Keywords: convergence to equilibrium, hypocoercive Markovian semi-groups, spectral decompositions, Ornstein-Ulhenbeck generator, L 2 -operator norms, kinetic evolution equations 35K65, secondary: 35H10, 47A10, 15A18, 15A60, 37A30, 60J60

For any a ą 0, consider the hypocoercive generators yB x `aB 2 y ´yB y and yB x ´axB y `B2 y ´yB y , respectively for px, yq P R{p2πZq ˆR and px, yq P R ˆR. The goal of the paper is to obtain exactly the L 2 pµ a q-operator norms of the corresponding Markov semi-group at any time, where µ a is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.

Spectral decompositions and L 2 -operator norms of toy hypocoercive semi-groups

Sébastien Gadat and Laurent Miclo Institut de Mathématiques de Toulouse, UMR 5219 Université de Toulouse and CNRS, France

Introduction

Convergence to equilibrium of Markovian semi-groups has been investigated a lot under various coercive assumptions on the generator, such as spectral gap or logarithmic Sobolev inequalities, especially in a reversible framework. Nevertheless, asymptotical exponential convergence to equilibrium is also encountered when the generator satisfies some hypoelliptic type conditions. This phenomenon has been called hypocoercivity (see the book of Villani [START_REF] Villani | [END_REF] for the history) and has recently attracted more and more attention, with e.g. the works of Desvillettes and Villani [START_REF] Desvillettes | On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation[END_REF], Eckmann and Hairer [START_REF] Eckmann | Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators[END_REF], Rey-Bellet and Thomas [START_REF] Rey-Bellet | Fluctuations of the entropy production in anharmonic chains[END_REF], Hérau and Nier [START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF], Hérau [START_REF] Hérau | Short and long time behavior of the Fokker-Planck equation in a confining potential and applications[END_REF], Villani [START_REF] Villani | [END_REF], Dolbeault, Mouhot and Schmeiser [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF], Ottobre, Pavliotis and Pravda-Starov [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF] and the references therein. Typically, a hypocoercivity result bounds the convergence to equilibrium (for instance in L 2 or entropy sense) by a term such as C expp´ctq, where t ě 0 is the time and C, c ą 0 are two constants depending on the problem at hand. But these constants are not easy to describe and generally not optimal in the literature mentioned above. Furthermore, the previous bound gives no information about the behavior of the underlying semi-group at small times, namely how it begins to go toward equilibrium. To try to clarify the situation, we will study in details in this paper two simple models of hypocoercivity, by computing exactly the corresponding distance to equilibrium in the L 2 -sense. Despite the scope may seem limited, some features will be intriguing, such as the appearance of discrete binomial and Poisson laws.

The state space of our first toy model is T ˆR, where T ≔ R{p2πZq stands for the usual circle. The coordinates of a generical element of T ˆR are denoted px, yq, where x and y are often interpreted respectively as a position and a speed (i.e. T ˆR is seen as the tangent bundle of T). For any given a ą 0, we are interested in the differential operator L a ≔ yB x `aB 2 y ´yB y .

and in the generated Markovian semi-group pP paq t q tě0 . Consider the product probability measure µ a ≔ λ b γ a , where λ is the normalized Lebesgue measure on T and where γ a is the normal distribution of mean 0 and of variance a. It is easy to check that µ a is invariant for L a : for any smooth function f on T ˆR with bounded derivatives, µ a rL a rf ss " 0. It follows that for any t ě 0, P paq t can be extended into a continuous operator on L 2 pµ a q with operator norm equal to 1. It is furthermore known that µ a is ergodic for the semi-group, in the sense that P paq t converges toward µ a in L 2 pµ a q for t large:

@ f P L 2 pµ a q, lim tÑ`8 › › ›P paq t rf s ´µa rf s › › › " 0,
where }¨} designates the L 2 pµ a q-norm.

Our goal is to recover and to quantify this convergence given by the next result.

Theorem 1 For any a ą 0 and t ě 0, we have

|||P paq t
´µa ||| " max ˆexpp´tq, exp

" ´a ˆt ´2 1 ´expp´tq 1 `expp´tq ˙˙,
where ||| ¨||| stands for the operator norm in L 2 pµ a q.

It is interesting to look at the behaviours of this operator norm for small and large times. As t goes to 0 `, ln ´|||P paq t ´µa ||| ¯" ´a 12 t 3 p1 `op1qq.

(

) 1
This shows that initially, the operator norm decreases quite slowly as a function of time, the power 3 should be seen as an order of the hypocoercivity of the operator L a . On the other side, as t goes to `8, ´ln ´|||P paq t

´µa |||

¯" " apt ´2 `Ope ´tqq , if a ď 1 t , if a ą 1, which reflects the exponential convergence to equilibrium of the semi-group pP paq t q tě0 . This kind of informations cannot be deduced from the bounds obtained in the literature. Indeed, note that the mapping ϕ : R `Q t Þ Ñ t ´2p1 `expp´tqq ´1p1 ´expp´tqq is strictly convex, so that the bound of Theorem 1 is equivalent to the family of inequalities, parametrized by s ě 0, @ t ě 0, |||P paq t ´µa ||| ď maxpexpp´tq, C s expp´c s tqq, where for all s ą 0, C s ≔ expp´apϕpsq ´sϕ 1 psqqq ą 1 and c s ≔ aϕ 1 psq ą 0. Up to scalings in time and in the speed variable and to a change of direction in position, we deduce immediately from Theorem 1:

Corollary 2 For any a, c ą 0 and b P Rzt0u, consider the operator L a,b,c ≔ byB x `aB 2 y ´cyB y , which admits µ a{c as invariant probability. We have for the corresponding semi-goup pP pa,b,cq t q tě0 , @ t ě 0, |||P pa,b,cq t ´µa{c ||| L 2 pµ a{c qý " max ˆexpp´ctq, exp

" ´ab 2 c 3 ˆct ´2 1 ´expp´ctq 1 `expp´ctq ˙˙.
In particular the associated asymptotical exponential rate is lim tÑ`8 ´1 t ln ´|||P pa,b,cq t ´µa{c ||| L 2 pµ a{c qý ¯" min ˆc, ab 2 c 2

˙.

It is instructive to draw a comparison with the heat semi-group pQ paq t q tě0 on T generated by the operator K a ≔ aB 2

x , which injects the same amount a of randomness per unit of time as any one of the generators L a,b,c , where b P R and c ą 0 are free parameters. Since K a is self-adjoint in L 2 pλq and admits a as spectral gap, we get @ t ě 0, |||Q paq t ´λ||| L 2 pλqý " expp´atq.

Thus it appears that if we had to choose between the Monte Carlo procedures pQ paq t q tě0 and pP pa,b,cq t q tě0 to sample according to λ, it would be better to use, with a tuning c ą a and b{c ą 1, the first coordinate for the latter Markov process, namely the primitive integral of an Ornstein-Ulhenbeck process. Of course both procedures require the sampling of the trajectory of a Brownian motion, which is more difficult to get than the sampling of a uniform variable on the circle, nevertheless this is another illustration of the paradigm that to go fast to equilibrium, it is better to resort to non-reversible Markov processes (see for instance [START_REF] Diaconis | On the spectral analysis of second-order Markov chains[END_REF], where this question was studied in the framework of second order finite Markov chains).

Our second toy model has R ˆR as state space and also depends on a parameter a ą 0: we are now interested in the differential operator r L a ≔ yB x ´axB y `B2 y ´yB y .

It is easy to check that the probability measure r µ a ≔ γ 1{a b γ 1 is invariant for r L a and we consider the associated semi-group p r P paq t q tě0 of Markov operators on L 2 pr µ a q. As it will be seen in the next section, in the first model, for all a ą 0, the operator L a is diagonalizable in L 2 pµ a q and its spectrum is real. For the second model, the value 1{4 is critical with this respect: for a P p0, 1{4q, r L a is diagonalizable in L 2 pr µ a q and its spectrum is real, while for a P p1{4, `8q, r L a is still diagonalizable in L 2 pr µ a q (complexified) but most of its eigenvalues are not real. In the critical case a " 1{4, r L a is not diagonalizable in L 2 pr µ a q and contains Jordan blocks of all orders. Nevertheless r µ a is always ergodic and the next result quantifies the convergence: Theorem 3 For any a ą 0 and t ě 0, we have ||| r P paq t ´r µ a ||| " C a ptq exp ˜´1 ´ap1 ´4aq

2 t ¸,
where |||¨||| stands for the operator norm in L 2 pr µ a q and where the factor C a ptq is described as follows: if a P p0, 1{4q, let θ ≔ ? 1 ´4a and define

C a ptq ≔ g f f f ee ´θt `1 ´θ2 2θ 2 p1 ´e´θt q 2 `1 ´e´2θt 2 ¨1 `1 θ d 1 `pθ ´2 ´1q ˆeθt ´1 e θt `1
˙2' .

If a P p1{4, `8q, let θ ≔ ? 4a ´1i and define

C a ptq ≔ d 1 `|e θt ´1| 2|θ| 2 ˆ|e θt ´1| `b|e θt ´1| 2 `4|θ| 2 ˙.
If a " 1{4, define

C a ptq ≔ g f f e 1 `t2 2 `td 1 `ˆt 2
˙2.

Again, let us look more precisely at the behaviors of this operator norm for small and large times. When t ą 0 goes to zero, we obtain as above a decrease of order t 3 : for a P p0, 1{4s, we have ||| r P paq t ´r µ a ||| " 1 ´ˆa 6 `1 ´4a 2 `1 ´?1 ´4a ˘˙t 3 `opt 3 q,

and for a P r1{4, `8q, ||| r P paq t ´r µ a ||| " 1 ´a 6 t 3 `opt 3 q.

When t goes to infinity, the behavior is different according to the position of a with respect to 1{4 (with an asymptotic exponential rate varying for a P p0, 1{4s): if a P p0, 1{4q, we have

||| r P paq t ´r µ a ||| " 1 θ exp ˆ´1 ´?1 ´4a 2 t ˙.
The factor in front of the exponential explodes with time if a " 1{4:

||| r P p1{4q t ´r µ 1{4 ||| " t exp ˆ´t 2 ˙.
If a ą 1{4, since the mapping

R `Q ν Þ Ñ 1 `ν 2p4a ´1q ´ν `aν 2 `4p4a ´1q ¯,
is increasing, it appears that the factor R `Q t Þ Ñ C a ptq is oscillating between the values 1 and a 1 `2p1 `2? aqp4a ´1q ´1 with period T a ≔ 2π{ ? 4a ´1. These oscillations are sufficiently moderate so that R `Q t Þ Ñ C a ptq expp´t{2q is non-increasing, as it is always the case for the L 2 pµqoperator norms of a Markovian semi-group admitting µ as invariant probability. The above periodicity admits a peculiar consequence: it follows from (4) that d dt C a ptq expp´t{2q| t"0 " 0 and in conjunction with @ k P Z `, @ t ě 0, C a pkT a `tq expp´pkT a `tqq " expp´kT a qC a ptq expp´tq, we get that the time derivative of ||| r P paq t ´r µ a ||| vanishes at all times of the form t " kT a , with k P Z `.

On the state space R ˆR, we can play with scalings in time, speed and position to deduce from Theorem 3: q tě0 , with the notation of Theorem 3,

@ t ě 0, ||| r P pa,b,c,dq t ´r µ a,b,c,d ||| L 2 pr µ a,b,c,d qý " C ab{d 2 pdtq exp ˜´1 ´ap1 ´4abd ´2q 2 dt ¸.
It follows that the asymptotic exponential rate of p r P pa,b,c,dq t q tě0 is p1 ´ap1 ´4abd ´2q `q{2. We are led to make a comparison with the semi-group p r Q than p r Q pa,b,c,dq t q tě0 to sample accordingly to γ bc{padq . Hence the remarks for the first model are still valid.

Instead of scaling position and speed variables as in Corollary 4, we could have considered appropriate linear transformations of R 2 and end up with operators associated to certain quadratic symbols. Hypocoercivity of general differential operators with quadratic symbols have been recently investigated by Ottobre, Pavliotis and Pravda-Starov [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF], who obtained bounds on L 2 -convergence which are relatively precise at the level of the exponential rate (showing that all rates strictly below those obtained above are admissible). But they provide no clue about the behavior of the operator norm for small times, while it would be very interesting to relate the order of hypercoercivity (the power 3 in (1), (3) or ( 4)) to the number of times one needs to take Lie brackets in order to get the full tangent space in Hörmander's condition [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF], even only in the framework of quadratic symbols. One first step in this direction would be to investigate finite chains of nearest-neighbor interacting harmonic oscillators coupled to one heat bath (see e.g. Eckmann and Hairer [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF] or Ottobre, Pavliotis and Pravda-Starov [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF], despite that these authors were not primarily interested in this situation).

Our approach is completely different from the pseudo-differential techniques of Ottobre, Pavliotis and Pravda-Starov [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF]. We begin by studying in details the spectral decomposition of the operators at hand. For the second model, it was already done by Risken [START_REF] Risken | The Fokker-Planck equation[END_REF] (see also the book of Helffer and Nier [START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF] or an unpublished paper of Kavian [15]). But we don't stop with the knowledge of the eigenvalues and of the eigenvectors, instead we investigate the scalar product of the eigenvectors: due to the fact that the above generators are not reversible, the eigenvectors cannot be all orthogonals. It appears that their geometric structure can be nicely described by L 2 scalar products with respect to classical discrete laws such as Poisson or binomial distributions. This leads to the construction of certain functions which well-behave under the action of the semi-groups and turn out to be the optimal functions for the computation of the operator norms. It should be noted that these optimal functions change with the time at which are computed the operator norm, explaining why the latter cannot have a simple exponential form.

Of course, one can hope for precise spectral decompositions only in a restricted framework of quadratic symbols (but see also Eckmann and Hairer [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF], where the spectrum of certain hypoelliptic generators is proven to be contained in a cusp). Nevertheless, our analysis put forward a simple Lie algebra structure associated to the above models which is "almost" shared by kinetic models corresponding to operators of the form yB x ´U 1 pxqB y `B2 y ´yB y , say on TˆR, where the potential U : T Ñ R is a smooth function. We believed the revealed structure could lead to a third order linear ordinary differential equation satisfied by the evolution of the L 2 -norm of the semi-group (applied to a generical function of mean zero with respect to the invariant measure), which is sufficiently coercive to imply hypocoercive bounds. Unfortunately this is not true and an idea is still missing with this respect. It was our initial motivation: to find at each time instantaneous informations on the evolution of L 2 -norm of the semi-group which locally describe the trend to equilibrium and globally imply hypocoercive bounds. This approach would be very convenient to deal with the time-inhomogeneous evolutions we have in mind (sampling and optimizing hypocoercive random algorithms) and it explains our interest in the small time behavior. The point of view is different from the traditional analytical approach to hypocoercivity, consisting in replacing the natural L 2norm by a more coercive norm, typically a norm which is comparable to an appropriately weighted H 1 -norm. The additional terms are chosen so that when differentiating with respect to time the evolution semi-group, one gets a first order differential inequality for this new norm (see for instance Villani [START_REF] Villani | [END_REF]). The kind of estimates we are looking for are not more provided by the probabilistic approach to hypocoercivity through Liapounov functions (see for instance Bakry, Cattiaux and Guillin [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF] and the references therein).

The paper is constructed on the following plan: in the next section (respectively Section 4) we investigate the spectral decomposition of the first model (resp. second model), which is used in Section 3 (resp. Section 5) to compute the corresponding operator norms. The last section is devoted to some observations about simple kinetic models and to the motivations sketched in the preceding paragraph.

Spectral decomposition of the first model

We compute here the spectral decomposition of a kinetic generator associated to the null potential on T. Despite it is among the simplest case of hypocoercivity, we did not find its detailed treatment in the literature. The manipulations we are to consider will be encountered again in Section 4, under a slightly modified form. Furthermore, a very helpful Poisson distribution will make a mysterious appearance in this continuous setting! So, for a ą 0, which is fixed for the whole section, we are interested in the operator L a ≔ yB x `aB 2 y ´yB y .

