ∆ = (τ + , τ 0 ) , where 1/ψ ′ (±∞) := lim m→±∞ m/ψ(m). Let a > 0. 1. Then, for every x ∈ ∆ lim ε→0 lim t→∞ 1 logt log Q a T (X) (t) logt ∈ [x -ε, x + ε] = -I (x) (14)
where

I (x) = sup m∈(m 0 ,m + ) {m -xψ(m)} . (15)

Moreover

lim A→∞ lim sup t→∞ 1 logt log Q a T (X) (t) logt > A = -∞ . ( 16
) 3. If either ∆ = (0, ∞) or the complement of ∆ is exponentially negligible, i.e. if ∀ε > 0 lim sup t→∞ 1 logt log Q a inf x∈ ∆ x - T (X) (t) logt > ε = -∞ (17)
then the family of distributions of T (X) (t)/ logt under Q a satisfies the LDP on [0, ∞) at scale logt with good rate function :

I (x) = I (x) if x ∈ ∆ , ∞ otherwise.
The following proposition describes some properties of I which are direct consequences of [START_REF] Kyprianou | Fluctuations of Lévy processes with applications[END_REF]. The proof is left to the reader. Proposition 1. 1. If ψ * denotes the Fenchel-Legendre dual of ψ defined by

ψ * (x) = sup m∈R mx -ψ(m) , (x ∈ R) , then I (x) = xψ * (x -1 ) , (x ∈ ∆ ) . ( 18
)
2. Let τ e = 1/Eξ 1 ∈ (τ + , τ 0 ). The function I is convex on ∆ , decreasing on (τ + , τ e ) , has a minimum 0 at τ e , and is increasing on (τ e , τ 0 ).

3. a. If m 0 is a true minimum, i.e. if m 0 > -∞ and ψ ′ (m 0 ) = 0, then τ 0 = ∞, and as x → ∞, I admits the asymptote y = -xψ(m 0 ) + m 0 . b. If m 0 = -∞ and ψ ′ (m 0 ) ∈ (0, ∞) , set lim m→m 0 ψ(m) -mψ ′ (m 0 ) = -b ∈ [-∞, 0). Then τ 0 < ∞ and I (τ 0 ) = bτ 0 and I ′ (τ 0 ) = -ψ(m 0 ) = ∞. c. If m 0 = -∞ and -∞ < ψ(m 0 ) < 0, hence ψ ′ (m 0 ) = 0, then τ 0 = ∞ and I (x)/x → -ψ(m 0 ) and I (x) + ψ(m 0 )x → -∞, as x → ∞. 4. a. If m + < ∞ and ψ(m + ) = ∞, then ψ ′ (m + ) = ∞. We get τ + = 0 and I (0) = m + with I ′ (0) = ∞. b. If m + = ∞ and ψ ′ (m + ) < ∞, set lim m→m + ψ(m) -mψ ′ (m + ) = -b ∈ [-∞, 0). Then τ + > 0, I (τ + ) = bτ + ≤ ∞ and I ′ (τ + ) = -ψ(m + ) = -∞. c. If m + = ∞, with ψ(m + ) = ∞ and ψ ′ (m + ) = ∞ then τ + = 0 and I (τ + ) = m + = ∞.
Proof. of Theorem 2 1) A slight adaptation of the Gärtner-Ellis method ([11] Th. 2.3.6) allows to deduce ( 14) from the asymptotic behaviour of the normalized log-Laplace transform of T (X) (t), given by the following lemma. Recall that ψ is increasing on (m 0 , m + ).

Lemma 1. For θ ∈ (-ψ(m + ), -ψ(m 0 )), set L t (θ ) = log E a exp(θ T (X) (t)). Then as t → ∞ we have 1 logt L t (θ ) → L(θ ) (19)
where

L(θ ) = -m ⇐⇒ θ = -ψ(m) . (20)
The function L is differentiable on (-ψ(m + ), -ψ(m 0 )) and satisfies L ′ (θ ) = 1/ψ ′ (m), so that the range of L ′ is precisely ∆ . Then the left-hand-side of ( 14) admits the limit I (x) = xθ -L(θ ) where θ is the unique solution of L ′ (θ ) = x i.e., thanks to [START_REF] Maulik | Tail asymptotics for exponential of Lévy processes[END_REF], I (x) = -xψ(m) + m which is exactly the right-hand-side of [START_REF] Küchler | Exponential families of stochastic processes[END_REF].

