Laurent Miclo

On the hypergroup property

Keywords: hypergroup property, finite reversible Markov processes, biaised Ehrenfest model, finite birth and death processes, Neumann eigenvectors, Dirichlet (minor) eigenvalues primary: 20N20, secondary: 60J05, 60J10, 15A18, 15A42, 60J80, 65F15

The hypergroup property satisfied by certain reversible Markov chains can be seen as a generalization of the convolution related features of class random walks on groups. Carlen, Geronimo and Loss [3] developed a method for checking this property in the context of Jacobi eigen-polynomials. A probabilistic extension of their approach is proposed here, enabling to recover the discrete example of the biased Ehrenfest model due to Eagleson [8]. Next a spectral characterization is provided for finite birth and death chains enjoying the hypergroup property with respect to one of the boundary points.

A theoretical result

There are several definitions of the hypergroup property for a reversible Markov kernel P . One of them, recalled in (4) below, is the non-negativity of certain sums of products of quantities related to the eigenfunctions associated to P . In the context of Jacobi polynomials, Carlen, Geronimo and Loss [START_REF] Carlen | On the Markov sequence problem for Jacobi polynomials[END_REF] developed a method in order to check this property. Here we begin by extending it, giving it a general probabilistic flavor by replacing in their criterion some mappings by Markov kernels. Next we will see how the resulting abstract condition can be applied to recover the first instance of the hypergroup property, namely the example of the biased Ehrenfest model due to Eagleson [START_REF] Eagleson | A characterization theorem for positive definite sequences on the Krawtchouk polynomials[END_REF]. We will investigate further the hypergroup property for birth and death Markov chains, by providing a spectral criterion in the last section. For general motivations relative to the notion of the hypergroup property, see for instance Diaconis and Griffiths [START_REF] Diaconis | Exchangeable pairs of Bernoulli random variables, Krawtchouck polynomials, and Ehrenfest urns[END_REF] or Bakry and Huet [START_REF] Bakry | The hypergroup property and representation of Markov kernels[END_REF].

Let p S, S, μ, P q be a reversible Markov framework: p S, Sq is a measurable space endowed with a probability measure μ and P is a self-adjoint Markovian operator on L 2 pμq. Recall that the Markov property consists in two assumptions: on one hand, for any non-negative function f P L 2 pμq, P rf s is non-negative, and on the other hand, P r1s " 1, where 1 P L 2 pμq is the constant function taking the value 1 (μ-a.s.).

Consider another measurable space pS, Sq, as well as a Markov kernel Q from p S, Sq to pS, Sq: it is a mapping from S ˆS to r0, 1s such that for any x P S, Qpx, ¨q is a probability distribution and for any A P S, Qp¨, Aq is a measurable mapping (for our purpose, the requirements with respect to the first variable only need to be satisfied μ-a.s.). The kernel Q can be seen as a Markov operator from BpSq, the space of bounded measurable functions defined on pS, Sq, to Bp Sq, via the formula @ f P BpSq, @ x P S, Qrf spxq ≔ ż f pxq Qpx, dxq

Denote by µ the image of μ by Q: @ A P S, µpAq ≔ ż Qpx, Aq μpdxq Then we have @ f P BpSq, µrf s " μrQrf ss and since by Cauchy-Schwarz inequality, pQrf sq 2 ď Qrf 2 s, it appears that Q can be extended into an operator of norm 1 from L 2 pµq to L 2 pμq. Denote by Q ˚the adjoint operator of Q, it is in particular an operator of norm 1 from L 2 pμq to L 2 pµq. In fact we have a little better:

Lemma 1 The operator Q ˚is Markovian.

Proof

To check the preservation of non-negativeness, it is sufficient to see that for any non-negative f P L 2 pμq and g P L 2 pµq, xQ ˚rf s, gy µ ě 0, where x¨, ¨yµ stands for the scalar product in L 2 pµq. This property is an immediate consequence of

xQ ˚rf s, gy µ " xf, Qrgsy μ ě 0
For the computation of Q ˚r1s, note that for any f P L 2 pµq,

xQ ˚r1s, f y µ " x1, Qrf sy μ " μrQrf ss " µrf s " x1, f y µ
Since this is valid for all f P L 2 pµq, we conclude that Q ˚r1s " 1.