Decomposition of the generator on stable subspaces

A priori it can be seen as an endomorphism on smooth functions defined on T ˆR, but for our purposes, it is better to consider its closure in L 2 pµ a q, where the invariant measure µ a " λ b γ a was presented in the introduction. Here we will mainly consider real Hilbert spaces, since a posteriori all the eigenvalues of L a will be real. If we were in a totally Gaussian setting, namely if T was replaced by R and yB x by yB x ´bxB y , for some constant b ą 0, it would be natural to observe the action of the above operator on tensor products of appropriately normalized Hermite polynomials, as it was done by Risken [START_REF] Risken | The Fokker-Planck equation[END_REF] (see also Section 4 below). In the present situation, it is rather tempting to replace the Hermite polynomials in the first variable (position x) by the usual trigonometric functions. For p P Z `, denote

@ x P T, $ ' & ' % ϕ p pxq ≔ 2 p p! ? p2pq! cosppxq, ψ p pxq ≔ 2 p p! ? p2pq!
sinppxq.

The factors are such that pϕ p , ψ p`1 q pPZ `is an orthonormal basis of L 2 pλq and they are obtained via Wallis' integrals. For the second variable y, it is natural to use the Hermite polynomials since they can be conveniently associated to the standard Gaussian distribution γ 1 . Recall that they are defined by @ q P Z `, @ y P R, h q pyq ≔ p´1q q ? q! exppy 2 {2q d q dy q expp´y 2 {2q,

(see for instance the book of Szegő [START_REF] Szegő | Orthogonal polynomials[END_REF], as well as for their basic properties used below). To get the orthonormal polynomials ph q,a q qPZ `associated to γ a , for any fixed a ą 0, we use the similitude of scale 1{ ? a:

@ q P Z `, @ y P R, h q,a pyq ≔ h q py{ ? aq.

The family ph q,a q qPZ `is then an orthonormal basis of L 2 pγ a q and then pϕ p b h q,a , ψ p`1 b h q,a q p,qPZ ìs an orthonormal basis of L 2 pµ a q. We compute that:

Lemma 5 For all p, q P Z `, we have L a rϕ p b h q,a s " ´qϕ p b h q,a ´?ap ? qψ p b h q´1,a ´?ap a q `1ψ p b h q`1,a , L a rψ p b h q,a s " ´qψ p b h q,a `?ap ? qϕ p b h q´1,a `?ap a q `1ϕ p b h q`1,a .

Proof

From the relations satisfied by the usual Hermite polynomials, we get that for any q P Z `and y P R, ah 2 q,a pyq ´yh 1 q,a pyq " ´qh q,a pyq, ? a a q `1h q`1,a pyq " yh q,a pyq ´?a ? qh q´1,a pyq.

We deduce that for all p, q P Z `and all px, yq P T ˆR, L a rϕ p b h q,a spx, yq " ϕ 1 p pxqyh q,a pyq ´qϕ p pxqh q,a pyq " ´p? aψ p pxqp a q `1h q`1,a pyq `?qh q´1,a pyqq ´aqϕ p pxqh q,a pyq " ´pqϕ p b h q,a `?ap ? qψ p b h q´1,a `?ap a q `1ψ p b h q`1,a qpx, yq.

The computation of L a rψ p b h q,a s is similar.

From these computations we get, on one hand that for q P Z `, ϕ 0 b h q,a is an eigenfunction of L a associated to the eigenvalue ´q and on the other hand that for p P N, the following vector subspaces V p and W p are stable by L a :

V p ≔ ClpVectpϕ p b h q,a , ψ p b h q`1 : q P 2Z `qq, W p ≔ ClpVectpψ p b h q , ϕ p b h q`1,a : q P 2Z `qq, where for any A Ă L 2 pµ a q, ClpAq and VectpAq stand respectively for the closure of A in L 2 pµ a q and for the vector space generated by A.

Spectral analysis of L a on V p

Since each V p and W p are stable subspaces of L a , we must now study the spectral decomposition of the restriction of L a to the Hilbert subspace V p (the same conclusions will also hold for W p ) , where p P N is fixed. Consider the orthonormal basis pe q q qPZ `given by e 0 ≔ ϕ p b h 0,a , e 1 ≔ ψ p b h 1,a , e 2 ≔ ϕ p b h 2,a etc. This basis enables us to identify V p with l 2 pZ `q, Z `being endowed with the counting measure. From Lemma 5, the (infinite) tridiagonal matrix M associated to the restriction of L a to V p described with the basis pe q q qPZ `is

‹ ‹ ‹ ‹ ‹ ‹ ‹ ' . (6)
It appears that this object is only parametrized by c ≔ ? ap. Let us write M " D `cS ´cS ˚, where D and cS are respectively the diagonal and upper-diagonal part of M , so that ´cS ˚corresponds to the lower-diagonal of M . Note that if S is interpreted as an (unbounded) operator of l 2 pZ `q, then S ˚is (the infinite matrix associated to) its adjoint operator in l 2 pZ `q. In the sequel, we won't make much difference between such matrices and their corresponding operators on l 2 pZ `q, but some preliminaries are needed in order to precisely define their domains. A priori the operators M , D, S and S ˚are well-defined on D, the subspace of real sequences pzpqqq qPZ `from l 2 pZ `q which admit only a finite number of non-zero coefficients. It is immediate to check that they are in fact closable and that the domains of their closures are given by DpSq " DpS ˚q " DpDq " DpM q " tpzpqqq qPZ `P l 2 pZ `q : ÿ qPZ `qz 2 pqq ă `8u.

It is natural to identify the operators M , D, S and S ˚with their respective closures. In particular the spectral decomposition of the restriction of L a to V p is then equivalent to the one of M . Nevertheless, it is more fruitful to look at the operators M , D, S and S ˚as endomorphisms of S, the subspace of sequences pzpqqq qPZ `from l 2 pZ `q which are such that for any r ě 0, ř qPZ `qr z 2 pqq ă `8. The advantage of this point of view is that we can compose the above operators without having to take care about their domains.

We can now state the main result of this paragraph wich describes the spectral analysis of M . Theorem 6 Let ξ 0 ≔ pξ 0 pqqq qPZ `be the element of S given by @ q P Z `, ξ 0 pqq ≔ p´1q t q`1 2 u c q ? q! expp´c 2 {2q.

Consider the elements of S defined by

@ n P Z `, ξ n " pcI ´S˚qn ξ 0 ,
where I is the identity operator. Then for any n P Z `, ξ n is an eigenvector of M associated to the eigenvalue ´c2 ´n. Furthermore pξ n q nPZ `is a (Hilbert) basis of l 2 pZ `q.

The proof will be based on the Lie algebra generated by the operators D, S and S ˚, whose structure is determined by the following computation:

Lemma 7 We have that rS, S ˚s " I, rD, Ss " S, rD, S ˚s " ´S˚.

Proof

Recall that we interpret the operators as endomorphism of S, so the above brackets are well-defined.

For any q P Z `, we have that SS ˚pe q q " Spp´1q q a q `1e q`1 q " p´1q q a q `1Spe q`1 q " p´1q q a q `1p´1q q a q `1e q " pq `1qe q .

Similarly, we get that S ˚Spe q q " qe q , so that rS, S ˚spe q q " pSS ˚´S ˚Sqpe q q " e q , namely rS, S ˚s " I. For any q P Z `, we also compute, with the convention e ´1 " 0, that DSpe q q " p´1q q`1 ? qDpe q´1 q " p´1q q ? qpq ´1qe q´1 , SDpe q q " ´qSpe q q " p´1q q q ? qe q´1 .

It follows that rD, Ss " pDS ´SDqpe q q " p´1q q`1 ? qe q´1 " Spe q q, hence rD, Ss " S. The last relation is an immediate consequence of the previous one, since D ˚" D:

rD, S ˚s " ´rD ˚, pS ˚q˚s˚" ´rD, Ss ˚" ´S˚ Let us denote by V the vector subspace of endomorphisms of S generated by D, S, S ˚and I. Since the latter operators are clearly independent, V is 4-dimensional. Furthermore, taking into account that rI, Ds " rI, Ss " rI, S ˚s " 0, the bracket r¨, ¨s endows V with a Lie algebra structure. This property of V suggests that to get informations about the spectral decomposition of M " D `cS ´cS ˚, it is interesting to first investigate the spectral decomposition of the adjoint operator of M (in the Lie algebra sense, see for instance the book of Hall [START_REF] Hall | Lie groups, Lie algebras, and representations[END_REF]), which is defined by ad

M : V Q X Þ Ñ rM, Xs P V.
This is the object of the next result:

Lemma 8 The kernel of the operator ad M is 2-dimensional and is generated by I and M . There are two other eigenvalues, 1 and ´1, whose corresponding eigenspaces are respectively generated by J `≔ cI `S and J ´≔ cI ´S˚.

Proof Indeed, with the help of Lemma 7 we compute that the matrix associated to ad M in the basis pI, D, S, S ˚q is given by ¨0 0 c c

0 0 0 0 0 ´c 1 0 0 ´c 0 ´1 ‹ ‹ ' .
This matrix is not difficult to diagonalize, its characteristic polynomial is X 2 pX 2 ´1q, and the announced results easily follow.

The interest of the operators J `and J ´is summarized as follows: if z P CS is an eigenvector of M associated to the eigenvalue l P C, then either J `pzq " 0 or J `pzq is an eigenvector of M associated to the eigenvalue l `1. Indeed, the relation rM, J `s " J `implies that M pJ `pzqq " J `pM pzqq `J`p zq " pl `1qJ `pzq.

Similarly, either J ´pzq " 0 or J ´pzq is an eigenvector of M associated to the eigenvalue l ´1. This observation will be the key to the spectral decomposition of M , but let us first notice that any eigenvalue l P C of M has a non-positive real part. To show this assertion, let z " pzpqqq qPZ This argument can be extended to the eigenvalues of any Markovian generator L in L 2 pµq, where µ is an invariant probability for L, and in particular to the eigenvalues of L a in L 2 pµ a q.

Thus if there exists an eigenvalue l P C of M associated to an eigenvector z P CSzt0u, then necessary we can find n P Z `such that J n `pzq " 0. Because otherwise, we would conclude that for any n P Z `, l `n is an eigenvalue of M and thus its real part is non-positive, which is not possible. This is a hint on how we can find some eigenvectors of M : by looking at the kernel of J `, whose computation is our next task.

Lemma 9

The kernel of J `: DpSq Ñ l 2 pZ `q is generated by the vector ξ 0 appearing in Theorem 6. The kernel of J ´: DpSq Ñ l 2 pZ `q is reduced to t0u.

Proof

More generally, let z " pzpqqq qPZ `be any sequence from R Z `, J `pzq can be defined as the sequence pJ `pzqpqqq qPZ `with @ q P Z `, J `pzqpqq ≔ czpqq `p´1q q a q `1zpq `1q, So the equation J `pzq " 0 is equivalent to @ q P Z `, zpq `1q " p´1q q`1 czpqq ? q `1 .

It appears that such a sequence z is determined by zp0q:

@ q P Z `, zpqq " p´1q pq`1qq 2 c q ? q! zp0q.
Remarking that for all q P Z `, p´1q

pq`1qq 2 " p´1q t q`1 2 u , we deduce that z is proportional to ξ 0 . The first announced result then follows from the fact that ξ 0 P DpSq. The kernel of J ´is obtained in a similar way, noting that J ´can also be extended to R Z `via @ z " pzpqqq qPZ `P R Z `, @ q P Z `, J ´pzqpqq ≔ czpqq ´p´1q q`1 ? qzpq ´1q.

Thus, starting with J ´pzqp0q " czp0q, if z is such that J ´pzq " 0, we get that zp0q " 0 and by iteration we end up with z " 0.

More precisely, we have ξ 0 P S and for any function f : Z `Ñ R `, we observe that ÿ

qPZ `f pqqξ 2 0 pqq " Erf pN c 2 qs (7)
where N c 2 is a Poisson distribution of parameter c 2 . This is why we have chosen the normalization ξ 0 p0q " expp´c 2 {2q, which implies that ξ 0 has norm 1 in l 2 pZ `q. It follows another important computational property of ξ 0 with respect to the operator algebra generated by S ˚. As a byproduct, we check that ξ 0 is an eigenvector of M , as this was suggested by the observations made before Lemma 9 (note that this is also a qualitative consequence of the facts that J `pM pξ 0 qq " M pJ `pξ 0 qq ´J`p ξ 0 q " 0 and that kerpJ `q is one-dimensional).

Lemma 10 We compute that for any n P Z `,

pS ˚qn pξ 0 q " 1 c n DpD `1qpD `2q ¨¨¨pD `n ´1qpξ 0 q. ( 8
)
It follows from the particular case n " 1 that M pξ 0 q " ´c2 ξ 0 .

Proof By the usual convention that a void product is equal to 1 or I, for n " 0, (8) reduces to ξ 0 " ξ 0 . Let us check it for n " 1, namely that S ˚pξ 0 q " 1 c Dpξ 0 q. For any q P Z `, we have S ˚pξ 0 qpqq " p´1q q`1 ? qξ 0 pq ´1q " p´1q q`1`t q 2 u ?

q c q´1 expp´c 2 {2q a pq ´1q! " ´q c ξ 0 pqq " 1 c Dpξ 0 qpqq,
where we have used that p´1q q`1`t q 2 u " ´p´1q t q`1 2 u . Since J `pξ 0 q " 0, we deduce directly that Spξ 0 q " ´cξ 0 . Recalling that M " D `cS ´cS ˚, it follows that M pξ 0 q " Dpξ 0 q ´c2 ξ 0 ´Dpξ 0 q " ´c2 ξ 0 .

Next we prove (8) by induction over n. So let us assume it for a given n P N, we write pS ˚qn`1 pξ 0 q " pS ˚qn D c pξ 0 q " 1 c prpS ˚qn , Ds pξ 0 q `DpS ˚qn pξ 0 qq .

Lemma 7 enables to compute the above bracket:

rpS ˚qn , Ds " pS ˚qn´1 rS ˚, Ds `pS ˚qn´2 rS ˚, DsS ˚`¨¨¨`rS ˚, DspS ˚qn´1
" pS ˚qn´1 S ˚`pS ˚qn´2 S ˚S˚`¨¨¨`S˚p S ˚qn´1 " npS ˚qn .

Putting together these computations, we get pS ˚qn`1 pξ 0 q " 1 c pnpS ˚qn pξ 0 q `DpS ˚qn pξ 0 qq " 1 c pD `nqpS ˚qn pξ 0 q " 1 c n`1 DpD `1qpD `2q ¨¨¨pD `n ´1qpD `nqpξ 0 q, as wanted.

Starting with the eigenvector ξ 0 P S, we construct the sequence of eigenvectors pξ n q nPZ `≔ pJ n ´pξ 0 qq nPZ `which are associated to the eigenvalues p´c 2 ´nq nPZ `, according to the discussion following the proof of Lemma 8. Indeed, none of the vectors J n ´pξ 0 q, for n P N, vanishes, because we have seen in Lemma 9 that the kernel of J ´is trivial.

Since the elements of the sequence pξ n q nPZ `are non-zero and associated to different eigenvalues, it is easy to see that any finite family of them is independent in l 2 pZ `q. It is more involved to check that the whole sequence pξ n q nPZ `is independent in l 2 pZ `q. To go in this direction, we present an isometry which will also play an important role in the next section. It gives a convenient way to deal with the fact that the vectors of the sequence pξ n q nPZ `are non-orthogonal.

Let Q be the subspace of V p consisting of vectors z which can be written as a linear combinaison of a finite number of elements of pξ n q nPZ `:

z " ÿ nPZ `f pnqξ n , (9)
where only a finite number of the real coefficients f pnq are non-zero. Due to the above observation, these coefficients are uniquely determined for z P Q. So we associate to such an element z P Q the polynomial

F pXq ≔ ÿ nPZ `f pnqX n . (10)
We also consider the function G defined on Z `by

@ n P Z `, Gpnq ≔ ˆ1 `1 c d dX ˙n F pXq ˇˇˇX "c , (11)
(where the power n corresponds to the composition of differential operators).