Proof. of Lemma 1

To compute the Laplace transform of T (X) (t), we use a Girsanov type change of probability.

For m ∈ (m -, m + ) let

ψ m (θ ) = ψ(m + θ ) -ψ(m) ,
and let {(ξ t ,t ≥ 0); P (m) } be a Lévy process starting from 0 whose exponent is ψ m (Esscher transform). Finally let {(X t ,t ≥ 0); (Q m a ) a>0 } be the associated pssMp.

Besides, from [START_REF] Ph | Exponential functionals of Lévy processes[END_REF] the following relation between the infinitesimal generators L ξ of ξ and L X of X under Q a :

L X f (x) = 1 x L ξ ( f • exp)(log x) ,
implies (see (2.7) in [START_REF] Ph | Exponential functionals of Lévy processes[END_REF]),

L X f m = ψ(m) f m-1 for f m (x) = x m so that Q m a | F t = X t a m exp -ψ(m) t 0 ds X s . Q a | F t , (t ≥ 0) . (21)
We deduce that

E a exp -ψ(m) t 0 ds X s = a m E m a (X t ) -m (22)
and, owing to the scaling property

E a exp -ψ(m) t 0 ds X s = a m t -m E m a/t (X 1 ) -m . (23)
Let us choose m such that ψ ′ (m) > 0.

We can know use the results on entrance boundary, for example [START_REF] Bertoin | The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes[END_REF]. Since the new Lévy process ξ (m) has a positive mean

Eξ (m) = ψ ′ m (0) = ψ ′ (m) > 0 then Theorem 1 i) therein entails that lim t↑∞ E m a/t (X 1 ) -m = E m 0 (X 1 ) -m . ( 24
)
Lemma 2. With the notations of Theorem 2, the quantity F(m) := E m 0 (X 1 ) -m is finite for m ∈ (m 0 , m + ).

From (23)

L t (θ ) = -m logt + m log a + log E m a/t (X 1 ) -m and then, as t → ∞, thanks to Lemma 2:

L t (θ ) logt → L(θ ) = -m . ( 25
)
This ends the proof of Lemma 1 and consequently the proof of part 1) in Theorem 2, up to the result of Lemma 2 whose proof is postponed.

2) The statement is a consequence of the Chernov inequality. Indeed for fixed

θ ∈ (0, -ψ(m 0 )) log Q a T (X) (t) logt > -A ≤ -θ A + L(θ )
for every θ ∈ (0, -ψ(m 0 )).

3) Let us assume that ∆ = (0, ∞). To get a full LDP, it is enough to prove that lim

A→∞ lim t→∞ 1 logt log P τ(t) logt > A = -∞ , (exponential tightness) but it is a consequence of the Chernov inequality: log Q a T (X) (t) logt > -A ≤ -θ A + L(θ )
for every θ ∈ (0, -ψ(m 0 )).

Assume now that the condition ( 17) is satisfied. For every x in the complement of ∆ we have

lim δ →0 lim t→∞ 1 logt log Q a T (X) (t) logt ∈ [x -δ , x + δ ] = -∞ , so that a weak LDP is satisfied on [0, ∞).
To strenghten it is enough to apply the above reasoning. ⊓ ⊔

Remarks

Remark 1. (Reciprocal pairs) The relation [START_REF] Letac | Natural exponential families with cubic variance[END_REF] between two rate functions is known to hold for pairs of inverse processes. This problem arose historically when people deduced a LDP for renewal processes whose interrival times have sums satisfying a LDP. It was extended to more general processes, see [START_REF] Duffield | Large deviations of inverse processes with nonlinear scalings[END_REF] and the bibliography therein.

If (ξ t ) is a spectrally negative Lévy process with Laplace exponent ψ, then t -1 ξ t satisfies the LDP at scale t with rate function ψ * . The subordinator

τ(u) = inf{t > 0 : ξ t > u} ( 26
)
has a Laplace exponent which is exactly L defined by he relation ( 20)

ψ(m) = -θ , L(θ ) = -m . ( 27
)
Then t -1 τ(t) satisfies the LDP at scala t with rate function x → xψ * 1

x . This can be seen as an application of Theorem 1 of [START_REF] Duffield | Large deviations of inverse processes with nonlinear scalings[END_REF]).