Define

P ≔ Q ˚P Q (1)
By composition, P is a Markov operator from L 2 pµq to L 2 pµq, which is clearly self-adjoint, by self-adjointness of P . To get a more interesting property of P , we need to introduce the following notion. A Markov operator G from L 2 pμq to itself is said to be Q-compatible if we have

QQ ˚GQ " GQ (2)
Lemma 2 If P is Q-compatible, then the operators P and P are intertwined through Q:

QP " P Q

Proof

By definition, we have

QP " QQ ˚P Q " P Q by Q-compatibility.
From now on, P is assumed to be Q-compatible. It seems that an important tool to investigate the Markov operator P intertwined with P is the set G of Markov operators G from L 2 pμq to itself which commute with P , G P " P G, and which are Q-compatible. This set G has the structure of a semi-group: for all G, G 1 P G, GG 1 P G. Indeed, GG 1 clearly commutes with P if both G and G 1 commute with P . If (2) is satisfied by G and G 1 , then the same is true for GG 1 , because

QQ ˚GG 1 Q " QQ ˚GQQ ˚G1 Q " GQQ ˚G1 Q " GG 1 Q
In particular G contains t P n : n P Z `u, the semi-group generated by P , but as it can be observed on the example of the next section, G can be larger than a temporal evolution semi-group. Under the above setting, to each G P G, we can associate a Markov operator K G on L 2 pµq, via

K G ≔ Q ˚GQ Proposition 3
For all G P G, K G and P commute.

Proof

The argument is similar to that used in the proof of Lemma 2: using Lemma 5 and Assumption (2), it appears that

K G P " Q ˚GQQ ˚P Q " Q ˚G P Q " Q ˚P GQ " Q ˚P QQ ˚GQ " P K G
It is time to come to the main application of the above considerations. Assume that S is a finite set of cardinal N P N (then up to lumping some of its elements together, there is no loss of generality in taking for S the σ-algebra consisting of all the subsets of S and up to removing the µ-negligible points from S, we furthermore assume that µ gives a positive weight to all the points of S). By symmetry, P is diagonalizable, let pϕ l q lP N be an orthonormal basis of L 2 pµq consisting of its eigenvectors. We make the hypothesis that all the eigenvalues of P are of multiplicity 1 and that there exists x 0 P S such that for all x 1 P S, there exists G P G such that

K G px 0 , ¨q " δ x 1 (3)
Theorem 4 Under the above conditions, P satisfies the hypergroup property with respect to x 0 , namely we have ϕ l px 0 q " 0 for all l P N and @ x, y, z P S, ÿ lP N ϕ l pxqϕ l pyqϕ l pzq ϕ l px 0 q ě 0 (4)

Proof

Fix l P N and denote θ l the eigenvalue associated to the eigenvector ϕ l . From P rϕ l s " θ l ϕ l and Proposition 3, we deduce that for any G P G,

P rK G rϕ l ss " K G rP rϕ l ss " θ l K G rϕs
namely, either K G rϕ l s " 0 or K G rϕ l s is an eigenvector of P associated to the eigenvalue θ l . Due the multiplicity 1 of this eigenvalue, we deduce that K G rϕ l s is proportional to ϕ l (this being also true if K G rϕ l s " 0), say K G rϕ l s " λpG, lqϕ l . Since this is true for all l P N , the spectral decomposition of K G is given by pλpG, lq, ϕ l q lP N and we have @ y, z P S, K G py, zq " ÿ lP N λpG, lqϕ l pyqϕ l pzqµpzq (5)

Fix x 1 P S and let G P G be as in [START_REF] Carlen | On the Markov sequence problem for Jacobi polynomials[END_REF]. We get from this equation that λpG, lqϕpx 0 q " K G rϕ l spx 0 q " ϕpx 1 q

If ϕ l px 0 q was to vanish, the same would be true of ϕ l px 1 q, for all x 1 P S, contradicting that ϕ is a vector of norm 1. Thus ϕ l px 0 q " 0 and λpG, lq " ϕpx 1 q{ϕpx 0 q. It follows from ( 5) that for all y, z P S,

ÿ lP N ϕ l px 1 qϕ l pyqϕ l pzq ϕ l px 0 q " ÿ lP N
λpG, lqϕ l pyqϕ l pzq " K G py, zq µpzq ě 0 because K G is a Markovian matrix. This is the wanted hypergroup property, because x 1 was chosen arbitrarily.

To put more flesh on the notion of Q-compatibility, let us present a traditional instance of the above setting. Instead of a Markov kernel Q, assume that we are given a measurable mapping q from p S, Sq to pS, Sq. It can also be seen as a "deterministic" Markov kernel from p S, Sq to pS, Sq, via @ x P S, Qpx, ¨q ≔ δ qpxq p¨q (6) so that the above development applies.