Proposition 11

The mapping Q Q z Þ Ñ G is an isometry with respect to the norms l 2 pZ `q and L 2 pPpc 2 qq, where Ppc 2 q stands for the Poisson distribution of parameter c 2 .

Proof By definition and Lemma 10, we have for any n P Z `,

ξ n " pcI ´S˚qn ξ 0 " ÿ lP 0,n ˆn l ˙cn´l p´1q l pS ˚ql ξ 0 " ÿ lP 0,n ˆn l ˙cn´l ˆ´1 c ˙l DpD `1q ¨¨¨pD `l ´1qξ 0 " c n ÿ lP 0,n ˆn l ˙c´2l p´Dqp´D ´1q ¨¨¨p´D ´l `1qξ 0 . (12)
We deduce that for any n, m P Z `,

xξ n , ξ m y " c n`m ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq xp´Dqp´D ´1q ¨¨¨p´D ´l `1qξ 0 , p´Dqp´D ´1q ¨¨¨p´D ´k `1qξ 0 y " c n`m ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq ÿ qPZ `qpq ´1q ¨¨¨pq ´l `1qqpq ´1q ¨¨¨pq ´k `1qξ 2 0 pqq " c n`m ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq ErN pN ´1q ¨¨¨pN ´l `1qN pN ´1q ¨¨¨pN ´k `1qs,
where N is a Poisson random variable of parameter c 2 (recall ( 7)

). It follows that if z " ř nPZ `f pnqξ n belongs to Q, then xz, zy " ÿ n,mPZ `f pnqf pmq xξ n , ξ m y " ÿ n,mPZ `f pnqf pmqc n`m ÿ lP 0,n ,kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq ErN pN ´1q ¨¨¨pN ´l `1qN pN ´1q ¨¨¨pN ´k `1qs,
where only a finite number of terms are non-zero. Note that we have for any l P Z `, we have

d l dX l F pXq ˇˇˇX "c " ÿ nPZ `f pnqnpn ´1q ¨¨¨pn ´l `1qc n´l .
Henw rewritting terms `n l ˘N pN ´1q ¨¨¨pN ´l `1q under the form `N l ˘npn ´1q ¨¨¨pn ´l `1q, we get

xz, zy " ÿ l,kPZ `dl dX l F pXq ˇˇˇX "c d k dX k F pXq ˇˇˇX "c c ´pl`kq E "ˆN l ˙ˆN k ˙ " E « ˆ1 `c´1 d dX ˙N F pXq ˇˇˇˇX "c ˆ1 `c´1 d dX ˙N F pXq ˇˇˇˇX "c ff " ErG 2 pN qs,
which is the wanted isometry relation.

In order to prove the independence of the family pξ n q nPZ `, we need to control the mapping associating F to G, this is the goal of next result.

Lemma 12 Using the notations introduced in ( 9), ( 10) and ( 11), we have

@ n P Z `, |f pnq| ď c n exp `p4c 2 `2 `c´2 q{2 ˘aErG 2 pN c 2 qs,
where

N c 2 is a Poisson random variable of parameter c 2 .
Proof By definition, we have for any n P Z `,

Gpnq " ÿ mP 0,n ˆn m ˙F pmq pcq c m (13)
" ÿ mPZ `npn ´1q ¨¨¨pn ´m `1q

F pmq pcq m! c m .
For any real x ą c, denote by H x the density of a Poisson distribution of parameter px ´cqc with respect to a Poisson distribution of parameter c 2 :

@ n P Z `, H x pnq " ˆpx ´cqc c 2 ˙n expp´px ´cqc `c2 q (14)
"

ˆx ´c c ˙n expp´cx `2c 2 q.
Its interest is that for any m P Z `, we have

ErN c 2 pN c 2 ´1q ¨¨¨pN c 2 ´m `1qH x pN c 2 qs " ErN cpx´cq pN cpx´cq ´1q ¨¨¨pN cpx´cq ´m `1qs " px ´cq m c m ,
where N r stands for a Poisson random variable of parameter r, for any r ą 0. Putting together the above relations, we get that

ErGpN c 2 qH x pN c 2 qs " ÿ mPZ `F pmq pcq m! px ´cq m " F pxq.
By analytic extension, this identity holds for any x P C, since both sides are easily seen to be holomorphic functions of x. In particular for any θ P r0, 2πq and n P Z `, we get

ˆ2 c exppiθq ˙n F ˆc exppiθq 2 ˙" ErGpN c 2 qp2{cq n expp´inθqH c exppiθq{2 pN c 2 qs.
An integration with respect to 1 r0,2πq pθqdθ{p2πq yields

f pnq " ż r0,2πq ˆ2 c exppiθq ˙n F ˆc exppiθq 2 ˙dθ 2π " ErGpN c 2 qJ n pN c 2 qs,
where for any m P Z `, we have

J n pmq " ż r0,2πq
p2{cq n expp´inθqH c exppiθq{2 pmq dθ 2π ( 15)

" 1 n! d n dX n H X pmq ˇˇˇX "0 " expp2c 2 q c m n! d n dX n pX ´cq m expp´cXq ˇˇˇX "0 " expp2c 2 q c m n! ÿ pP 0,n ˆn p ˙dp dX p pX ´cq m d n´p dX n´p expp´cXq ˇˇˇˇˇX "0 " expp2c 2 q c m n! ÿ pP 0,n ˆn p ˙mpm ´1q ¨¨¨pm ´p `1qpX ´cq m´p p´cq n´p expp´cXq ˇˇˇˇˇX "0 " expp2c 2 q c m n! ÿ pP 0,n ˆn p ˙mpm ´1q ¨¨¨pm ´p `1qp´cq m`n´2p " p´1q m`n c n expp2c 2 q n! ÿ pP 0,n ˆn p ˙mpm ´1q ¨¨¨pm ´p `1qc ´2p .
Using Cauchy-Schwarz inequality, we obtain for any n P Z `,

|f pnq| ď a ErJ 2 n pN c 2 qs a ErG 2 pN c 2 qs.
To bound the first factor, we write that for any m P Z `,

expp´2c 2 q |J n pmq| " c n n! ÿ pPZ `ˆn p ˙mpm ´1q ¨¨¨pm ´p `1qc ´2p " c n n! ÿ pPZ `ˆm p ˙npn ´1q ¨¨¨pn ´p `1qc ´2p (16) ď c n ÿ pPZ `ˆm p ˙c´2p " c n `1 `c´2 ˘m .
Thus we get as announced that

ErJ 2 n pN c 2 qs ď c 2n expp4c 2 qE « ˆ1 `1 c 2 ˙2N c 2 ff " c 2n expp4c 2 q exp ˜c2 ˆ1 `1 c 2 ˙2 ´c2 ¸.
The independence in l 2 pZ `q of the family pξ n q nPZ `now follows without difficulty: it is equivalent to the fact that if ř nPZ `f pnqξ n " 0, where the sum in l.h.s. is converging in l 2 pZ `q, then f pnq " 0 for all n P Z `. But for n P Z `, consider z n ≔ ř mP 0,n f pmqξ m P Q, vector to which we associate the function G n as in [START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF]. The convergence of the sequence pz n q nPZ `to zero in l 2 pZ `q is equivalent to

lim nÑ8 ErG 2 n pN c 2 qs " 0.
According to Lemma 12, we have for any m ď n,

|f pmq| ď c m exp `p4c 2 `2 `c´2 q{2 ˘aErG 2 n pN c 2 qs,
so letting n going to infinity, we get f pmq " 0, for any given m P Z `, as required.

Remark 13 Denote by φ the map that associates to any G P L 2 pP c 2 q the formal series F pXq ≔ ř nPZ `f pnqX n , where

@ n, m P Z `, f pnq ≔ ErGpN c 2 qJ n pN c 2 qs J n pmq ≔ p´1q m`n c n expp2c 2 q n! ÿ pP 0,n ˆn p ˙mpm ´1q ¨¨¨pm ´p `1qc ´2p .
The previous proof shows that the bound of Lemma 12 is valid in this context, so the convergence radius of F is at least 1. But the above arguments can be improved to get that F define in fact a holomorphic function in the whole plane. More precisely, in ( 16), we can rather use the bound

npn ´1q ¨¨¨pn ´p `1q n! " 1 pn ´pq! ď 1 pn ´mq! ,
with the convention that pn ´mq! " 1 if m ě n. Consequently we have

@ n, m P Z `, |J n pmq| ď expp2c 2 q c n pn ´mq! ˆ1 `1 c 2 ˙m . It follows that, for n P N, ErJ 2 n pN c 2 qs ď c 2n expp4c 2 qE « 1 ppn ´Nc 2 q!q 2 ˆ1 `1 c 2 ˙2N c 2 ff ď c 2n expp4c 2 qE « 1 pn ´Nc 2 q! ˆ1 `1 c 2 ˙2N c 2 ff ď c 2n expp4c 2 q n! E « n N c 2 ˆ1 `1 c 2 ˙2N c 2 ff " c 2n expp4c 2 q n! exp ˜c2 ˆ1 `1 c 2 ˙2 n ´c2 ¸.
We conclude that

@ n P N, |f pnq| ď c n expp2c 2 q ? n! exp ˜˜ˆc `1 c ˙2 n ´c2 ¸{2 ¸aErG 2 pN c 2 qs,
and this bound is sufficient to insure that F P HpCq. Note that for n " 0, the above computations have to be slightly modified, starting with J 0 pmq " p´1q m expp2c 2 q and ending with |f p0q| ď expp2c 2 q a ErG 2 pN c 2 qs. One can go further and check that (13) holds for any G P L 2 pP c 2 q. Indeed, first consider the holomorphic function R defined by

@ x P C, Rpxq ≔ ErGpN c 2 qH x pN c 2 qs ´F pxq.
By definition of F , see also (15), we have

@ n P Z `, 1 2π ż Cp0,c{2q
x ´nRpxq dx " 0, where Cp0, c{2q is the circle of radius c{2 centered at 0. By holomorphy, this implies that

@ x P C, F pxq " ErGpN c 2 qH x pN c 2 qs. ( 17
)
Next we compute that for any n, n 1 P Z `(recall (14)),

ÿ mP 0,n ˆn m ˙1 c m d m dx m H x pn 1 q ˇˇˇx "c " exppc 2 q ÿ mP 0,n ˆn m ˙1 c m ÿ pP 0,m ˆm p ˙dp dx p ˆx ´c c ˙n1 ˇˇˇˇx "c d m´p dx m´p expp´cpx ´cqq ˇˇˇx "c " 1 0,n pn 1 q exppc 2 q ÿ mP 0,n ˆn m ˙1 c m`n 1 ˆm n 1 ˙n1 !p´cq m´n 1 " 1 0,n pn 1 q exppc 2 qc ´2n 1 ÿ mP n 1 ,n
npn ´1q ¨¨¨pn ´m `1q pm ´n1 q! p´1q m´n 1 " 1 0,n pn 1 q exppc 2 q npn ´1q ¨¨¨pn ´n1 `1q c 2n 1 ÿ lP 0,n´n 1 ˆn ´n1 l ˙p´1q l " δ n pn 1 q n!c ´2n exppc 2 q " pPpc 2 qrnsq ´1δ n pn 1 q.

Thus ( 13) is obtained by applying the operator

ř mP 0,n `n m ˘1 c m d m
dx m ˇˇx"c to [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF]. Remark

14 Due to the independence of pξ n q nPZ `in l 2 pZ `q, the linear morphism

Q Q z Þ Ñ G can be extended to the closure ClpQq of Q in l 2 pZ `q, let
us call ψ this mapping. It is an isometry between ClpQq and the closure of ψpQq in L 2 pPpc 2 qq. We deduce from (13) that the image of Q by ψ is the space of the restrictions to Z `of polynomial mappings, which is well-known to be dense in L 2 pPpc 2 qq. Thus ψ is an isometry between ClpQq and L 2 pPpc 2 qq. It appears that the inverse of ψ is ϕ ˝φ, where φ is defined at the beginning of Remark 13 and where ϕ associates to any series ř nPZ `f pnqX n from φpL 2 pPpc 2 qqq the element

ř nPZ `f pnqξ n of ClpQq.
It is time to check that ClpQq " l 2 pZ `q, this will end the Proof of Theorem 6 Indeed, by the above results, the density of Q in l 2 pZ `q will enable us to conclude that pξ n q nPZ ìs a Hilbert basis of l 2 pZ `q. Thus it remains to show that if z P l 2 pZ `q is such that xz, ξ n y " 0 for all n P Z `, then z " 0. So let z " pzpqqq qPZ `be such an element. Since ξ n " J n ´pξ 0 q and J ´" cI ´S˚, this vector z also satisfies

@ n P Z `,
xz, S ˚npξ 0 qy " 0, and according to Lemma 10, this is also equivalent to @ n P Z `, xz, D n pξ 0 qy " 0, or

@ n P Z `, ÿ qPZ `qn zpqqp´1q t q`1 2 u c q ? q! expp´c 2 {2q " 0. ( 18
)
Let us denote m " pmpqqq qPZ `the signed measure on Z `with @ q P Z `, mpqq ≔ zpqqp´1q t q`1 2 u c q ? q! expp´c 2 {2q.

Let m `and m ´stand respectively for the non-negative and non-positive parts of m, so that m " m `´m ´. From ( 7) and for all r ě 0, we have

ÿ qPZ `expprqq |mpqq| ď d ÿ qPZ `expp2rqqξ 2 0 pqq d ÿ qPZ `z2 pqq " exppc 2 pexpp2rq ´1q{2q a xz, zy ă `8
and thus m `and m ´are non-negative measures admitting exponential moments of all order. Furthermore [START_REF] Rey-Bellet | Fluctuations of the entropy production in anharmonic chains[END_REF] shows that all the usual moments of m `and m ´coincide, so we can apply the moment characterizing theorem (see for instance the section XV4 of the book of Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF]) to get that m `" m ´, namely m " 0. It follows that z " 0 as wanted.

Eigenvectors properties

Let us now compute more explicitely the eigenvectors ξ n , for n P Z `, Proposition 15 For any n P Z `, the mapping ξ n defined in Theorem 6 is given, as a function of px, yq P T ˆR, almost everywhere by

ξ n px, yq " 2 p p! a p2pq! ? n!ℜ ˆin h n ˆy ? a ´2ic ˙exppippx `yqq " 2 p p! a p2pq! ÿ lP 0,n ˆn l ˙p2cq n´l ? l!h l,a pyqℜpi l exppippx `yqqq.
Thus ξ n px, yq is an appropriate linear combination of terms of the types y m cospppx `yqq and y m sinpppx `yqq for m P 0, n .

Proof

For any given n P Z `, let us write ξ n " ÿ qPZ `ξn pqqe q , with @ q P Z `, ξ n pqq ≔ ppcI ´S˚qn ξ 0 qpqq.

Taking into account [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF] and the definition of ξ 0 given in Theorem 6, we get that for any q P Z `, ξ n pqq " c n ÿ lP 0,n ˆn l ˙c´2l pp´Dqp´D ´1q ¨¨¨p´D ´l `1qξ 0 qpqq " c n ÿ lP 0,n ˆn l ˙c´2l qpq ´1q ¨¨¨pq ´l `1qp´1q t q`1 2 u c q ? q! expp´c 2 {2q.

Denoting k p ≔ 2 p p! ? p2pq!
, it is not difficult to check from the definition of the orthonormal basis pe q q qPZ `that @ q P Z `, @ px, yq P T ˆR, e q px, yq " k p ℜpp´1q t q`1 2 u i q exppipxqh q,a pyqq.

Rigorously speaking, such equalities have to be understood almost everywhere in px, yq P T ˆR, since we are dealing with functions from L 2 pµ a q. Putting these expansions together, it appears that

ξ n px, yq (20)
" expp´c 2 {2qk p c n ℜ ¨ÿ qPZ `ÿ lP 0,n ˆn l ˙c´2l qpq ´1q ¨¨¨pq ´l `1q c q ? q! i q exppipxqh q,a pyq '.