The relation (6) between A (ξ ) u and τ (ξ ) (t):

τ (ξ ) (t) = inf{u ≥ 0 : A (ξ ) u = u 0 e ξ s ds ≥ t} , (t ≥ 0)
is more involved than [START_REF] Yor | Exponential functionals of Brownian motion and related processes[END_REF]. An alternative proof of our Theorem 2 about LDP for (logt) -1 τ (ξ ) (t), in the spirit of Theorem 1 of [START_REF] Duffield | Large deviations of inverse processes with nonlinear scalings[END_REF]), would require an LDP for

t -1 log A (ξ ) t
which is far from obvious. At first glance, the lower bound of large deviations seems accessible since we have for convenient x and δ ,

(ξ s /s ∈ [x -δ , x + δ ] ∀s ∈ [0, u]) ⇒ A (ξ ) u ∈ e (x-δ )u -1 (x -δ ) , e (x+δ )u -1 (x + δ ) .
Conversely, we can apply Theorem 1 of [START_REF] Duffield | Large deviations of inverse processes with nonlinear scalings[END_REF] to get Proposition 2. The family of distributions of t -1 log A (ξ ) t (under P) satisfies the LDP at scale t, with good rate function ψ * .

To end this first remark, let us add two more comments:

• The statement of Lemma 1, viewed in terms of ξ can be rephrased as the striking relation

lim t→∞ 1 logt E exp θ τ (ξ ) (t) = log E exp θ τ(1) .
• The relation ( 27) is also well known in the study of exponential families (inverse families or reciprocal pairs) , see [START_REF] Küchler | Exponential families of stochastic processes[END_REF] Section 5.4 and [START_REF] Letac | Natural exponential families with cubic variance[END_REF] Section 5A.

Remark 2. (Another approach for the LDP) We saw in a previous section that the core of the proof of the LLN is the convergence of the occupation measure 1 t t 0 δ U s ds → µ followed by a scaling and by the choose of f (x) = x -1 as a test function. It could be natural to look for a LDP according the same line of reasoning. Such a proof would consist of three steps. First establish an LDP for the law of the occupation measure under Q 0 . The rate functional is expressed with the help of the infinitesimal generator of U. Then apply the contraction by the (non-continous) mapping ν → ν( f ), hence solve a variational problem. This would give an LDP for the law of

1 t t 0 ds U s = 1 t e t 1 ds X s , under Q 0 .
It would remain to convert it into an LDP under Q a . In the Appendix of [START_REF] Yor | Large deviations for the Bessel clock[END_REF], the authors give the first rate functional and solved the variational problem, but with the lack of justification for the contraction principle. The result fits with the rate function of Theorem 2.

Remark 3. (Functional LDP) In the Bessel clock, a functional LDP was stated ( [START_REF] Yor | Large deviations for the Bessel clock[END_REF] Th. 4.1). Here it is possible to consider the same problem, i.e. the study of the LDP for the sequence of processes

u ∈ [0, 1] → 1 n T (X) (e nu )
under Q a . The rate function involves an action functional built on I and a recession function which will be x → rx where r = -ψ(m 0 ) (when it is finite).

Remark 4. (Central Limit

Theorem) It is known for the Bessel clock (Theorem 1.1 in [START_REF] Yor | Large deviations for the Bessel clock[END_REF]) but seems unknown otherwise. We can conjecture that logt

T (X) (t) logt - 1 Eξ 1 =⇒ N (0; ψ ′′ (0)/ψ ′ (0) 3 ) (28)
as soon as E(ξ 1 ) 2 < ∞ (or ψ ′′ (0) < ∞).

Proof of Lemma 2

From (10) applied to the Esscher transform we have for m > m 0

F(m) = E (m) (I m-1 ∞ ) .
The finiteness of F(m) is a consequence of the following Lemma summarizing properties of the moments of the exponential functionals of Lévy processes which we detail for completeness.