Let T be the σ-field generated by q. In this context, the operator Q is an isometry from L 2 pµq to L 2 pμq, because for all f P BpSq, pQrf sq 2 " Qrf 2 s (this identity is in fact a characterization of the μ-a.s. determinism of Q). A convenient property of Q ˚is: Lemma 5 The Markov operator QQ ˚corresponds to the conditional expectation with respect to T .

Proof By composition, QQ ˚is a Markov operator from L 2 pμq to itself. To show that it corresponds to the conditional expectation with respect to T , we must prove that

@ f, g P L 2 pμq, xQQ ˚rf s, Qrgsy μ " xf, Qrgsy μ
This comes from the fact that Q is an isometry, which implies that the l.h.s. is equal to

μrQrQ ˚rf sgss " µrQ ˚rf sgs " xQ ˚rf s, gy µ " xf, Qrgsy μ The notion of Q-compatibility (2) of a Markov kernel G is then equivalent to GQ is a Markov operator from L 2 pµq to L 2 pμ, T q (7)
where L 2 pμ, T q is the subspace of L 2 pµq consisting of functions measurable with respect to T . Thus in the context of a deterministic Q, Lemma 2 amounts to the famous criterion of Dynkin [START_REF] Dynkin | Markov processes. Vols. I, II[END_REF] insuring that a function of a Markov chain is itself a Markov chain.

An example

Eagleson [START_REF] Eagleson | A characterization theorem for positive definite sequences on the Krawtchouk polynomials[END_REF] proved that the biased Ehrenfest model satisfies the hypergroup property, let us show how Theorem 4 enables to recover this result.

We begin by recalling the underlying birth and death Markov transition kernel P on S ≔ 0, N , with N P N ˚(so there is a slight modification of the notations of Theorem 4: the cardinal of S is now N `1), parametrized by p P p0, 1q:

@ x, y P 0, N , P px, yq ≔ $ ' ' & ' ' % N ´x N p , if y " x `1 x N p1 ´pq , if y " x ´1 1 ´p ´p1 ´2pq x N , if y " x 0 , otherwise (8)
This birth and death kernel is irreducible and its unique reversible probability measure µ is given by

@ x P 0, N , µpxq " ˆN x ˙px p1 ´pq N ´x (9)
This can be computed directly or deduced, as in the previous section, from the existence of a simple reversible Markov framework p S, S, μ, P q "above" P . Indeed, take S " t0, 1u N , endowed with the σ-field S of all its subsets, and consider the mapping q going from S to S defined by

@ x ≔ px l q lP 0,N P S, qpxq ≔ ÿ lP N xl
The probability measure μ is given by @ x P S, μpxq ≔ p qpxq p1 ´pq qpxq

For any l P 0, N , consider the Markov transition matrix Pl defined by

@ x ≔ px k q kP 0,N , y ≔ pȳ k q kP 0,N P S, Pl px, ȳq ≔ $ & % p , if ȳl " 1 and ȳk " xk , for k " l 1 ´p , if ȳl " 0 and ȳk " xk , for k " l 0 , otherwise
The measure μ is clearly reversible for Pl , as well as for

P ≔ 1 N ÿ lP N

Pl

As in the end of last section, we reinterpret the mapping q as the Markov kernel from p S, Sq to pS, Sq given in [START_REF] Diaconis | On characterizations of Metropolis type algorithms in continuous time[END_REF]. The associated σ-field T Ă S consists of the events which are left invariant by all the permutations of the indices. In particular, Assumption (2) is satisfied, P Q being clearly a Markov operator from L 2 pµq to L 2 pμ, T q. Furthermore the Markovian matrix P defined in ( 1) is given by ( 8) and the image of μ by q coincides with µ described in [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]. Thus µ is necessarily reversible with respect to P . But the interest of the above construction is that it enables to recover the hypergoup property of P via Theorem 4. From now on, assume that p P p0, 1{2s (if p P p1{2, 1q, reverse the order of the segment 0, N to come back to the situation where p P p0, 1{2q). For l P N , consider the Markov transition matrix H l defined by @ x ≔ px k q kP N , y ≔ pȳ k q kP N P S,

H l px, ȳq ≔ $ ' ' & ' ' % 1
, if xl " 1, ȳl " 0 and ȳk " xk , for k " l p{p1 ´pq , if xl " 0, ȳl " 1 and ȳk " xk , for k " l p1 ´2pq{p1 ´pq , if xl " 0, ȳl " 0 and ȳk " xk , for k " l 0 , otherwise