Interpreting again the term and W p qpq ´1q ¨¨¨pq ´l `1qc q´2l as c ´l d l dX l X q ˇˇX "c

, we have

ÿ lP 0,n ˆn l ˙c´2l qpq ´1q ¨¨¨pq ´l `1qc q " ˆ1 `1 c d dX ˙n X q ˇˇˇX "c
, so that ÿ qPZ `ÿ lP 0,n ˆn l ˙c´2l qpq ´1q ¨¨¨pq ´l `1q c q ? q! i q h q,a pyq " ˆ1 `1 c d dX

˙n ÿ qPZ `X q ? q! i q h q,a pyq ˇˇˇˇˇX "c
.

To go further, recall that Hermite polynomials satisfy @ r P C, @ y P R, ÿ qPZ `rq h q pyq ? q! " exppry ´r2 {2q.

Thus we deduce that ÿ qPZ `X q ? q! i q h q,a pyq " exp

ˆiXy ? a `X2 2
" exp ˜´1 2 ˆiX ´y ? a ˙2 `y2 2a ¸.
Recalling the definition (5) of the Hermite polynomials, the previous formulation leads to ÿ qPZ `ÿ lP 0,n ˆn l ˙c´2l qpq ´1q ¨¨¨pq ´l `1q c q ? q! i q h q,a pyq where we have used another property of Hermite polynomials:

@ n P Z `, @ r, s P C, ? n!h n pr `sq " ÿ lP 0,n ˆn l ˙sn´l ? l!h l prq.
This relation, parity properties of the Hermite polynomials and equation ( 20) lead immediately to the announced expressions.

Remark 16 A posteriori, the last assertion of Proposition 15, as well as the spectrum of the restriction of L a to V p , could have been obtained in the following way. Consider the change of variables T ˆR Q px, yq Þ Ñ pz, yq P T ˆR with z " x `y (in T). Acting on functions of the form gpz, yq, the generator L a can be rewritten under the form

p L a ≔ aB 2 z `2aB z B y `aB 2 y ´yB y .
Consider next functions g of product type g 1 b g 2 , with

g 1 : T Q z Þ Ñ exppαzq P T,
where α P iZ. The relation B z g 1 pzq " αg 1 pzq implies that p L a rg 1 b g 2 s " g 1 b p L a,α rg 2 s, where p L a,α is the Sturm-Liouville differential operator acting on functions h of the real variable y through p L a,α rhspyq ≔ ah 2 pyq ´py ´2aαqh 1 pyq `aα 2 hpyq.

It is not difficult to check that this operator admits a family pp q q qPZ `of polynomials with complex coefficients such that: for any q P Z `, p q is of degree q and p L a,α rp q s " paα 2 ´qqp q (the factor aα 2 ´q is imposed by the coefficient of highest degree of p L a,α rp q s). Thus we easily recover all the spectal information contained in Theorem 6 and Proposition 15. But relations such as those described in Lemma 7 will be encountered again in Sections 4 and 6, indeed, they are the starting point of all our developments. 20

Spectral analysis of L a on W p

The spectral decomposition of the restriction of L a to W p , for fixed p P N, is similar. This is due to the fact that the restriction of L a to W p is conjugate to the restriction of L a to V p . More precisely, consider the basis pe 1 q q qPZ `of W p given by

e 1 0 ≔ ψ p b h 0,a , e 1 1 , ≔ ´ϕp b h 1,a , e 1 2 ≔ ψ p b h 2,a , e 1 3 , ≔ ´ϕp b h 3,a , . . . . . .
Then the matrix of the restriction of L a to W p in the basis pe 1 q q qPZ `is also given by ( 6). Thus Theorem 6 and Proposition 15 are still valid, after obvious modifications (note for instance that [START_REF] Risken | The Fokker-Planck equation[END_REF] remains true is we replace e q by e 1 q and the real part ℜ by the imaginary part ℑ):

Proposition 17 For n P Z `, consider ξ 1 n ≔ pcI ´S˚1 q n ξ 1 0 , where S ˚1 and ξ 1 0 have the same coefficients as S ˚and ξ 0 , introduced in Theorem 6, but in the basis pe 1 q q qPZ `instead of pe q q qPZ `.

Then pξ n q nPZ `is a Hilbert basis of W p consisting of eigenvectors associated respectively to the eigenvalues p´c 2 ´nq nPN of the restriction of L a to W p . Coming back to functional notations, we have for all n P N, @ px, yq P T ˆR, ξ 1 n px, yq "

2 p p! a p2pq! ? n!ℑ ˆin h n ˆy ? a ´2ic ˙exppippx `yqq ˙.
This result completes the spectral decomposition of L a in L 2 pµ a q. This operator is diagonalizable, the set of its eigenvalues is Λ a ≔ t´c 2 ´n : p, n P Z `u, and the multiplicity of any l P Λ a is 1 Z `plq `2cardptpp, nq P N ˆZ`: l " ´pc 2 `nquq (in particular if a is not rational, the multiplicity of l P Λ a is 1 or 2, according to l P Z `or not).

Remark 18

The above conclusions do not extend to the Gaussian framework, where one is rather interested in the (closure in L 2 pγ b ´1 b γ a q of the) operator

r L a,b ≔ yB x ´bxB y `aB 2 y ´yB y ,
where a, b ą 0. As it will be seen in Section 4 (considering the scalings x Þ Ñ x{ ? a and y Þ Ñ y{ ? a), r L a,b is diagonalizable only if b " 1{4 (for b " 1{4, Jordan blocks of all orders appear), while for b ą 1{4, some of the eigenvalues are not real. In some sense, the appearance of complex eigenvalues facilitates the convergence to equilibrium (see the end of Section 5) and here we are far from this situation, if we look at L a as an ersatz of r L 1,b as b Ñ 0 `.

2.5

Link with hypocoercivity

The above spectral decomposition of L a is not sufficient to deduce its hypocoercivity. More precisely, let pP paq t q tě0 be the Markovian semi-group associated to L a in L 2 pµ a q, according to Hille-Yosida's theory [START_REF] Yosida | Functional analysis[END_REF]. Formally, we have for all t ě 0, P paq t " expptL a q, which corresponds to the evolution equation B t P paq t pf q " P paq t pL a pf qq, valid at least for all f in the domain DpL a q of L a in L 2 pµ a q. Probability theory provides a regular version of this semi-group. Consider the stochastic differential equation in T

ˆR " dX t " Y t dt dY t " ´Yt dt `?2dB t , (22)
where pB t q tě0 is a standard real Brownian motion. Assume that initially pX 0 , Y 0 q takes the deterministic value px, yq P T ˆR. It is well-known that the above stochastic differential equation admits a solution (which is almost surely (a.s.) unique with respect to the law of the Brownian motion, see for instance the book of Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]). Then for any t ě 0 and f P L 2 pµ a q, we have µ a -a.s. in px, yq P T ˆR,

P paq t pf qpx, yq " E x,y rf pX t , Y t qs,
where the subscript x, y of the expectation indicates that we started with pX 0 , Y 0 q " px, yq.

As already alluded to, hypocoercivity concerns the exponentially fast convergence of pP paq t q tě0 toward its equilibrium µ a , here in L 2 pµ a q. It was proven that given f P L 2 pµ a q, one can find two numbers Cpf q ě 0 and α ą 0 such that

@ t ě 0, › › ›P paq t pf q ´µa pf q › › › ď Cpf q expp´αtq, ( 23
)
where }¨} stands for the L 2 pµ a q norm. The constant α depends on a but not on f , see for instance Villani [START_REF] Villani | [END_REF] or Dolbeault, Mouhot and Schmeiser [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF] for this kind of hypocoercive bounds.

A straightforward consequence of the spectral analysis of our simple model is that it is sufficient to study hypocoercivity on V p , for p P Z `. Indeed, for q P Z `, denote U q the line in L 2 pµ a ) generated by ϕ 0 b h q,a . The subspaces U q , V p , W p 1 , for q P Z `and p, p 1 P N are mutually orthogonal and their Hilbert sum is equal to whole space L 2 pµ a q. If A is one of these subspaces, let Π pAq be the orthogonal projection on A and remark that Π pAq commutes with the elements of the semi-group. Denote by pP pa,Aq t q tě0 the semi-group generated by the restriction of L a on A, we have for all t ě 0, P pa,Aq t " P paq t Π pAq " Π pAq P paq t Π pAq . It follows that for any t ě 0 and for any f P L 2 pµ a q with µ a pf q " 0,

› › ›P paq t pf q › › › 2 " ÿ qPZ `› › ›P pa,Uqq t pf q › › › 2 `ÿ pPN › › ›P pa,Vpq t pf q › › › 2 `ÿ p 1 PN › › ›P pa,W p 1 q t pf q › › › 2 .
Since µ a pf q " 0, we have Π pU 0 q pf q " 0. The other terms of the first sum are also easy to estimate: @ t ě 0, @ q P N, P pa,Uqq t pf q " expp´qtqΠ pUqq pf q.

(24)

We deduce that for all t ě 0,

ÿ qPZ `› › ›P pa,Uqq t pf q › › › 2 ď expp´tq ÿ qPN › › ›Π pUqq pf q › › › 2 .
If we were able to estimate

› › ›P pa,Vpq t pf q › › › 2
, for p P N, then a similar bound would also be valid for

› › ›P pa,Wpq t pf q › › › 2
, because the action of P pa,Wpq t is isometrically conjugate to that of P pa,Vpq t . Thus to deduce bounds such as [START_REF] Yosida | Functional analysis[END_REF], it is enough to know how to deal with the quantity

› › ›P pa,Vpq t pf q › › › 2
, for p P N and t ě 0. This is not obvious, because the eigenvectors pξ n q nPZ `of P pa,Vpq t (described in Theorem 6 and Proposition 15) are not orthogonal. Indeed, we computed their scalar products in the proof of Proposition 11: for any m, n P Z `,

xξ n , ξ m y " c n`m ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq E " N plq c 2 N pkq c 2 ı ą 0, (25)
where we recall that c " ? ap, that N c 2 is a Poisson variable of parameter c 2 and where we used the notation @ n P Z `, @ m P Z `, n pmq ≔ npn ´1q ¨¨¨pn ´m `1q.

To any function f P V p , we can associate a sequence of coefficients pf pnqq nPZ `so that f " ř nPZ `f pnqξ n in L 2 pµ a q. Their interest is that for all t ě 0,

P pa,Vpq t pf q " expp´c 2 tq ÿ nPZ `expp´ntqf pnqξ n ,
and computations similar to those of the proof of Proposition 11 lead to

› › ›P pa,Vpq t pf q › › › 2 " expp´2c 2 tq ÿ m,nPZ `f pnqf pmq expp´pn `mqtq xξ n , ξ m y " expp´2c 2 tqErG 2 t pN c 2 qs,
where

@ t ě 0, n P Z `, G t pnq ≔ ˆ1 `1 c expptq d dX ˙n ÿ mPZ `f pmqX m ˇˇˇˇˇX "c expp´tq .
Unfortunately we have not been able to directly relate this quantity and the same expression at time t " 0. This is why we develop another approach in the next section, where the important role will rather be played by "Poisson distributions with negative parameters".

Computation of L 2 -operator norms

Our purpose here is to prove Theorem 1. From the considerations of the end of last section, this requires to compute the operator norm of P pa,Vpq t in L 2 pµ a q, for any given a ą 0, p P N and t ě 0.

Indeed, remark that

Lemma 19 For any a ą 0 and t ě 0, we have

|||P paq t ´µa ||| " max ˆexpp´tq, max pPN |||P pa,Vpq t ||| ˙.

Proof

From the orthogonality of the subspaces U q , V p , W p 1 , for q P Z `and p, p 1 P N, and from their stability by the operators P paq t , for all t ě 0, we get

|||P paq t ´µa ||| " |||P paq t ´P pa,U 0 q t ||| " max ˜sup qPN |||P pa,Uqq t |||, sup pPN |||P pa,Vpq t |||, sup p 1 PN |||P pa,W p 1 q t ||| ¸,
where we also used that µ a " Π pU 0 q " P pa,U 0 q t where µ a is seen as an endomorphism of L 2 pµ a q. From (24), we deduce that @ t ě 0, @ q P N, |||P pa,Uqq t

||| " expp´qtq, and by conjugacy we have @ t ě 0, @ p P N, |||P pa,Vpq t

||| " |||P

pa,Wpq t |||.

Lower bound of |||P

pa,V p q t ||| So let a ą 0 and p P N be fixed and denote again c ≔ ? ap. By isometry, for any t ą 0, the operator norm of P pa,Vpq t in L 2 pµ a q coincides with that of expptM q in l 2 pZ `q, where M is defined in [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF]. We have seen in Theorem 6 that the spectrum of M consists of the sequence p´c 2 ´nq nPZ `and that a corresponding Hilbert basis of eigenvectors is pξ n q nPZ `, where @ n P Z `, @ q P Z `, ξ n pqq " c n ÿ lP 0,n ˆn l ˙c´2l q plq p´1q t q`1 2 u c q ? q! expp´c 2 {2q, (see [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF]). We have already computed their scalar product in (25), but let us give another expression which will be more convenient.

Lemma 20

We have for all n, m P Z `,

xξ n , ξ m y " p2cq n`m exppp4c 2 q ´1qErn pN 1{p4c 2 q q m pN 1{p4c 2 q q s where N 1{p4c 2 q is a Poisson random variable of parameter 1{p4c 2 q.

Proof

We have seen in ( 25) that for all n, m P Z `,

xξ n , ξ m y " c n`m ÿ pPZ `ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙pplq p pkq c ´2pl`kq`2p p! expp´c 2 q.
To go further, let us introduce two free variables X and Y and interpret in the above formula, for p, l, k P Z `,

p plq " d l dX l X p ˇˇˇX "1 and p pkq " d k dY k Y p ˇˇˇY "1
.

We are thus lead to compute at X " 1 " Y the expression

c n`m expp´c 2 q ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq d l`k dX l dY k ÿ pPZ `X p Y p c 2p p! " c n`m expp´c 2 q ÿ lP 0,n , kP 0,m ˆn l ˙ˆm k ˙c´2pl`kq d l`k dX l dY k exppc 2 XY q " c n`m expp´c 2 q ÿ lP 0,n ˆn l ˙c´2l d l dX l ˆ1 `1 c 2 d dY ˙m exppc 2 XY q " c n`m expp´c 2 q ÿ lP 0,n ˆn l ˙c´2l d l dX l p1 `Xq m exppc 2 XY q " c n`m expp´c 2 q ÿ lP 0,n
ˆn l ˙c´2l ÿ qP 0,l ˆl q ˙ˆd q dX q p1 `Xq m ˙ˆd l´q dX l´q exppc 2 XY q " c n`m expp´c 2 q ÿ lP 0,n ˆn l ˙c´2l ÿ qP 0,l ˆl q ˙mpqq p1 `Xq m´q pc 2 Y q l´q exppc 2 XY q.

For X " 1 " Y , we get

xξ n , ξ m y " c n`m ÿ lP 0,n ˆn l ˙c´2l ÿ qP 0,l ˆl q ˙mpqq 2 m´q c 2pl´qq " 2 m c n`m ÿ l,qPZ `ˆn l ˙ˆl q ˙p2c 2 q ´qm pqq .
Interpreting again m pqq as d q dX q X m ˇˇX"1 , we have

ÿ qPZ `ˆl q ˙p2c 2 q ´qm pqq " ÿ qPZ `ˆl q ˙p2c 2 q ´q d q dX q X m ˇˇˇˇˇX "1 " ˆ1 `1 2c 2 d dX ˙l X m ˇˇˇˇX "1 , so that xξ n , ξ m y " 2 m c n`m ÿ lPZ `ˆn l ˙ˆ1 `1 2c 2 d dX ˙l X m ˇˇˇˇˇX "1 " 2 m c n`m ˆ2 `1 2c 2 d dX ˙n X m ˇˇˇX "1 " p2cq n`m ˆ1 `1 4c 2 d dX ˙n X m ˇˇˇX "1 " p2cq n`m ÿ lPZ `ˆn l ˙1 p4c 2 q l d l dX l X m ˇˇˇˇˇX "1 " p2cq n`m ÿ lPZ `ˆn l ˙1 p4c 2 q l m plq " p2cq n`m ÿ lPZ `nplq m plq p4c 2 q ´l l! .
This expression can be written under the form given in the above lemma.