Lemma 3. Let ζ be a Lévy process of Laplace exponent ϕ given by

E exp(λ ζ t ) = exp(tϕ(λ ))
such that ϕ ′ (0) ∈ (0, ∞) and let

I ∞ = ∞ 0 e -ζ s ds . Then 1) EI s ∞ < ∞ for all s ∈ [-1, 0] and all s > 0 : ϕ(-s) < 0 . (29) 2) For r > 0, if ϕ(r) < ∞ and EI -r ∞ < ∞, then EI -r-1 ∞ = ϕ(r) r EI -r ∞ . ( 30
)

Proof. of Lemma 3

The statement 1) comes from a rephrased part of a lemma in Maulik and Zwart on the existence of moments of exponential functionals (see [START_REF] Maulik | Tail asymptotics for exponential of Lévy processes[END_REF] Lemma 2.1). The statement 2) comes from a a recursive argument due to Bertoin and Yor (see [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF], Theorem 2 (i) and Theorem 3), which we detail here for the sake of completeness. Set, for t ≥ 0

J t = ∞ t e -ζ s ds (t ≥ 0) ; J 0 = I ∞ = ∞ 0 e -ζ s ds
For all r > 0 we have

J -r t -J -r 0 = r t 0 e -ζ s J -r-1 s ds (31)
Besides, from the properties of Lévy processes, we deduce

J s = e -ζ s Î(s) and Î(s) = ∞ 0 e -(ζ s+u -ζ s ) du (law)
= I ∞ with Î(s) independent of J s . Plugging into (31) we get

e rζ t Î(t) -r -I -r ∞ = r t 0 e rζ s Î(s) -r-1 ds (32)
Assume that ϕ(r) < ∞ and EI -r-1 = ∞, taking expectations on both sides of the inequality e rζ t Î(t) -r > r t 0 e rζ s Î(s) -r-1 ds we would get EI -r = ∞. Consequently, if EI -r < ∞, we have EI -r-1 < ∞. Moreover, taking expectations on both sides of (32) we get (30). Now, to end the proof of Lemma 2, we apply Lemma 3 with ϕ = ψ m . For m > 0, we choose

s = m -1 hence ϕ(-s) = ψ(1) -ψ(m) which is negative if m > 1. So from 1) of Lemma 3, F(m) is finite for m > 0. Now, if m ∈ [m 0 , 0], set k = ⌊-m⌋. Then m + k ∈ [-1, 0],
and from 1) of Lemma 3, E (m) I m+k ∞ is finite. Besides, we have ψ(m) < ψ(j) for all integers j such that 0 ≤ j ≤ k, hence ϕ(-mj) = ψ(j) -ψ(m) > 0 and applying (30) recursively, we have

F(m) = ϕ(-m)ϕ(-m -1) • • • ϕ(-m -k) (-m)(-m -1) • • • (-m -k) E (m) I m+k ∞ ( 33
)
and this quantity is finite.

Examples

For all the following examples, we give the Laplace exponent ψ and the parameters τ + , τ e . In Sections 5.1, and 5.2 we obtain an explicit expression for τ 0 and I . In Section 5.3 we do not have an explicit expression for I (and sometimes for τ 0 ). For all examples (but one) we identify the behaviour at the boundary, according to the classification of Proposition 1.

The Brownian process with drift (Bessel clock [27])

For ν > 0, we consider the Lévy process

ξ t = 2B t + 2νt .
The pssMp (X t ) is the squared Bessel process of dimension d = 2(1 + ν). Its index is α = 1 and it is the only continuous pssMp (of index 1). The clock is related to the scale function of a diffusion in random environment (see [START_REF] Yor | Exponential functionals of Brownian motion and related processes[END_REF] chap. 10).

We have

ψ(m) = 2m(m + ν), m ± = ±∞, ψ(m ± ) = +∞ .
Here m 0 = -ν/2 and ψ(m 0 ) = -ν 2 /4, so that

τ + = 0, τ 0 = ∞, τ e = 1 2ν , ∆ = (0, ∞) .
The rate function is

I (x) = (1 -2νx) 2 8x , (x > 0) .
Notice that here the function L may be obtained by explicit inversion