It is immediate to check that

Pl " pI `p1 ´pqH l where I is the identity matrix (seen as the motionless Markov kernel). In particular, H l commutes with Pl and with P . More generally, for A Ă N , let H A be given by

H A ≔ ź lPA H l
(in r.h.s. the order of the compositions of the Markov kernels does not matter, because they commute among themselves). Again, H A is a Markov kernel commuting with P . Nevertheless, it lacks symmetry to belong to G. So for any l P N , consider

G l ≔ 1 `N l ˘ÿ AĂ N : cardpAq"l H A
which is easily seen to belong to G. This leads us to consider the Markov kernel K G l on 0, N . It appears without difficulty that

@ l P N , K G l pN, ¨q " δ N ´lp¨q
This observation enables to apply Theorem 4 to get that P satisfies the hypergroup property with respect to the point N (if p ě 1{2, P satisfies the hypergroup property with respect to the point 0).

To investigate the extent of the applicability of the approach of the previous section, it would be interesting to study the multidimensional Krawtchouk polynomials, which are a multidimensional extension of the above example, cf. Diaconis and Griffiths [START_REF] Diaconis | Exchangeable pairs of Bernoulli random variables, Krawtchouck polynomials, and Ehrenfest urns[END_REF][START_REF] Diaconis | An introduction to multivariate Krawtchouk polynomials and their applications[END_REF]. Nevertheless, to generalize the result of this section, staying in the one-dimensional setting of finite birth and death chains, already presents surprising challenges, as we are now going to see.

On birth and death chains

Instead of working with a "covering Markov framework" p S, S, μ, P q, where hidden symmetries in the initial model pS, S, µ, P q are more obvious, one can also try to find directly the commuting Markov kernels. We investigate here the situation of finite birth and death chains, by providing a spectral characterization of the hypergroup property with respect to the left boundary point. This enables to construct a practical algorithm for checking this property. Next we conjecture two seemingly natural discrete versions of Achour-Trimèche's theorem [START_REF] Achour | Opérateurs de translation généralisée associés à un opérateur différentiel singulier sur un intervalle borné[END_REF] (see also Bakry and Huet [START_REF] Bakry | The hypergroup property and representation of Markov kernels[END_REF]), asserting the hypergroup property under certain log-concavity of the reversible measure. Using numerical implementations of the proposed algorithm, it appears they are both wrong.

We begin by recalling the framework of finite birth and death chains. For some N P N, we take S ≔ 0, N endowed with its total σ-field S and an irreducible birth and death Markov kernel P , i.e. whose permitted transitions are those to the nearest neighbors, S being given its usual discrete line graph structure (with self-connecting loop at each vertex, to allow non-zero diagonal entries for P ). Then there exists a unique invariant probability measure µ for P and it is reversible. Our purpose is to investigate the set of Markov kernels commuting with P , namely the set K ≔ tK P M : KP " P Ku where M is the set Markov kernels on S. Note that the elements of K admits µ as invariant probability. Indeed, we have

µKP " µP K " µK
This shows that µK is invariant by P , so that µK " µ.

We are looking for conditions on P which ensure that for any probability distribution µ 0 , there exists a Markov kernel K µ 0 P K such that K µ 0 p0, ¨q " µ 0 , namely we are trying to check the hypergroup property with respect to 0. By convexity of K, this amounts to find, for any given x 1 P S, K x 1 P K such that K x 1 p0, ¨q " δ x 1 p¨q, since we can next take for any probability distribution µ 0 ,

K µ 0 " ÿ x 1 PS µ 0 px 1 qK x 1 (10)
Remark 6 The commutation relation KP " P K can be seen as a discrete wave equation in K, by interpreting the first (respectively, second) variable in the matrix K as a time (resp., space) variable. More precisely, denote k the density kernel associated to K: where for i P t1, 2u, L piq stands for the generator acting on the i-th variable as L. A least formally, one recognizes a wave equation. Thus our objective is to see when a wave equation starting from a non-negative initial condition remains non-negative.