This formulation enables to compute quite efficiently the norm of expptM qz for z in a dense subspace of l 2 pZ `q. It is based on the remark that ErN pnq s " ρ n if n P Z `and N is a Poisson distribution of parameter ρ ą 0. In fact we will also use the underlying computation with negative ρ: `r f pnq p f pmqp2cq n`m exppp4c 2 q ´1qErn pN 1{p4c 2 q q m pN 1{p4c 2 q q s " exppp4c 2 q ´1qE » -¨ÿ nPZ `p2cq

Lemma
n r f pnqn pN 1{p4c 2 q q '¨ÿ mPZ `p2cq m p f pmqm pN 1{p4c 2 q q 'fi fl ,

where N 1{p4c 2 q is still a Poisson random variable of parameter 1{p4c 2 q.

For any fixed N P Z `, we have

ÿ nPZ `p2cq n r f pnqn pN q " ÿ nPZ `p2cq n n pN q r ρ n n! " ÿ nPZ `,něN p2cq n r ρ n pn ´N q! " p2cr ρq N ÿ nPZ `p2cq n r ρ n n! " p2cr ρq N expp2cr ρq.
Thus it appears that xr z, p zy " exppp4c 2 q ´1q expp2cpr ρ `p ρqqE " p4c 2 r ρp ρq N 1{p4c 2 q ı " expp2cpr ρ `p ρqq exppr ρp ρq.

In particular, if z P l 2 pZ `q is given by z " ř nPZ `ρn n! ξ n , with ρ P R, we get

}z} 2 " exp `ρ2 `4cρ ˘.
The interest of this formula is that for such z, we have

@ t ě 0, expptM qz " expp´c 2 tq ÿ nPZ `expp´ntq ρ n n! ξ n , (27)
so that @ t ě 0, }expptM qz} 2 " expp´2c 2 tq exp `expp´2tqρ 2 `4 expp´tqcρ ˘.

We deduce a lower bound on the operator norm ||| expptM q||| in l 2 pZ `q:

Lemma 22 For any t ě 0, we have

|||

|||

To get a matching upper bound of ||| expptM q|||, consider the subspace Z of z P l 2 pZ `q which are finite linear combinaisons of vectors of the previous type, namely that can be written under the form

z " ÿ nPZ `ÿ lP r ν l ρ n l n! ξ n , (29)
where r P N and ν l , ρ l are real numbers, for l P r .

Lemma 23

The subspace Z is dense in l 2 pZ `q.

Proof Consider z P l 2 pZ `q orthogonal to Z, we want to show that z " 0. The orthogonality of z to Z is equivalent to the fact that for any

ρ P R, C z, ÿ nPZ `ρn n! ξ n G " 0
This means that the series ř nPZ `xz,ξny n! ρ n vanishes for all real values of ρ and it is possible only if @ n P Z `, xz, ξ n y " 0.

But we have seen in Proposition 6 that this implies that z " 0.

The previous result suggests that any z P l 2 pZ `q can be written under the form

z " ÿ nPZ `ż ρ n n! νpdρqξ n , (30)
for an appropriate signed measure ν on R. But we won't push the investigation in this direction (see also the last remark of this section), since what is interesting for us is that by density

@ t ě 0, ||| expptM q||| " sup zPZzt0u }expptM qz} }z} , (31)
and that the norm }z} can be computed for z P Z:

Lemma 24 Let z P Z be given by ( 29).Then we have

}z} 2 " ν 1 Apρqν,
where ν (respectively ν 1 ) is the column (resp. line) vector of coordinates pν l q lP r and where Apρq is the r ˆr-matrix given by @ k, l P r , A k,l pρq ≔ exp pρ k ρ l `2cpρ k `ρl qq .

Proof

This is an immediate computation: let us denote for l P r ,

z l ≔ ÿ nPZ `ρn l n! ξ n , so that z " ř lP r ν l z l and xz, zy " ÿ l,kP r ν l ν k xz l , z k y " ÿ l,kP r ν l A l,k pρqν k ,
according to Lemma 21.

The advantage of the decomposition (29) is that it well-behaves under the action of the semi-group under consideration:

@ t ě 0, expptM qz " expp´c 2 tq ÿ nPZ `ÿ lP r ν l pexpp´tqρ l q n n! ξ n .
The above lemma then implies that xexpptM qz, expptM qzy " expp´2c 2 tqν 1 Apexpp´tqρqν.

(32)

To treat the r.h.s., we need the following result.

Lemma 25 For any t ě 0, any r P N and any ν " pν k q kP r , ρ " pρ k q kP r P R r , we have

ν 1 Apexpp´tqρqν ď exp ˆ´4c 2 1 ´expp´tq 1 `expp´tq ˙ν1 Apρqν.

Proof

Fix the dimension r P N and the time t ą 0 (for t " 0 the announced result is trivial), and consider

ρ 0 ≔ ´2c 1 `e´t , (33)
which is the maximizer in (28) (we omit the dependence on t in the sequel). Define next the vector h " ph k q kP r P R r by

@ k P r , h k ≔ ρ k ´ρ0 ,
where ρ " pρ k q kP r P R r is a vector given as in the statement of the lemma. We compute that for any k, l P r , e ´2t ρ k ρ l `2e ´tcpρ k `ρl q " ´4c 2 e ´t 2 `e´t p1 `e´t q 2 `2e ´tc 1 `e´t ph k `hl q `e´2t h k h l , and

ρ k ρ l `2cpρ k `ρl q " ´4c 2 1 `2e ´t p1 `e´t q 2 `2e ´tc 1 `e´t ph k `hl q `hk h l .
Note that the terms h k `hl have the same factor in the two last expressions. This leads us to introduce the vector η whose coordinates are given by

@ k P r , η k ≔ exp ˆ2e ´t 1 `e´t h k ˙νk ,
so that we can write

ν 1 Ape ´tρqν " exp ˆ´4c 2 e ´t 2 `e´t p1 `e´t q 2 ˙η1 Bpe ´thqη ν 1 Apρqν " exp ˆ´4c 2 1 `2e ´t p1 `e´t q 2 ˙η1 Bphqη,
where Bphq is the r ˆr-matrix given by @ k, l P r , B k,l phq ≔ expph k h l q.

Since ´4c 2 1 `2e ´t p1 `e´t q 2 `4c 2 e ´t 2 `e´t p1 `e´t q 2 " ´4c 2 1 ´e´t 1 `e´t , it remains to prove that for any η " pη k q kP r P R r and any h " ph k q kP r P R r , η 1 Bpe ´thqη ď η 1 Bphqη.

To get this bound, it is sufficient to expand these expressions:

η 1 Bpe ´thqη " ÿ k,lP r exppe ´2t h k h l qη k η l " ÿ k,lP r ÿ nPZ `e´2nt n! ph k h l q n η k η l " ÿ nPZ `e´2nt n! ¨ÿ kP r h n k η k '2 ď ÿ nPZ `1 n! ¨ÿ kP r h n k η k '2 " η 1 Bphqη.
Coming back to (31) and (32), we get that for any t ě 0,

||| expptM q||| ď exp ˆ´c 2 ˆt ´2 1 ´expp´tq 1 `expp´tq ˙˙,
and in conjunction with Lemma 22, it follows that

||| expptM q||| " exp ˆ´c 2 ˆt ´2 1 ´expp´tq 1 `expp´tq ˙˙.
Coming back to the notations of the beginning of this section, we have that for any a ą 0, any p P N and any t ě 0,

|||P pa,Vpq t ||| " exp ˆ´ap 2 ˆt ´2 1 ´expp´tq 1 `expp´tq ˙˙.
Injecting this quantity in Lemma 19, Theorem 1 is proved.

Final remarks

To finish this section, let us make explicit the functions for which the operator norms of the semigroup are reached. It will appear a posteriori that there is a faster way to justify the introduction of such functions of the form presented in Lemma 21.

From the above computations, it follows that if t ą 0 is such that |||P

T ˆR Q px, yq Þ Ñ exp ˆ´2iy 1 `expp´tq `ipx `yq ˙,
are maximizers for the computation of |||P paq t ´µa ||| in the complexified L 2 pµ a q. Lemma 26 leads us to consider for any p P N and for any ρ P R the mapping From here it is possible to follow our previous arguments (computing instead µ a rF p,r ρ F p,p ρ s for r ρ, p ρ P R, namely values of the characteristic function associated to µ a ) to get the same proof of Theorem 1. So the above manipulations of functions of the form F p,ρ are a short way to avoid the spectral decomposition of L a . This approach could also be considered in our second model (or for some quadratic symbol operators), but it would be more tricky, because in the end the corresponding maximizing functions for the computation of the operator norms will be linear and not of the form R ˆR Q px, yq Þ Ñ exppiαx `iβyq, where α, β P R.

F p,ρ : T ˆR Q px, yq Þ Ñ exppiρy `ippx `yqq. If R `Q t Þ Ñ ρ t P R is a smooth function, define @ t ě 0, @ px, yq P T ˆR, G t px, yq ≔ F p,
Remark 27 After some elementary manipulations, Lemma 26 enables to translate the question asked around (30) into the following one: is there a set M of complex measures on R such that the relation

@ y P R, f pyq " ż R exppiyρq νpdρq,
induces a bijection between functions f P L 2 pµ a q (complexified) and measures ν P M? Whatever it is, M contains all finite linear combinations of Dirac masses. Note that since L 2 pµ a q is not included in the space S 1 of tempered distributions, the usual Fourier transform in S 1 does not give the answer. 4

Spectral decomposition of the Gaussian case

We treat here the spectral decomposition of our second model. Despite it is already known (see for instance Risken [START_REF] Risken | The Fokker-Planck equation[END_REF]), we will proceed differently, rather following an approach based on a decomposition of the generator similar to our roadmap used in Section 2. Apart from underlying the analogies and differences between our two models, this will put us in good position to compute the operators norms.

Decomposition of the generator on stable subspaces

So for fixed a ą 0, we are interested in the operator r L a defined in [START_REF] Desvillettes | On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation[END_REF]. Since the coefficients of r L a are affine and the associated invariant measure r µ a " γ 1{a b γ is Gaussian, it is natural to check how r L a acts on the Hermite polynomials, renormalized to be orthogonal in L 2 pr µ a q. The definition of the orthogonal polynomials associated to γ were recalled in [START_REF] Eckmann | Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators[END_REF]. To simplify notations, we sightly modify those adopted in Section 2 and rather consider @ p P N, @ x P R, h p,a pyq ≔ h p p ? axq.

The family ph p,a b h q q p,qPN is then an orthogonal basis of L 2 pr µ a q. In analogy with Lemma 5, we begin by

Lemma 28 For all p, q P N, we have r L a rh p,a b h q s " ? aph p´1,a b h q`1 ´?aqh p`1,a b h q´1 ´qh p,a b h q .

Proof Taking into account the following classical relations, valid for all q P N and y P R (with the convention h ´1 " 0), h 2 q pyq ´yh 1 q pyq " ´qh q pyq h 1 q pyq " qh q´1 pyq h q`1 pyq " yh q pyq ´qh q´1 pyq, we compute that for all p, q P N and x, y P R, we have r L a rh p,a b h q spx, yq " y ? aph p´1,a pxqh q pyq ´axqh p,a pxqh q´1 pyq ´qh p,a pxqh q pyq " ? aph p´1,a pxqph q`1 `qh q´1 qpyq ´?aqph p`1,a `ph p´1,a qpxqh q´1 pyq ´qh p,a pxqh q pyq " p ? aph p´1,a b h q`1 ´?aqh p`1,a b h q´1 ´qh p,a b h q qpx, yq.

This formula leads us to introduce, for n P N, the subspace H n of L 2 pr µ a q generated by h p,a b h n´p , for p P 0, n . Indeed, ´?an 0

H
‹ ‹ ‹ ‹ ‹ ‹ ‹ ' . ( 34
)
In order to diagonalize this matrix, it is fruitful to decompose it into its diagonal , above-diagonal and below-diagonal parts, i.e. Ă M n " r D n `?aS n ´?aS n, with @ p, q P 0, n , r D n pp, qq ≔ " ´pn ´pq , if p " q 0 , otherwise.

@ p, q P 0, n , S n pp, qq ≔ " a pp `1qpn ´pq , if q " p `1 0 , otherwise.

and where S n stands here for the transposed matrix associated to S n . The next point is crucial to understand the spectral structure of Ă M n :

Lemma 29 For any n P N, the commutators of S n , S n and r D n are given by

rS n , S n s " ´2 r D n ´nI n rS n , r D n s " S n rS n, r D n s " ´Sn ,
where I n is the n ˆn identity matrix.

Proof

The two first relations are just direct computations: for any p, q P 0, n , we have rS n , S n spp, qq " S n S npp, qq ´Sn S n pp, qq " S n pp, p `1qS n pq, q `1qδ q`1"p`1 ´Sn pp ´1, pqS n pq ´1, qqδ q´1"p´1

" ppp `1qpn ´pq ´ppn ´p `1qqδ q"p " pn ´2pqδ q"p " ´2 r D n pp, qq ´nI n pp, qq.

In a similar way, we have for any p, q P 0, n , rS n , r D n spp, qq " S n r D n pp, qq ´r D n S n pp, qq " S n pp, p `1q r D n pq, qqδ q"p`1 ´r D n pp, pqS n pq ´1, qqδ p"q´1

" ´ppn ´p ´1q a pp `1qpn ´pq ´pn ´pq a pp `1qpn ´pqqδ q"p`1 " a pp `1qpn ´pqδ q"p´1 " S n pp, qq.

The last equality is a consequence of the previous one:

rS n , r D n s " r r D n, pS n q ˚s" r r D n , S n s " ´rS n , r D n s " ´Sn .
Thus it appears that the vector space r V n generated by the four matrices I n , r D n , S n and S n is a real Lie subalgebra of glpn `1, Rq, stable by transposition and containing Ă M n . It is not difficult to check that for n P Nzt0u, the four matrices I n , r D n , S n and S n are independent, so that dimp r V n q " 4 (the case n " 0 is different: r V 0 " R and dimp r V 0 q " 1). To avoid trivialities, we assume that n P Nzt0u in the discussion that follows. It is possible to reduce the dimension to 3, by considering the next slight modifications. Define

D n ≔ r D n `n 2 I n M n ≔ Ă M n `n 2 I n ,
and let V n be the vector space generated by the three matrices D n , S n and S n. We deduce immediately from the above lemma that as the elements of usual basis of slp2, Rq given by

e 1 ≔ 1 2 ˆ1 0 0 ´1 ˙e2 ≔ 1 ? 2 ˆ0 1 0 0 ˙e3 ≔ 1 ? 2 ˆ0 0 1 0 ˙.
For n " 1, we even have equality between these elements and if we rather see the V n , for n P N, as complex vector spaces, then pV n q nPN is the family of all irreducible representations of slp2, Cq (see for instance Section 4.4 of the book of Hall [START_REF] Hall | Lie groups, Lie algebras, and representations[END_REF]).

Spectral analysis of La

The Lie algebra structure of V n suggests that to get informations about the spectral decomposition of M n " ? aS n ´?aS n `Dn P V n , it is interesting to first investigate the spectral decomposition of the adjoint operator at M n , which is defined by

ad Mn : V n Q X Þ Ñ rM n , Xs P V n .
This is the object of the next result, where V n and ad Mn are replaced by their natural complexifications.

Lemma 30 Let n P Nzt1u be fixed. The kernel of the operator ad Mn is generated by M n . For a " 1{4, there are two other eigenvalues, θ and ´θ where

θ ≔ " ? 1 ´4a , if a P r0, a{4q ? 4a ´1i , if a ą 1{4.
The corresponding eigenspaces are respectively generated by

J `" 4 ? aD n `p1 ´θqS n ´p1 `θqS n (35)
J

´" 4 ? aD n `p1 `θqS n ´p1 ´θqS n.