L(θ ) = -ν + √ ν 2 -2θ 2 , (θ < ν 2 /2) .
The boundary τ + = 0 is in the situation 4c) of Proposition 1 and the boundary τ 0 is in situation 3a, with asymptote y = ν 2 2 x -ν 2 . The minimum of I , reached in τ e , corresponds to the LLN (Theorem 1) which is in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF] Exercise (4.23) Chap. IV and in [START_REF] Yor | Large deviations for the Bessel clock[END_REF] Theorem 1.1. Let us now comment the change of probability. Under P m , the processes (ξ t ) and (X t ) remain in the same family as under P : they are Brownian motion with drift and squared Bessel, respectively, but with ν replaced by ν + 2m. Moreover, it is known from [START_REF] Yor | Exponential functionals of Brownian motion and related processes[END_REF] Chap. 8 (or equivalently [START_REF] Ph | On exponential functionals of certain levy processes[END_REF]), that under P,

I ∞ (law) = (2Z ν ) -1
where Z ν is a gamma variable of parameter ν. This allows to see directly that the assertion of Lemma 2 holds.

Lévy processes of the form ξ

t = ±dt ± Pois(β , γ) t
Here Pois(a, b) t is the compound Poisson process of parameter a whose jumps are exponential r.v. of parameter b. They are studied in Section 8.4.3 of [START_REF] Yor | Exponential functionals of Brownian motion and related processes[END_REF] (or equivalently [START_REF] Ph | On exponential functionals of certain levy processes[END_REF]), with very informative Tables. In particular, these families are invariant by changes of probability and the law of the exponential functional I ∞ is known, which allows again to see directly that the assertion of Lemma 2 holds.

ξ

t = dt + Pois(β , γ) t with d ≥ 0
(The particular case of the compound Poisson process corresponds to the case d = 0). We have

ψ(m) = m d + β γ -m , m -= -∞ , m + = γ , ψ(m + ) = ∞ ,
Here m 0 = -∞ and ψ(m 0 ) = -∞ so that

τ + = 0 , τ 0 = d -1 , τ e = γ γd + β , ∆ = (0, d -1 ) .
The rate function is

I (x) = γ(1 -dx) -β x 2 , (0 ≤ x ≤ d -1
) . To check the assumption [START_REF] Lamperti | Semi-stable Markov processes[END_REF] we go back to the process ξ . Notice that ξ s ≥ ds for every s ≥ 0, so that τ (ξ ) (t) ≤ d -1 log(1 + dt), whence, for every ε > 0 P τ (ξ ) (t) logt > τ 0 + ε = 0 for t large enough, which is equivalent to [START_REF] Lamperti | Semi-stable Markov processes[END_REF] thanks to [START_REF] Ph | On exponential functionals of certain levy processes[END_REF].

The boundary τ + is in situation 4a and τ 0 in situation 3b, i.e. vertical tangents.

In the following examples, with no loss of generality we may and will assume that d = 1, since (with obvious notations)

τ (ξ ;d,β ) (t) (law) = d -1 τ (ξ ;1,β /d) (dt) . 5.2.2 ξ t = -t + Pois(β , γ) t with 0 < γ < β We have ψ(m) = m -1 + β γ -m , m -= -∞ , m + = γ , ψ(m + ) = ∞ .
Here

m 0 = γ -β γ , ψ(m 0 ) = -( √ γ -β ) 2 , so that τ + = 0 , τ 0 = ∞ , τ e = γ β -γ , ∆ = (0, ∞) .
The rate function is

I (x) = γ(1 + x) -β x 2 (x > 0) .
The boundary τ + is in situation 4a and τ 0 in situation 3a with asymptote y = ( β -√ γ) 2 x + γβ γ.

The saw-tooth process

It is a particular case of the Cramer-Lundberg risk process ([15] Section 1.3.1). 0 < β < γ. The Lévy process ξ is defined by

ξ t = t -Pois(β , γ) t . (34)
The self-similar process X t is described precisely in ( [START_REF] Ph | Beta-gamma random variables and intertwining relations between certain Markov processes[END_REF] p. 327). The Lévy exponent of (ξ t ) is given by:

ψ(m) := m γ -β + m γ + m , m -= -γ , m + = ∞ , ψ(m + ) = ∞ , ψ ′ (m + ) = 1 . (35)
We have easily

m 0 = -γ + β γ , ψ(m 0 ) = - √ γ -β 2 so that τ + = 1 , τ 0 = ∞ , τ e = γ γ -β , ∆ = (1, ∞) .
The rate function is

I (x) = γ(x -1) -β x 2 , (x ≥ 1) .
The condition ( 17) is fulfilled since by definition ξ t ≤ t for all t > 0, so we have τ (ξ ) (t) ≥ logt for all t > 0. The minimum of the rate function I is reached at x = γ γ-β in accordance with the law of large numbers ([9], Theorem 4.7 ii)). The boundary τ + = 1 is in situation 4b and the boundary τ 0 in situation 3a, with an asymptote of equation y = ( √ γβ ) 2 x + β γγ.