@ t
The biased Ehrenfest birth and death processes of the previous section with p P r1{2, 1q provide examples of the Markov kernels we want to characterize. We will denote by M p the Markov matrix defined in (8) with p P r1{2, 1q. Here is a simpler example where the discrete wave interpretation is particularly obvious:

Example 7 Consider the birth and death random walk on 0, N : its Markov kernel M 0 (not to be confused with the notation M p , for p P r1{2, 1q, defined above) is given by @ x, y P 0, N , M 0 px, yq "

$ & % 1
, if px, yq " p0, 1q or px, yq " pN, N ´1q 1{2 , if |x ´y| " 1 and x P t0, N u 0 , otherwise

For any x 0 P 0, N and ε P t´1, 1u, let pψ x 0 ,ε pxqq xPZ `the deterministic and discrete time evolution in 0, N constructed in the following way: ψ x 0 ,ε p0q " x 0 and if x 0 P 1, N ´1 , then we take ψ x 0 ,ε p1q to be x 0 `ε. If x 0 " 0 (respectively x 0 " N ), we take ψ x 0 ,ε p1q " 1 (resp. ψ x 0 ,ε p1q " N ´1). Next for x P N, if ψ x 0 ,ε px ´1q and ψ x 0 ,ε pxq have been constructed with d x 0 ,ε pxq ≔ ψ x 0 ,ε pxq ´ψx 0 ,ε px 1q P t´1, 1u, then we take ψ x 0 ,ε px `1q " ψ x 0 ,ε pxq `dx 0 ,ε pxq, except if it is not possible (i.e. ψ x 0 ,ε pxq P t0, N u), in which case we consider ψ x 0 ,ε px `1q " ψ x 0 ,ε pxq ´dx 0 ,ε pxq. Visually, it corresponds to a trajectory of a particle issued from x 0 , starting to go to x 0 `ε and keeping in the same direction until it is reflected on one of the "walls at ´1 and N `1".

We leave to the reader as an exercise to check that for any x 0 P 0, N , the Markov kernel K x 0 defined by @ x, y P 0, N , K x 0 px, yq " 1 2 ´δψ x 0 ,1 pyq `δψ x 0 ,´1 pyq does commute with M 0 .

We come back to the situation of a general irreducible birth and death Markov kernel P . To describe the main theoretical result of this section, we need some further notations. For n P 0, N ´1 , denote by ´1 ă θ n,0 ă θ n,1 ă ¨¨¨ă θ n,n´1 ă 1 the n eigenvalues of the minor of P corresponding to the rows and columns indexed by 0, n ´1 .

This polynomial has degree n and the last sum over n n is empty except if n n´1 " 1, because for any l P 0, n , n l ď n ´l. The interest of R n comes from Lemma 10 For any probability distribution µ 0 , there exists a unique matrix K µ 0 commuting with P and whose first line coincides with µ 0 . It is given by

@ n P 0, N , K µ 0 pn, ¨q " µ 0 R n pP qp¨q (11)

Proof

We begin by showing that a solution K satisfying the two requirements of this proposition is necessarily given by the above formula. To simplify the notations, we consider the case where µ 0 " δ x 1 , with x 1 P 0, N given. Fix some n P 1, N . From the commutation relation, we get that P n K " KP n . The first line of this matrix identity reads ÿ mP 0,n P n p0, mqKpm, ¨q " P n px 1 , ¨q because P n p0, mq " 0 for m P n `1, N . It follows that Kpn, ¨q " 1 apn, nq ¨P n px 1 , ¨q ´ÿ mP 0,n´1 apn, mqKpm, ¨q' which provides an iteration formula for the computations of Kpn, ¨q, starting from Kp0, ¨q " δ x 1 .

It leads without difficulty to the announced expression, Kpn, ¨q " δ x 1 R n pP qp¨q. These arguments extend to the situation of a general probability measure µ 0 . Conversely, the matrix defined by (11) satisfies on one hand, K µ 0 p0, ¨q " µ 0 p¨q and on the other hand, for all n P 0, N , K µ 0 pn, ¨q " 1 apn, nq ¨µ0 P n p¨q ´ÿ mP 0,n´1 apn, mqK µ 0 pm, ¨q' namely ÿ mP 0,n P n p0, mqK µ 0 pm, ¨q " µ 0 P n p¨q or equivalently, we have the equality of the first line of P n K µ 0 and K µ 0 P n :

δ 0 P n K µ 0 " δ 0 K µ 0 P n
Since this is true for all n P 0, N , we deduce that

@ n P 1, N , δ 0 P n´1 P K µ 0 " δ 0 P n´1 K µ 0 P
Note that the support of the measure δ 0 P n´1 is exactly 0, n ´1 , thus by iteration, it follows that all the lines of P K µ 0 coincide with the corresponding ones of K µ 0 P , i.e. K µ 0 commutes with P .