For a " 1{4, the operator ad Mn is not diagonalizable and its matrix is equal to the 3 ˆ3 Jordan block ¨0 1 0 0 0 1 0 0 0 ' associated to the eigenvalue 0, in the basis pM n , D n ´2? aS n, ´2? aS nq.

Proof

Due to the fact that rM n , M n s " 0, we already know that M n is an eigenvector associated to the eigenvalue 0 of ad Mn . Using the above bracket relations, we compute that the matrix associated to ad Mn in the basis pD n , S n , S n q is given by

¨0 ´2? a ´2? a ? a ´1 0 ? a 0 1 '.
It characteristic polynomial is ´XpX 2 ´1 `4aq, so for a " 1{4, ad Mn admits three distinct eigenvalues which are 0, θ and ´θ, defined in the above statement. Computing associated eigenvectors, we get the announced results, for a " 1{4. For a " 1{4, since the characteristic polynomial is ´X3 , it appears that 0 is the only possible eigenvalue. Furthermore it is clear that the above matrix has rank 2 (in fact for any a ě 0), so ad Mn is necessarily similar a 3 ˆ3 Jordan block associated to the eigenvalue 0. Already knowing that M n is in the kernel of ad Mn , it is not difficult to complete it into a basis in which the matrix associated to ad Mn has the required form, e.g. the basis given in the lemma.

For the remaining of this section, the case a " 1{4 will often be excluded from our study. This value is critical for the spectra of the M n , n P Nzt1u, to be real. More precisely, we will see that for a P p0, 1{4s, the spectrum of M n is real (so a posteriori complexification was not necessary), while for a P p1{4, `8q, it does contain non-real eigenvalues. First we present a simple but very useful technical result.

Lemma 31 For a P p0, `8qzt1{4u, kerpJ `q, the kernel of J `, is of dimension 1.

Proof

From (35) we remark that J `is a tridiagonal matrix, whose supdiagonal has only non-vanishing entries (namely the values a app `1qpn ´pqpθ ´1q, for p P 0, n ). So if f " pf p q pP 0,n is a vector belonging to kerpJ `q and if f 0 " 0, we deduce by iteration that f " 0: indeed the equation J `p0, 0qf 0 `J`p 0, 1qf 1 " 0, implies f 1 " 0, next the equation J `p1, 0qf 0 `J`p 1, 1qf 1 `J`p 1, 2qf 2 " 0 enables us to see that f 2 " 0, etc., in the end the nullity f n is a consequence of the last but one equation. It follows that kerpJ `q is at most of dimension 1, otherwise we could find a non-zero vector in kerpJ `q whose first coordinate is zero.

To see that kerpJ `q is not reduced to t0u, let be given λ an eigenvalue of (the complexification of) M n and denote by ϕ " 0 a corresponding eigenvector. Since we have

M n J `ϕ " J `Mn ϕ `θϕ " pλ `θqϕ.
we get that either λ `θ is an eigenvalue of M n or J `ϕ " 0. If the latter condition is not satisfied, we iterate this operation to see that either λ `2θ is an eigenvalue of M n or J 2 `ϕ " 0. But λ `pθ cannot be an eigenvalue of M n for all p P N, so necessarily there exists p P N with J p `ϕ " 0 and J p`1 `ϕ " 0, i.e. J p `ϕ P kerpJ `qzt0u.

By extending to Jordan-type subspaces the latter argument, we will prove the following important result.

Proposition 32 For a " 1{4, the matrix M n diagonalizable and all the eigenvalues have multiplicity 1. More precisely if λ is an eigenvalue of M n such that λ `θ is not an eigenvalue of M n , then the spectrum of M n is the set tλ ´kθ : k P 0, n u. Furthermore, for k P 1, n , J (respectively J ´) the spectral line associated to λ ´kθ (resp. λ ´pk ´1qθ) into the spectral line associated to λ ´pk ´1qθ (resp. λ ´kθ).

Proof

We define that a subspace V of C n`1 is of type pl, dq, with l P C and d P N, if there exists a basis pϕ 0 , ϕ 1 , ..., ϕ d q of V such that

M n ϕ 0 " lϕ 0 , @ p P 1, d , M n ϕ p " lϕ p `ϕp´1 .
The Jordan decomposition implies that M n is diagonalizable if and only if there is no pl, 1q-type subspace for any l P C (by taking into account that maximal pl, dq-type subspaces lead to Jordan blocks, which contains pl, 1q-type subspaces if d ě 1). We are to proceed by a contradictory argument to show that M n is diagonalizable. First consider V a pl, 1q-type subspace endowed of a basis pϕ 0 , ϕ 1 q as above. The relation M n J `" J `Mn `θJ `implies that

M n J `ϕ0 " pl `θqJ `ϕ0 , M n J `ϕ1 " pl `θqJ `ϕ1 `J`ϕ0 . (36)
Thus if kerpJ `qXV " t0u, we get that J `ϕ0 and J `ϕ1 must be independent, so J `pV q is a pl`θ, 1qtype subspace. In particular l `θ is an eigenvalue of M n . Next let λ be as in the statement of the proposition and assume there exists a pλ, 1q-type subspace V , endowed of a basis pϕ 0 , ϕ 1 q as above. Necessarily kerpJ `q Ă V , otherwise the above argument would lead to fact that λ `θ is an eigenvalue of M n . So let f P kerpJ `qzt0u be given. The relation M n J `f " J `Mn f `θJ `f implies that J `Mn f " 0, namely M n f P kerpJ `q. Lemma 31 then shows that M n f is proportional to f , i.e. f is an eigenvector of M n . The only eigenvectors of M n belonging to V are proportional to ϕ 0 , thus we deduce that J `ϕ0 " 0. But (36) (with l replaced by λ) implies that either λ `θ is an eigenvalue of M n , which is forbidden by our choice of λ, either J `ϕ1 " 0, which is not more possible, because it would lead to dimpkerpJ `qq ě 2. It follows that a pλ, 1q-type subspace does not exist. Nevertheless, a eigenvector ϕ " 0 associated to λ exists and necessarily J `ϕ " 0. A consequence of this property and of Lemma 31 is that for any l " λ and any pl, 1q-subspace V , we have kerpJ `q X V " t0u. As it was already shown, we then get that J `pV q is a pl `θ, 1q-type subspace. If l `θ " λ, we reiterate this procedure. Necessarily we end up with a integer p P Nzt0u such that l `pθ " λ, otherwise we would construct an infinity of eigenvalues. But another contradiction appears, because J p `pV q is in fact a pλ, 1q-type subspace. In conclusion, there is no pl, 1q-type subspace: M n is diagonalizable.

The other assertions of the proposition are proven in a similar way: first kerpJ `q is necessarily the eigenspace associated to λ, which by consequence is of multiplicity 1. Next any non-zero eigenvector ϕ associated to an eigenvalue l " λ of M n is such that J `ϕ is a non-zero eigenvector associated to l `θ. Iterating again, we deduce there exists p P Nzt0u such that l `pθ " λ and J p `ϕ belongs to the line eigenspace associated to λ. Another application of Lemma 31 shows that the dimension of the eigenspace associated to l was necessarily 1 (otherwise you could find ϕ " 0 in this eigenspace and k P 1, p such that J k´1 `ϕ belongs to kerpJ `q but not to the eigenspace associated to λ, which is not permitted). This is only possible if the spectrum of M n coincides with the set tλ ´kθ : k P 0, n u and if for k P 1, n , J `transforms the spectral line associated to λ ´kθ into the spectral line associated to λ ´pk ´1qθ. Rather working with J ´instead of J leads to the corresponding statement for J ´.

To end the determination of the spectrum of M n , we point out another particular feature of this matrix: M n is skew-centrosymmetric, i.e. T pM n q " ´Mn , where for any pn `1q ˆpn `1q matrix M " pM k,l q k,lP 0,n , we define @ k, l P 0, n , pT pM qq k,l ≔ M n´k,n´l .

This transformation T also applies to vectors by @ f " pf k q kP 0,n , T pf q ≔ pf n´k q kP 0,n and it is easily checked that for any matrix M and vector f , T pM f q " T pM qT pf q, (for general references about (skew) centrosymmetric matrices, see for instance the papers of Weaver [START_REF] Weaver | Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors[END_REF] and Lee [START_REF] Lee | Centro-Hermitian and skew-centro-Hermitian matrices[END_REF]). An important consequence of the skew-centrosymmetry of M n is that its spectrum is symmetric with respect to 0. Indeed, if λ is an eigenvalue of M n and if ϕ is a corresponding eigenvector, we get, by using that T is a linear involution, that

M n T pϕq " T pT pM n qϕq " ´T pM n ϕq " ´λT pϕq.
This shows that ´λ is also an eigenvalue of M n , an associated eigenvector being T pϕq. In conjunction with Proposition 32, this observation leads to the determination of the spectrum of M n .

Proposition 33 For a " 1{4, the spectrum of M n is tpk ´n{2qθ : k P 0, n u. For a " 1{4, M n is similar to the Jordan block of size n `1 associated to the eigenvalue 0 (in particular M n is not diagonalizable for n ě 1).

Proof

The first assertion is an immediate consequence of Proposition 32 and of the symmetry of the spectrum of M n . Note that as a " 1{4 goes to 1{4, θ and the eigenvalues of M n converge to zero. A usual result on perturbation of spectrum (cf. for instance the chapter 2 of book of Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF]) then implies that the spectrum of M n for a " 1{4 is reduced to t0u. But the arguments of the proof Lemma 31 also apply to the tridiagonal matrix M n to show that the dimension of kerpM n q is at most 1. By the Jordan decomposition, it follows that M n is necessarily similar to the Jordan block of size n `1 associated to the eigenvalue 0.

In view of this result it is natural to make the convention that θ " 0 when a " 1{4. Recalling that for all n P N, Ă M n " M n ´n 2 I n is (the matrix associated to) the restriction of r L a to H n and that L 2 pr µ a q " Â nPN H n , where the H n , n P N are mutually orthogonal, we get

Corollary 34 For any a ą 0, the spectrum of r

L a in L 2 pr µ a q is ! ´n 2 `pk ´n{2qθ : n P N, k P 0, n ) .
For a " 1{4, r L a is diagonalizable in L 2 pr µ a q, but r L 1{4 is not diagonalizable and it contains Jordan blocks of all dimensions.

For a " 1{4 and n P N, we have seen that a family pξ p q pP 0,n of eigenvectors associated to the eigenvalues ppp ´n{2qθq pP 0,n of M n is given by

@ p P 0, n , ξ p ≔ J p `ξ0 ,
where ξ 0 is a normalized vector generating the kernel of J

´.

Using this information, it is possible to make explicit the eigenvectors of r L a , which are polynomial. But as seen in Section 3 for our first model, to obtain hypocoercive bounds, it is more crucial to compute the scalar products of the eigenvectors than to known them exactly. This is the objective of next section.

Norms of hypocoercive Gaussian semi-groups

We are going to prove Theorem 3, by following the approach of Section 3, namely by investigating scalar products of underlying eigenvectors.

Let a ą 0 be fixed. Since, on one hand the orthogonal decomposition L 2 pr µ a q " Â nPN H n , introduced in the previous section, is left stable by all the elements of the semi-group p r P paq t q tě0 , and on the other hand r µ a correspond to the orthogonal projection on H 0 , the space containing the constant functions, we have for all t ě 0, By the isometries introduced at the beginning of Section 4, we have for any n P N and t ě 0, ||| r P paq t ||| Hný " ||| exppt Ă M n q|||, where Ă M n is the pn `1q ˆpn `1q matrix defined in (34) and where ||| ¨||| stands for the operator norm with respect to the canonical Hermitian norm on C n`1 . We are thus brought back to the finite dimensional setting of Section 4, n P N being fixed.

||| r P

Identification of eigenvectors when a P p0, 1{4q

For the first part of this section, we restrict ourself to the case a P p0, 1{4q, so that θ is real and belongs to p0, 1q. Recall that Furthermore, ξ 0 is a normalized vector generating the kernel of J ´and for all p P 1, n , ξ p ≔ J p `ξ0 , so that pξ p q pP 0,n is a family of eigenvectors of M n associated to the eigenvalues ppp ´n{2qθq pP 0,n . We begin by checking that ξ 2 0 ≔ pξ 2 0 ppqq pP 0,n is the binomial distribution of parameter p1 ´θq{2. Lemma 35 We can take @ p P 0, n , ξ 0 ppq "

D
d ˆn p ˙ˆ1 ´θ 2 ˙p ˆ1 `θ 2 ˙n´p . ( 37
)
Proof Let ξ be the vector whose coordinates are given by the r.h.s. of (37). It is sufficient to show that J ´ξ " 0. By definition, we have for any p P 0, n (with the convention ξp´1q " ξpn `1q " 0), Following the same arguments of Section 3, we are now looking for a family pQ p q pP 0,n of polynomials, with Q p of degree p for p P 0, n , such that

D
@ p P 0, n , ξ p " Q p pD n qξ 0 . (38)
To do so, we will need two preliminary computations and the first equality follows at once. The second equality is trivial for p " 0 since by convention J 0 `is the pn `1q ˆpn `1q identity matrix I n , so that rJ ´, J 0 `s " 0. For p " 1, we are going to show that rJ ´, J `s " 8θM n .

Let us first remark a priori from the actions of J ´and J `on the eigenspaces of M n , that J ´J`, J `J´a nd rJ ´, J `s are functions of M n , at least for a " 1{4, when all the eigenvalues of M n are distinct. Indeed, recalling that rD n , S n s " ´Sn , rD n , S n s " S n and rM n , J ´s " ´θJ Hence, from (39) we deduce that for any p P N, rJ ´, J p `s " rJ ´, J `sJ p´1 ``J `rJ ´, J `sJ p´2

``¨¨¨`J p´1 `rJ ´, J `s " 8θpM n J p´1 ``J `Mn J p´2 ``¨¨¨`J p´1 `Mn q.

Applying this formula to the vector ξ 0 and taking into account that J q `ξ0 is an eigenvector of M n associated to the eigenvalue pq ´n{2qθ for all q P 0, n (and that the relation M n J q `ξ0 " pq ´n{2qθJ q `ξ0 is also true for q ą n, since then J q `ξ0 " 0), we obtain for all p P N, rJ ´, J p `sξ 0 " 8θ 2 ppp ´1 ´n{2q `pp ´2 ´n{2q `¨¨¨`p´n{2qqJ p´1 `ξ0

" 4ppp ´1 ´nqθ 2 J p´1 `ξ0 , as announced.

We can now find pQ p pXqq pě0 such that (38) is satisfied.

Lemma 37 Consider the family of polynomials pQ p pXqq defined by the recurrence relation:

Q 0 pXq " 1 and Q 1 pXq " ´ˆ2 ? a θ 2 X `n ? a θ ˙, @ p P 1, n ´1 , Q p`1 pXq " ˆ´2 ? a θ 2 X `2p ´n ? a θ ˙Qp pXq `4θ 2 ppn `1 ´pqQ p´1 pXq.
Then for any p P 0, n , Q p is of degree p and (38) is fulfilled.

Proof

Of course ξ 0 " Q 0 pD n qξ 0 and if we assume that for some p P 0, n ´1 , ξ q " Q q pD n qξ 0 for all q P 0, p , then we can write, using the first relation of Lemma 36, and that J ´ξ0 " 0

ξ p`1 " J `pJ p `ξ0 q " ˆ8a ´2 ? a D n `2 ? a M n ´J´˙J p `ξ0 " 8a ´2 ? a D n Q p pD n qξ 0 `2 ? a ´p ´n 2 ¯θJ p `ξ0 ´rJ ´, J p `sξ 0 ´Jp `J´ξ0 " ˆ8a ´2 ? a D n `2 ? a θ ´p ´n 2 ¯˙Q p pD n qξ 0 ´rJ ´, J p `sξ 0 .
If p " 0, since rJ ´, J 0 `s " 0, this gives ξ 1 " Q 1 pD n qξ 0 with Q 1 the polynomial described in the lemma, recalling that θ 2 " 1 ´4a. For p ą 1, the second relation of Lemma 36 enables to replace rJ ´, J p `sξ 0 by 4ppp´1´nqθ 2 J p´1 `ξ0 " 4ppp ´1 ´nqθ 2 Q p´1 pD n qξ 0 , due to our iterative assumption. So we end up with the announced recurrence relation for the family pQ p pXqq pP 0,n , which clearly implies that for any p P 0, n , Q p is of degree p.