Three more examples

The first and the second may be seen for instance in [START_REF] Patie | Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration[END_REF]. The third one is a general class (see [START_REF] Kyprianou | The extended hypergeometric class of Lévy processes[END_REF]).

First

For α ∈ (1, 2), let X ↑ be the spectrally negative regular α-stable Lévy process conditioned to stay positive. It is a pssMp of index α and its associated Lévy process ξ ↑ has Laplace exponent

ψ(m) = c Γ (m + α) Γ (m) and m -= -α , m + = ∞ , ψ(m + ) = ∞ , ψ ′ (m + ) = ∞
where c is a positive constant. We have

ψ ′ (m) ψ(m) = Ψ (m + α) -Ψ (m)
where Ψ is the Digamma function. Then m 0 = γ α where γ α is the (unique) solution in (-1, 0) of the equation

Ψ (γ + α) = Ψ (γ) ,
and then ψ(γ α ) = c Γ (γ α +α) Γ (γ α ) < 0. We have

τ + = 0 , τ 0 = ∞ , τ e = 1 cΓ (α) , ∆ = (0, ∞) .
The boundary τ + is in situation 4c and τ 0 in situation 3a with asymptote. Notice that when α ↑ 2 we end up with the Laplace exponent of a Brownian motion with drift 1/2.

Second

Let κ ∈ (0, 1] and δ > κ/(1 + κ). Let X be the continuous state branching process with immigration whose branching mechanism (see [START_REF] Patie | Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration[END_REF]

Lemma 4.8.) is ϕ(u) = - c κ u κ+1
and immigration mechanism is

χ(u) = cδ κ + 1 κ u κ .
It is a pssMp of index κ and the associated Lévy process has Laplace exponent

ψ(m) = c(κ -(κ + 1)δ -m) Γ (-m + κ) Γ (-m) , m -= -∞ , m + = κ , ψ(m + ) = ∞ .
First, we see that ψ ′ (0) = lim ψ(m)/m = c((κ + 1)δ -κ)Γ (κ) > 0 and τ e = 1/ψ ′ (0). Since as m → -∞ we have ψ(m) ∼ c(-m) κ → ∞, we deduce the existence of m 0 > -∞ such that ψ ′ (m 0 ) = 0 (and ψ(m 0 ) < 0). Besides, when m ↑ κ or h = κm ↓ 0 then ψ ′ (m) ∼ (κ + 1)δΓ ′ (h) Γ (-κ) ↑ ∞ .

We have then τ + = 0 , τ 0 = ∞ , ∆ = (0, ∞) .

The boundary τ + = 0 is in situation 4c and τ 0 is in situation 3a with asymptote.

Hypergeometric-stable process

The modulus of a Cauchy process in R d for d > 1 is a pssMp of index 1 with infinite lifetime. Actually the associated Lévy process is a particular case of hypergeometricstable process of index α as defined in [START_REF] Caballero | Explicit identities for Lévy processes associated to symmetric stable processes[END_REF], with α < d. The characteristic exponent is given therein by Theorem 7, hence the Laplace exponent is : We have then τ + = 0 , τ 0 = ∞ , ∆ = (0, ∞) .

ψ(m) = -
In the particular case α We have no close expression for the rate function. The asymptote as x → ∞ is the line y = 2x π + 1. Notice that in [START_REF] Ph | Exponential functionals of Lévy processes[END_REF], the authors announced a study of the modulus of a multidimensional Cauchy process. They found the expression (36) but never published it ( [START_REF] Petit | Personal Communication[END_REF]).

Fig. 1

 1 Fig. 1 Example 5.1: The rate function I for ν = 1.

Fig. 2

 2 Fig. 2 Example 5.2.1: The rate function I for β = 2, γ = 1, d = 1.

Fig. 3

 3 Fig. 3 Example 5.2.2: The rate function I for β = 2, γ = 1.

Fig. 4

 4 Fig. 4 Example 5.2.3: The rate function I for β = 1, γ = 3.

  2 α Γ ((-m + α)/2) Γ (-m/2) Γ ((m + d)/2) Γ ((m + d -α)/2) and m -= -d , m + = α , ψ(m + ) = ∞ . Since mΓ (-m/2) = -2Γ ((2m)/2) we have ψ ′ (0) = lim m→0 ψ(m)/m = 2 α-1 Γ (α/2)Γ (d/2) Γ ((d -α)/2) > 0 .Moreover, for m = 0ψ ′ (m) ψ(m) = -Ψ (((-m + α)/2)) +Ψ ((m + d)/2) +Ψ (-m/2) -Ψ (m + d -α)/2) ,where Ψ is the digamma function. It is then easy to see that ψ ′ vanishes at m = m 0 := (αd)/2 and ψ(m 0 ) = -2 α Γ ((d + α)/4) Γ ((d -α)/4) 2 .

= 1 and d = 3 ,

 3 applying the identityΓ (x)Γ (1x) = π sin πx we have ψ(m) = (m + 1) tan πm 2(36)with m -= -3 , m + = 1 and ψ(-1) = -2/π by continuity. A direct computation givesψ ′ (m) = sin πm + πm + π 2 cos 2 πm 2 so that ψ ′ (0) = π/2 , m 0 = -1 , θ 0 = -ψ(m 0 ) = 2/π and ∆ = (0, ∞)

Fig. 5

 5 Fig. 5 Example 5.3.3: The rate function I (x) for α = 1, d = 3.

Acknowledgements: The authors want to thank Frédérique Petit for valuable conversations on the Cauchy clock.


Jean-François Lupi

Michèle Ude

Stanislaw Trzesien

Wilfried Blanc

Borut Lenardič

Hervé Guillon

Bernard Dussardier
email: bernard.dussardier@unice.fr

FIBRE EN SILICE À COEUR VITROCÉRAMIQUE DOPÉ ERBIUM ET MAGNÉSIUM PAR LA MÉTHODE DE « FLASH VAPORIZATION »

Keywords: Erbium, fibre, silice, nanoparticules, vitrocéramique

Une fibre en silice dopée à l'erbium et au magnésium a été fabriquée pour la première fois par FCVD et dopée par une méthode de « vaporisation flash ». Le coeur de la fibre contient des nanoparticules de taille < 50 nm, avec une densité volumique de 10 20 à 10 21 m -3 . Les pertes de fond sont inférieures à 0,6 dB/m. Nous étudions les propriétés d'absorption et de fluorescence de l'erbium autour de 1,55 µm.

INTRODUCTION

Développer des nouvelles fibres optiques dopées aux ions de terre rare (TR) pour les amplificateurs de puissance et les lasers nécessite des améliorations continues des propriétés spectroscopiques des fibres en plus de l'optimisation de la compacité et de l'efficacité économique des dispositifs. Certaines applications de fibres dopées TR souffrent de limitations en termes de propriétés spectroscopiques résultant de l'aggrégation des TR ou d'un environnement local inapproprié quand ils sont insérés dans la silice. Une solution intéressante consiste à utiliser la silice comme matériau hôte et support mécanique du guide d'onde, et à intégrer dans celui-ci des nanoparticules (NP) de structure et de composition appropriées, et dopées TR. La méthode originale proposée par le LPMC est basée sur le procédé MCVD (Modified Chemical Vapour Deposition). Comme les systèmes silicatés ont une forte immiscibilité stable quand ils contiennent des oxydes de métaux divalents, on profite des traitements thermiques inhérents à ce procédé pour obtenir des NP par le mécanisme de séparation de phase. En apportant les éléments séparateurs de phase et les ions TR par la méthode dite de dopage en solution, les NP croissent in situ dans le matériau lorsque des ions d'alcalino-terreux (Mg 2+ , Ca 2+ ou Sr 2+ ) sont incorporés en même temps que les TR [START_REF] Blanc | Fabrication of Rare Earth-Doped Transparent Glass Ceramic Optical Fibers by Modified Chemical Vapor Deposition[END_REF]. La distribution de taille des NP, les pertes par diffusion induite et les propriétés spectroscopiques des ions de TR dépendent des paramètres de fabrication [START_REF] Acapito | Different Er 3+ environments in Mg-based nanoparticledoped optical fibre preforms[END_REF]. Le terme 'vitrocéramique' est utilisé ici par simplicité, que les NP soient cristallines ou amorphes.