Remark 38 Let us define Q n`1 by extending the above recurrence:

Q n`1 pXq " ˆ´2 ? a θ 2 X `n ? a θ ˙Qn pXq `4θ 2 nQ n´1 pXq.
The computations of the above proof show that Q n`1 pD n qξ 0 " J n`1 `ξ0 " 0. It follows that for any p P 0, n , Q n`1 pD n qξ p " Q n`1 pD n qQ p pD n qξ 0 " Q p pD n qQ n`1 pD n qξ 0 " 0. Thus Q n`1 pD n q " 0, because pξ p q pP 0,n is a basis of the underlying vector space. The operator D n has n `1 distinct eigenvalues (given by the elements of its diagonal), so its minimal polynomial has degree n `1 and Q n`1 must be proportional to it. Indeed, an immediate analysis of the leading monomials proves that

Q n`1 pXq " ˆ´2θ 2 ? a ˙n`1 ź pP 0,n ´X `n 2 ´p¯.
A similar observation leads to the uniqueness property of the family pQ p pXqq pP 0,n : assume that for some p P 0, n , ξ p " r Q p pD n qξ 0 , where r Q p is a polynomial of degree p. Then necessarily we have r Q p " Q p . Indeed, p Q p ≔ Q p ´r Q p would be a polynomial of degree less than p such that p Q p pD n qξ 0 " 0 and by the above arguments it appears that p Q p must be proportional to Q n`1 of degree n `1. This is only possible if p Q p " 0 as wanted. Since it will be more convenient to work with polynomials whose leading term is 1 and to shift the eigenvalues of D n by n{2 (to end up with the set 0, n , which is the support of the binomial law ξ 2 0 ), we define @ p P 0, n , P p pXq "

ˆ´2θ 2 ? a ˙´p Q p pX ´n{2q.
In the sequel, we denote

υ ≔ 1 θ ´1.
It is easy to see that the recurrence relation holds P 0 pXq " 1, @ p P 0, n ´1 , P p`1 pXq " ´X ´p `´n 2 ´p¯υ¯P p pXq `1 4 ppn `1 ´pqpυ 2 `2υqP p´1 pXq.

Note that the term P ´1 is not necessary to determine P 1 . These modifications also prompt us to exchange the pξ p q pP 0,n for the pζ p q pP 0,n defined by

@ p P 0, n , ζ p ≔ ˆ´2θ 2 ? a ˙´p ξ p " P p ´Dn `n 2 I n ¯ξ0 . (40)
The family pζ p q pP 0,n still consists of a basis of eigenvectors of M n (associated to the family of eigenvalues ppp ´n{2qθq pP 0,n ), its advantage is encapsulated in the next result.

Lemma 39 For any p, q P 0, n , we have xζ p , ζ q y " β p1´θq{2 rP p P q s, where β p1´θq{2 is the binomial distribution of parameter p1 ´θq{2.

Proof

This is a direct computation: for any p, q P 0, n , we have xζ p , ζ q y " ÿ mP 0,n ζ p pmqζ q pmq " ÿ mP 0,n ´Pp ´Dn `n 2 I n ¯ξ0 ¯pmq ´Pq ´Dn `n 2 I n ¯ξ0 ¯pmq " ÿ mP 0,n P p pmqP q pmqpξ 0 pmqq 2 " β p1´θq{2 rP p P q s, where we have used ( 40) and (37).

The recurrence relation satisfied by pP p pXqq pP 0,n may lead the reader to think that they are the orthogonal polynomials associated to some law on 0, n . This law cannot be β p1´θq{2 , because the family pζ p q pP 0,n is not orthogonal: this is the heart of the subject and the motivation for the computations of this section.

It is time now to provide an explicit formula for the polynomials P p pXq, p P 0, n . In analogy with Section 3 again, it is more convenient to express them in the basis pΠ p q pP 0,n , where

@ p P 0, n , Π p pXq ≔ ź kP 0,p´1 pX ´kq,
(slightly abusing notations, the r.h.s. could also be written X ppq ).

Lemma 40 For any p P 0, n , we have

P p pXq " ÿ kP 0,p p pp´kq pn ´kq pp´kq pp ´kq! ´υ 2 ¯p´k Π k pXq. (41)

Proof

After computing the first elements of the family pP p pXqq pP 0,n , one guesses that they will be of the form

P p pXq " ÿ kP 0,p α p,k υ p´k Π k pXq,
for some coefficients pα p,k q pP 0,n ,kP 0,p independent of the parameter υ. Putting such a form in the recurrence relation, it appears that to be conserved for P p`1 (assuming it is true for P p and P p´1 ), one must have α p,k " ppn`1´pq 2pp´kq α p´1,k . Since necessarily α p,p " 1, this leads to the announced formula. Once the latter is suspected, it is sufficient to check it by induction: assuming that (41) is true for P p and P p´1 (this is immediate for P 0 and P Remark 41 It is possible to give a compact formula for the r.h.s. of (41): introduce two free variables Z 1 and Z 2 and consider the following interpretations:

pn ´kq pp´kq " p´1q p´k d p´k dZ p´k 1 1 Z n´p`1 1 ˇˇˇˇZ 1 "1 , Π k pXq " d p dZ p 2 Z X 2 ˇˇˇZ 2 "1 .
The r.h.s of (41) can then be seen as

ÿ kP 0,p ˆp p ´k˙ˆ´υ 2 ˙p´k d p´k dZ p´k 1 d p dZ p 2 Z X 2 Z n´p`1 1 ˇˇˇˇˇZ 1 "1,Z 2 "1 " ˆd dZ 2 ´υ 2 d dZ 1 ˙p Z X 2 Z n´p`1 1 ˇˇˇˇZ 1 "1,Z 2 "1
.

and considering the same expression at time t " 0, it appears that sup

Λ θ,t ≔ e ´2θt `1 ´θ2 2 ˆ1 ´e´θt θ ˙2 `1 ´e´2θt 2 ¨1 `1 θ d 1 `1 ´θ2 θ 2 ˆeθt ´1 e θt `1 ˙2' . (45)
Proof Consider J the pn`1q-diagonal matrix with J k,k " p´1q k for k P 0, n and r A ≔ JAJ, whose entries are non-negative. Being conjugate, the two matrices B ´1A ˚BA and JB ´1A ˚BAJ " B ´1 r A ˚B r A have the same spectrum. The advantage of B ´1 r A ˚B r A is that all its entries are positive, so Perron-Frobenius theorem asserts that if we can find an eigenvector vector of B ´1 r A ˚B r A whose coordinates are positive, then the corresponding eigenvalue is the largest one. In view of the entries of r A and B, it is natural to try vectors η ≔ pη l q lP 0,n P R n`1 whose coordinates are powers, namely of the form

@ l P 0, n , η l " r l ,
where r ą 0 is to be determined so that η is an eigenvector vector of B ´1 r A ˚B r A. We compute that for any k P 0, n , p r Aηq k " ÿ lP 0,n ˆk l ˙pe θt ´1q k´l e θlt r l " ´eθt ´1 `eθt r ¯k .

To simplify notation, let s ≔ e θt ´1 `eθt r. where we recall that θ ≔ ? 1 ´4a and that Λ θ,t was defined in (45).

Proof

We have seen at the beginning of this section that for any a P p0, 1{4q and any t ě 0, Since we know a priori that the l.h.s. is less or equal to 1, necessarily the quantity expp´p1´θqtqΛ θ,t is less or equal to 1 and the above supremum is attained for n " 1.

|||

We remark that for any fixed time t ě 0, as θ ą 0 goes to zero, Λ θ,t converges toward

Λ 0,t ≔ 1 `t2 2 `td 1 `ˆt 2
˙2.

Since on one hand, for any fixed n P N and t ě 0, the operator expptM n q is a continuous function of the (hidden) parameter a ą 0 and on the other hand, (47) is always true, the previous theorem can be extended for the value a " 1{4:

Corollary 46 For any t ě 0, we have

||| expptL 1{4 q ´r µ 1{4 ||| L 2 pr µ 1{4 qý " a Λ 0,t exp ˆ´1 2 t ˙.

Exact computation of ||| r P

paq t ´r µ a ||| L 2 pr µ a qý , a ą 1{4
For the remaining part of this section, we consider the situation where a ą 1{4. The parameter θ is now purely imaginary and we choose θ " ? 4a ´1i. Most of the previous arguments can be extended and we will only insist on the main changes.

First (37) is still valid if we rather rewrite it under the form @ p P 0, n , ξ 0 ppq " d ˆn p ˙ˆ?

1 ´θ ? 2 ˙p ˆ?1 `θ ? 2 ˙n´p ,
where the signs of the two complex numbers ? 1 ´θ and ? 1 `θ are chosen so that their product is equal to 2 ? a. Anyway, the important object is p|ξ 0 ppq| 2 q pP 0,n , which is just pp1`|θ| 2 q{4q n{2 ´`n p ˘¯pP 0,n , since |1 `θ| " |1 ´θ| " b 1 `|θ| 2 . Indeed, Lemmas 36, 37 and 40 don't need to be modified, since they only deal with algebraic properties of ξ 0 , J ´and J `. Similarly, we consider the family pζ p q pP 0,n defined by (40). The next change comes with Lemma 39, which must rather state that for any p, q P 0, n , we have

xζ p , ζ q y " p1 `|θ| 2 q n{2 β 1{2 rP p P q s, (48)
where β 1{2 is the binomial distribution of parameter 1{2 (note that x¨, ¨y now stands for the usual Hermitian product on C n`1 . expression which coincides with the announced one.

Fix any complex number ρ whose norm is different from 0 and 1. Then pzpρ k qq kP 0,n is a basis of C n`1 and any z P C n`1 can be written under the form (44), where the ν k , for k P 0, n , are the appropriate complex coordinates.

Using Lemma 47 and the notations introduced there, we obtain as in Lemma 43 that

Proof

If ν " 0, A is just the identity matrix, so the result is immediate. Assume that ν " 0. As in Lemma 44, the wanted largest eigenvalue is also the largest eigenvalue of B ´1 r A ˚B r A and it is sufficient to find a corresponding positive eigenvector. Again we look for a vector η ≔ pη l q lP 0,n P R n`1 whose coordinates are of the form @ l P 0, n , η l " r l , where r ą 0 is to be determined to insure that η is an eigenvector vector of B ´1 r A ˚B r A. Let us denote s " ν `r, so that for any k P 0, n , Finally, it appears that for any k P 0, n , pB ´1 r A ˚B r Aηq k " s k ˆ1 `νs γ ˙n´k .

Thus η will be the wanted eigenvector, with Λ ≔ p1 `νpν `rq{γq n as associated eigenvalue, if the following equation has a positive solution r: ν `r " r ˆ1 `νpν `rq γ ˙.

Since ν " 0, this is equivalent to r 2 `νr ´γ " 0, which admits r " p´ν `aν 2 `4γq{2 as positive solution. Expanding Λ, we end up with the announced result.

The arguments of the proof of Theorem 45 enable to conclude the computation of L 2 -operator norms of the hypocoercive semi-group associated to the generator r L a , for a P p1{4, 8q:

Theorem 49 For any a ą 1{4 and any t ě 0, we have ´1ˇˇ2 `4p4a ´1q

|||
˙.

Let us finish this section by noting that for all a ą 0, the maximizing functions for the computation of the operator norms of the semi-group p r P paq t q tě0 belong to H 1 , namely are linear mapping (but they are not eigenfunctions of r L a ). This justifies the assertion made before Remark 27.

Concluding remarks

One common feature of the previous analysis of L a or r L a , for a ą 0, is that the underlying L 2 space was decomposed into ' pPP V p , where the subspaces V p are orthogonal and left invariant by the generator at hand. In the first model the index set P is Z `\ N\N and Z `in the second model. These decompositions were maximal, in the sense that each of the V p , p P P, cannot be non-trivially decomposed further (due to the non-orthogonality of all the eigenvectors belonging to V p ). Inside each of the V p , p P P, the restriction of the generator was written under the form K p `Rp ´Rp , where K p is self-adjoint in V p and where the brackets of the operators K p , R p and R p have nice forms (especially rK p , R p s " R p , which implies that there is a basis consisting of eigenvectors of K p in which the matrix of R p has a below-diagonal form, thus among decompositions of the type K p `Rp ´Rp , R p is in some sense minimal). Indeed, everything was deduced from the relations satisfied by these brackets. So it is natural to wonder if something is left of these observations for more general models.

First we note that the decompositions of the restriction of generator to the subspace V p , p P P, can be lifted into a decomposition K `R ´R˚o f the initial generator, where K ≔ ' pPP K p and R ≔ ' pPP R p . More precisely, in the first model we get K " aB 2 y ´yB y R " yB x ´aB x B y R ˚" ´aB x B y , with rK, Rs " R, rR, R ˚s " aJ, where J " B 2

x is a coercive operator on T (and for any α, β ą 0, αJ `βK is coervive on T ˆR). Similarly, in the second model we have K " B 2 y ´yB y R " yB x ´Bx B y R ˚" axB y ´Bx B y , with rK, Rs " R, rR, R ˚s " J ´aK, where J ≔ B x B x " B 2 x ´axB x is a coercive Ornstein-Ulhenbeck operator on R. In the literature about hypocoercivity, the authors have often a predilection for brackets of first order operators (this is maybe due to the importance of Hörmander's condition in hypoellipticity), but it seems that in the considered toy models, the key is given by brackets between second order operators. More generally, let be given a smooth potential U : T Ñ R and consider on T ˆR the kinetic operator L ≔ yB x ´U 1 pxqB y `B2 y ´yB y .

The following remarks can be adapted to the situation of potentials defined on R, under appropriate conditions. The associated invariant probability is µ ≔ ν ˆγ1 , where ν is the Gibbs measure on T whose density with respect to the Lebesgue measure λ is proportional to expp´U q. As above, we can write L " K `R ´R˚, where K " B 2 y ´yB y R " yB x ´Bx B y R ˚" U 1 pxqB y ´Bx B y

The operator K is still self-adjoint in L 2 pµq and R ˚is adjoint to R. We furthermore have rK, Rs " R, rR, R ˚s " J ´U 2 K

where J ≔ B x B x " B 2

x ´U 1 pxqB x is the usual coercive Langevin operator associated to U on T ( note that the formulation L " K `R ´R˚i s different from the one proposed by Villani L " A ˚A `B in the first chapter of [START_REF] Villani | [END_REF] since our operator K is a second order one).