Récemment, des méthodes alternatives de dopage aux TR et aux modificateurs de la silice (principalement Al) pendant le procédé MCVD ont été developpées pour satisfaire les spécifications particulières des fibres pour lasers de très forte puissance. Parmi celles-ci, celles impliquant le dopage en phase gazeuse jusque vers la zone de réaction et de dépôt dans le tube substrat de silice offrent en principe une grande souplesse, notamment en terme de choix d'éléments chimiques.

Dans cet article nous mettons en oeuvre pour la première fois la méthode de dopage en phase gazeuse par « vaporisation flash » (Flash Vaporization) [START_REF] Lenardic | Fabrication of specialty optical fibers using flash vaporization method[END_REF] pour créer une fibre à coeur en vitrocéramique dopée Er 3+ et étudier le potentiel de cette méthode pour contrôler les propriétés optiques des fibres dopées TR. Ces caractérisations préliminaires permettent d'identifier qualitativement les contributions respectives des ions Er 3+ encapsulés dans les NP de ceux présents dans la matrice environnante.

PRÉPARATION ET CARACTÉRISATION DES FIBRES OPTIQUES

CONCLUSION

La méthode de dopage de préformes MCVD par un système de dopage par Vaporisation Flash a permis de créer un coeur de type vitrocéramique dopé aux ions de terre rare. Les caractérisations préliminaires indiquent qu'une fraction majoritaire des ions Er 3+ seraient encapsulées dans les nanoparticules. Toutefois, la sur-concentration locale en ions Er 3+ dans les NP provoquerait un quenching important. Ceci pourra être amélioré, par exemple en diminuant la concentration globale en ions Er 3+ , par rapport à la concentration d'agents séparateurs de phase (ici, Mg) tout en conservant les paramètres de génération de NP offrant une transparence meilleure que 0,6 dB/m. Ces hypothèses nécessitent d'être confirmées par d'autres caractérisations optiques, compositionnelles et structurelles, dont certains résultats seront présentés lors de la conférence.

RÉFÉRENCES

Fig. 1 :

 1 Fig. 1 : Contraste chimique de la zone de coeur sous MEB. (a) : préforme polie, (b) fibre clivée. Les points blanc les plus nets correspondent à des NP à la surface de l'échantillon. Barres d'échelle : 100 nm. Les analyses EDX indiquent que le germanium, le magnésium et l'erbium sont répartis sur toute la section du coeur. L'erbium a toutefois une distribution hétérogène. La résolution de l'EDX (~1 µm) est supérieure à la taille d'une NP et même de l'intervalle interparticulaire moyen. Ceci ne permet donc pas de conclure quantitativement sur une éventuelle partition (incorporation totale) des ions Er 3+ et Mg 2+ dans les NP, comme nous l'avons observé ailleurs sur le même système avec la méthode de dopage par la solution [4]. Toutefois, on note une forte élévation de concentration en erbium lorsque le faisceau électronique irradie les zones riches en NP par rapport aux zones pauvres en, ou dépourvues de NP. Ceci indiquerait une surconcentration en erbium dans les NP, bien supérieure à la concentration moyenne indiquée plus haut : c'est une situation propice aux aggrégats, et donc aux transferts d'énergie entre ions Er 3+ et au quenching (atténuation voire annulation) de la fluorescence infrarouge. La Fig. 2 montre les spectres d'absorption et d'émission de la fibre. Ils sont comparés aux spectres de fibres dopées Er 3+ caractéristiques : types I, II et III, respectivement germano-silicate, alumino-germano-silicate et aluminosilicate [5]. Le spectre d'absorption de la fibre étudiée ressemble à une fibre de type III, alors que celui d'émission est caractéristique d'une fibre germanosilicate.

Fig. 2 :

 2 Fig. 2 : Spectres d'absorption (a) et d'émission (b) de la fibre codopée Er+Mg, comparés aux courbes d'absorption et d'émission de fibres de types I, II et III.