We are wondering if these properties could not be used to deduce, in a direct manner, hypocoercive bounds for the semi-group pP t q tě0 associated to L. So let f P L 2 pµq be given with µrf s " 0 and denote for t ě 0, F t ≔ µrpP t rf sq 2 s. Since we expect behaviors such as (1), ( 3) and (4) to be valid again, it is natural to look for inequalities satisfied by F t , F 1 t , F 2 t and F 3 t . So let us compute formally (a justification would require regularity assumptions on f ) these derivatives: using the relation rK, Rs " R, we get that for all t ě 0, where f t is a short hand for P t rf s and where x¨, ¨y stands for the scalar product in L 2 pµq. In view of rR, R ˚s, which brings the missing coercivity through J, hope is rising. We first tried to find three constants A, B, C ą 0 such that for regular functions f and for all t ě 0,

F
AF t `BF 1 t `CF 2 t `F 3 t ď 0.
It is sufficient to prove such a differential inequality with t " 0. Interpreting AF 0 `BF 1 0 `CF 2 0 `F 3 0 as a quadratic form in f , we would like to find A, B, C ą 0 so that it is non-positive definite. In fact, we were able to attain this objective in the case U " 0, then up to appropriate changes of the constants A, B, C ą 0 (where would enter the supremum norms of U 1 and U 2 ), it could be extended to all smooth potential U . That is where we are brought back to the first toy model (with a " 1). Unfortunately, in this simple case, we can show that there is no choice of the constants A, B, C ą 0 so that the quadratic form AF 0 `BF 1 0 `CF 2 0 `F 3 0 is non-positive definite. Despite the fact that for any p P N, it is possible to find "constants" A p , B p , C p ą 0 such that the restriction to V p (and to W p , with the notations of Section 2) of the quadratic form A p F 0 `Bp F 1 0 `Cp F 2 0 `F 3 0

Corollary 4

 4 For any c, d ą 0 and a, b P R with ab ą 0, consider the operator r L a,b,c,d ≔ byB x ´axB y `cB 2 y ´dyB y which admits r µ a,b,c,d ≔ γ bc{padq b γ c{d as invariant probability. We have for the corresponding semi-goup p r P pa,b,c,dq t

  pa,b,c,dq t q tě0 on R generated by r K a,b,c,d ≔ cB 2 x ´da b xB x , whose amount of injected randomness is the same as r L a,b,c,d and whose reversible probability is γ bc{padq , the first marginal law of r µ a,b,c,d . Up to scalings of space and time, r K a,b,c,d is an Ornstein-Ulhenbeck generator whose spectral gap is da{b. It follows that the asymptotical exponential rate of p r Q pa,b,c,dq t q tě0 is da{b. So if d ), it is more efficient to use the first coordinate of p r P pa,b,c,dq t q tě0

paq t ´r µ a ||| 2 L 2

 22 pr µaqý " sup nPN ||| r P paq t ||| 2 Hný .

"

  s k .It follows that for any k P 0, n ,

  expptM q||| ě exp ˆ´c 2 ˆt

			1 `expp´tq ´2 1 ´expp´tq	˙˙.
	Proof
	Since by definition, for any t ě 0,
	||| expptM q||| ≔	sup zPl 2 pZ `qzt0u	}expptM qz} }z}	,
	expp´2c 2 tq exp expp´2c 2 tq exp	ρPR ˆ´4pexpp´tq ´1q 2 c 2 `pexpp´2tq ´1qρ 2 `4pexpp´tq ´1qcρ ˘" " expp´2tq ´1 ˆ4c 2 p1 ´expp´tqq expp´tq `1 ˙.
	3.2 Upper bound of |||P

we deduce from the above computations that ||| expptM q||| 2 ě expp´2c 2 tq sup ρPR exp `pexpp´2tq ´1qρ 2 `4pexpp´tq ´1qcρ ˘(28) " expp´2c 2 tq exp ˜sup pa,V p q t

  ProofRecall from Proposition 17, that almost everywhere in px, yq P T ˆR, ξ n px, yq is the real part of

			ÿ lP 0,n	ˆn l	˙l!p2cq n´l h l,a pyq ? l!	i l exppippx `yqq,
	where c "	?	ap. We deduce that zpx, yq is a.e. the real part of
			2 p p! a p2pq! " a exppippx `yqq 2 p p! p2pq! exppippx `yqq ÿ nPZ `ρn ÿ ÿ lP 0,n lPZ `il h l,a pyq p2cq n´l pn ´lq! ? l! ÿ i l h l,a pyq ? l! něl p2cq n´l pn ´lq! " 2 p p! a p2pq! exppippx `yqq ÿ lPZ l h l,a pyq ? expp2cρq l! `piρq " a p2pq! exppippx `yqq expp2cρq exp ? a 2 `ρ2 2 p p! ˆiρy ˙, ρ n
	where (21) was taken into account.
	Thus, when |||P
	paq t any element of U 1 zt0u is a maximizing function for the computation of |||P ´µa ||| " expp´tq, then paq t ´µa |||, for instance the mapping T ˆR Q px, yq Þ Ñ y. This no longer true if t ą 0 is such that |||P paq t ´µa ||| ą expp´tq, in which case z t ≔ ř nPZ `ρn t n! ξ n P V 1 is a maximizing function, where ρ t ≔ ´2? ap1 `expp´tqq ´1 is the quantity defined in (33) when p " 1 (let z 1 t ≔ ř nPZ `ρn t n! ξ 1 n P W 1 , where the ξ 1 n are defined in Proposition 17, with p " 1, then any non-null linear combination of z zpx, yq " a p2pq! exp apρ `ρ2 2 ? a `ppx `yq 2 p p! ˆ2? ˙cos ˆρy ˙.

t and z 1 t is also maximizing). So let us compute z t and more generally: Lemma 26 For any p P N and ρ P R, consider z ≔ ř nPZ `ρn n! ξ n P V p . Then we have, almost everywhere in px, yq P T ˆR, paq t ´µa ||| ą expp´tq, functions proportional to

  n is left invariant by r L a . Let us consider the matrix Ă M n associated to the restriction of r L a to H n in the orthonormal basis ph p,a b h n´p q pP 0,n . It is the tridiagonal matrix given by

	Ă M n "	¨´n ´?an 0	? ´pn ´1q an ´aa2pn ´1q	0 a2pn ´1q ´pn ´2q a . . .	¨¨¨0 . . . . . . 0 . . . ? an
		0	¨¨¨0

  The variational caracterization of eigenvalues then implies the second part of the above lemma.The matrix B being non degenerate, the largest eigenvalue of B ´1{2 A ˚BAB ´1{2 is also the largest eigenvalue of B ´1A ˚BA. Next result determines it:Lemma 44 The largest eigenvalue of B ´1A ˚BA is Λ n θ,t expp2θntq, where

	zPR n`1 zt0u	xexpptM n qz, expptM n qzy xz, zy	" exp p´nθtq sup ηPR n`1 zt0u " exp p´nθtq sup ηPR n`1 zt0u " exp p´nθtq sup ηPR n`1 zt0u	xAη, BAηy xη, Bηy @ AB ´1{2 η, BAB ´1{2 η xη, ηy @ η, B ´1{2 A ˚BAB ´1{2 η D D xη, ηy	.

  Then for any k P 0, n , we have Thus in the end, we get for any k P 0, n , Expanding the above condition, we end up with the second order equation in r: pe θt ´1qre θt r 2 `pe θt ´1 ´γr1 `eθt sqr ´γe θt s " 0.

		pB ´1 r A ˚B r Aηq k " ´eθt s	¯k	ˆ1	`pe θt ´1qs γ	˙n´k	.
	It appears that the vector η is an eigenvector for B ´1 r A ˚B r A if and only if we have
		e θt s ˆ1	`pe θt ´1qs γ	˙´1	" r,
	and in this case the corresponding eigenvalue will be
		Λ ≔ ˆ1 " ˆ1	˙n pe θt ´1 `eθt rq ˙n . `pe θt ´1qs γ `pe θt ´1q γ	(46)
	For t ą 0, this equation admits a positive solution as required, namely r " 1 2 e ´θt ˆ1 ´eθt `γr1 `eθt s `bpe θt ´1 ´γr1 `eθt sq 2 `4γe 2θt 1 2 e ´θt ¨1 ´eθt `γr1 `eθt s `p1 `eθt q g f f e pγ `1q ˜γ `ˆe θt ´1 " e θt `1 ˙2¸' .
	Inserting this value in (46), we obtain
	Λ " ¨1 " ¨1	`pe θt ´1q 2γ `pe θt ´1q 2 ¨eθt ´1 `p1 `eθt q ¨γ 2γ `e2θt ´1 2 ¨1 `g f f e ˆ1	`g f f e pγ `1q ˜γ `1 γ ˙˜1 `1 γ ˆeθt `ˆe θt e θt ´1 e θt `1 ˙2¸' 'n ´1 `1 ˙2¸' '' n ,
	and this leads to the assertion of the lemma.
	Now we can come back to our project of computing the L 2 -operator norms of the hypocoercive semi-group associated to the generator r L a , at least for a P p0, 1{4q.
	p r A ˚B r Aηq k " " γ n ÿ lP 0,n ˆeθt ˆl k ˙pe θt ´1q l´k e θkt e θt ´1 ˙k 1 k! ÿ " γ n ˆeθt e θt ´1 ˙k 1 k! ˆpe θt ´1qs ˆn l ˙γn´l s l γ ˙k n pkq " γ n ˆeθt s γ ˙k n pkq k! ˆ1 γ `pe θt ´1qs ˙n´k ÿ lěk . lěk γ l pkq ˆpe θt ´1qs ˙l n plq pn ´kq pl´kq pl ´kq! l! Theorem 45 For any a P p0, 1{4q and any t ě 0, we have ||| r P paq t ´r µ a ||| L 2 pr µaqý " a Λ θ,t exp ˆ´1 t ´θ 2 ˙,	γ ˆpe θt ´1qs	˙l´k

  According to Lemmas 43 and 44, we have||| expptM n q||| 2 " pexppθtqΛ θ,t q n ,

	so that
	||| r P t paq	´r µ a ||| 2 L 2 pr µaqý " sup

r P paq t ´r µ a ||| 2 L 2 pr µaqý " sup nPN expp´ntq||| expptM n q||| 2 . (47) nPN pexpp´p1 ´θqtqΛ θ,t q n .

  ). Definition 42 can be extended to any ρ P C, but Lemma 42 must be replaced by

								'' n
								p ρ
								θ
	Lemma 47 For any r ρ, p ρ P C, we have xzpr ρq, zpp ρqy " ´1 `|θ| 2 ¯´n{2	˜γ `ˆ1	`r ρ δ	˙ˆ1	`p ρ δ	˙¸n	,
	where
	γ ≔ |θ| 2	and	δ ≔	2θ 1 `|θ| 2 .

  As in the proof of Lemma 43, we also compute that for any t ě 0,xexpptM n qz, expptM n qzy " ´1 `|θ| 2 ¯´n{2 |expp´nθt{2q| 2 ÿThus, if the matrices A and B are defined in the same way as in Lemma 43, it appears the L 2operator norm of expptM n q is equal to the square root of the largest eigenvalue of the Hermitian matrix B ´1{2 A ˚BAB ´1{2 , where A ˚is the Hermitian adjoint matrix associated to A. If exppθtq " 1, consider the diagonal matrices r We can write A " r C r A p C, where the entries of r A are just the absolute values of the entries of A. The interest is that B ´1{2 A ˚BAB ´1{2 " p C ´1B ´1{2 r A ˚B r AB ´1{2 p C is conjugate to the symmetric matrix B ´1{2 r A ˚B r AB ´1{2 , whose entries are positive. We can then use Perron-Frobenius theorem to get Lemma 48 The largest eigenvalue of B ´1{2 A ˚BAB ´1{2 is Λ n θ,t , where

	with ν ≔ ˇˇe θt ´1ˇˇ.	Λ θ,t ≔	$ & %	1 1	`ν2 2|θ| 2 ˆ1 `b1 `4 |θ| 2 ν 2 ˙, if ν " 0 , if ν " 0
		pP 0,n " ´1 `|θ| 2 ¯´n{2 ÿ pP 0,n	ˆn p ˆn p	˙γn´p ˇˇˇˇˇÿ kP 0,n ˙γn´p |η p | 2 ,	ˆ1	`ρk δ	˙p ν k	ˇˇˇˇˇ2
	with
		@ l P 0, n ,	η l ≔	ÿ kP 0,n	ˆ1	`ρk δ	˙l ν k .
	" ´1 `|θ| 2 ¯´n{2 ÿ pP 0,n	ˆn p	pP 0,n ˙γn´p ˇˇˇˇˇÿ lP 0,p	ˆn p ˆp l ˙γn´p ˇˇˇˇˇÿ lP 0,p ˙p1 ´exppθtqq p´l exppθltqη l ˆp l ˙p1 ´exppθtqq p´l exppθltqη l ˇˇˇˇˇ2 .	ˇˇˇˇˇ2
					C and p C defined by
		@ k P 0, n ,	r C k,k ≔	p1 ´eθt q k |1 ´eθt |

xz, zy " ´1 `|θ| 2 ¯´n{2 ÿ k , p C k,k ≔ p1 ´eθt q ´ke θkt |p1 ´eθt q ´1| k .

  r P paq t ´r µ a ||| 2 L 2 pr µaqý " C a ptqe ´t,

	with
	C a ptq ≔ 1	`ˇˇe ? p4a ´1q 4a´1it ´1ˇˇ2 ˆˇˇe ?	4a´1it ´1ˇˇˇ`b ˇˇe ?	4a´1it

  1 t " 2 xKf t , f t y , F 2 K 3 f t , f t D ´24 xKf t , Rf t y ´12 xf t , Rf t y `4 xrR, R ˚sf t , f t y .

	t F 3 t	" 4 " 8	@ @	K 2 f t , f t	D	´4 xf t , Rf t y ,

Unfortunately, it is not obvious to obtain the wanted recurrence relation from this expression.

From the two last lemmas, we can deduce an explicit formula for xζ p , ζ q y, p, q P 0, n , but it is not so easy to handle. It rather suggests to consider certain particular vectors.

Exact computation of

From (41), we have for any m P 0, n and p P 0, n , P p pmq " ÿ kPZ `ppp´kq pn ´kq pp´kq pp ´kq! ´υ 2 ¯p´k m pkq .

Hence, using the relation p pp´kq {pp!q " 1{pk!q and exchanging sums, we obtain

Coming back to (43), it appears that 2 n xzpr ρq, zpp ρqy is equal to

where

and

as announced.

The main interest of vectors of the form zpρq, for ρ P R, is that they well-behave under the action of the semi-group associated to M n , more precisely:

To take advatange of this property, let us consider a basis of R n`1 , of the form pzpρ k qq kP 0,n : indeed, classical Vandermonde determinants show that such a family will be a basis as soon all the ρ k are distinct. Since powers play an important role in the kind of formulas that we have obtained so far, it is convenient to chose a basis of the form pzpρ k qq kP 0,n , where ρ is a real different from ´1, 0 and 1. Then any z P R n`1 can be written under the form

where the ν k , for k P 0, n , are the appropriate coordinates.

Lemma 43 With the notation (44), we have

, where γ ≔ θ 2 {p1 ´θ2 q ą 0. It follows that for any given t ě 0, the operator norm of expptM n q is equal to the square root of the largest eigenvalue of the symmetric matrix exp p´nθtq B ´1{2 A ˚BAB ´1{2 , where A and B are respectively the triangular and diagonal matrices defined by @ k, l P 0, n , A k,l ≔ ˆk l ˙p1 ´exppθtqq k´l exppθltq,

Proof

From Lemma 42 and by definition of γ, we get that for any k, l P 0, n , A zpρ k q, zpρ l q

Thus, if we using (44) and expand xz, zy, we get , where we have defined

Since ρ R t´1, 1u, Vandermonde determinant insures that the linear morphism R n`1 Q pν k q kP 0,n Þ Ñ η ≔ pη l q lP 0,n P R n`1 is bijective. Using the matrices A, B defined in the lemma, we can write xexpptM n qz, expptM n qzy " pexp p´θtq p1 ´θ2 qq n xAη, BAηy ,

Proof

The first part of the proof of Lemma 42 and (48) show that xzpr ρq, zpp ρqy

is non-positive definite (an analogous statement is valid in the Gaussian case). Thus an idea is missing to push further this alternative approach. Furthermore, these considerations are maybe not without links with the traditional approach, where the L 2 norm is modified by the addition of terms, since among them, xf, Rf y " xB x f, B y f y plays a major role (see for instance Villani [START_REF] Villani | [END_REF]).

Aknowledgments: We are grateful to Francis Nier for pointing out the papers of Hérau [START_REF] Hérau | Short and long time behavior of the Fokker-Planck equation in a confining potential and applications[END_REF] and Ottobre, Pavliotis and Pravda-Starov [START_REF] Ottobre | Exponential return to equilibrium for hypoelliptic quadratic systems[END_REF] and for related conversations. We also aknowledge fruitful discussions with Dominique Bakry.