It is natural to wonder if, for given n P 0, N , the polynomial R n is uniquely by the property (11). Indeed, assume that r R n is another polynomial of degree n satisfying the same equation. Since it must be true for all probability measure µ 0 , we get that R n pP q " r R n pP q. Thus a priori, R n is only determined up to an additional term belonging to the ideal generated by the unital minimal polynomial Q associated to the matrix P . Since P is an irreducible birth and death transition kernel, it is diagonalizable and all its eigenvalues are different.This implies that Q is of degree N . Thus if n P 0, N ´1 , R n is uniquely determined, due to the fact that its degree is n. But this argument doesn't seem to work for n " N . There is a more convenient way to see that R n is uniquely determined for all n P 0, N , even under an apparently weaker requirement, as we are to see.

Note that if µ 0 " δ 0 , the identity matrix I is a trivial solution to the problem corresponding to K 0 . By the uniqueness statement of Lemma 10, we conclude that K 0 " I. In the wave equation interpretation, K 0 corresponds to a wave initially localized at 0 and which travels at speed 1 to the right, until it reaches N at time N . The polynomial R n is in fact characterized by (11) with µ 0 " δ 0 :

Lemma 11 For all n P 0, N , there is a unique polynomial R n of degree n such that

δ n p¨q " δ 0 R n pP qp¨q (12)

Proof

Let us fix n P 0, N and write

R n pXq " ÿ pP 0,n r p X p
Since the support of the probability measure δ 0 P p is 0, p , for all p P 0, n , we deduce from (12) applied at n that 1 " r n P n p0, nq, namely r n " 1{apn, nq. Next applying (12) at n ´1, we deduce that 0 " r n´1 P n´1 p0, n ´1q `rn P n p0, n ´1q, i.e. r n´1 " ´apn, n ´1q{papn, nqapn ´1, n ´1qq. It appears that we can deduce iteratively the values of r n´2 , r n´3 , ..., r 0 .

The previous result enables an interesting interpretation of R n , for fixed n P 0, N , from which Proposition 8 follows at once. Consider the matrix r P n , indexed by 0, n ˆ 0, n and given by @ k, l P 0, n , r P n pk, lq ≔

" P pk, lq , if k P 0, n ´1 δ n plq , if k " n
It is a Markov transition matrix absorbed at n. Its eigenvalue are θ n,n ≔ 1 and the eigenvalues ´1 ă θ n,0 ă θ n,1 ă ¨¨¨ă θ n,n´1 ă 1 introduced before Proposition 8 and corresponding to eigenvectors vanishing at n.

Lemma 12 For n P 0, N fixed as above, we have R n pXq " 1 apn, nq pX ´θn,0 qpX ´θn,1 q ¨¨¨pX ´θn,n´1 q

Proof

Since P is a birth and death transition matrix, we have @ x P 0, n , δ 0 R n pP qpxq " δ 0 R n p r P n qpxq thus reinterpreting (12) on 0, n , we get

δ n p¨q " δ 0 R n p r P n qp¨q (13)
The same arguments as in the proof of Lemma 11 show that this equation determine the polynomial R n pXq, in particular the coefficient of X n is 1{apn, nq.

Consider the polynomial

QpXq ≔ pX ´θn,0 qpX ´θn,1 q ¨¨¨pX ´θn,n´1 q

Hamilton-Cayley theorem says that Qp r P n qp r P n ´Iq " 0 and in particular δ 0 Qp r P n qp r P n ´Iq " 0, which means that δ 0 Qp r P n q is an invariant measure for r P n . Since the invariant measures of r P n are proportional to δ n , we deduce that there exists a constant c n P R such that c n δ n p¨q " δ 0 Qp r P n qp¨q

Applying this inequality at n, we get that c n " P n p0, nq " apn, nq and the announced result is a consequence of the uniqueness statement of Lemma 11.

Proposition 8 suggests the following algorithm to check for the hypergroup property at 0 of a finite birth and death Markov kernel P : for all n P 0, N ´1 , one computes the eigenvalues θ n,0 , θ n,1 , . . . , θ n,n´1 and checks the non-negativity of its entries pP ´θn,0 qpP ´θn,1 q ¨¨¨pP ´θn,n´1 q. From a theoretical point of view, it may seem simpler to compute the normalized eigenvectors pϕ n q nP 0,N and to check directly the hypergroup property as it stated in (4) (with x 0 " 0 and the appropriate change of indices of the eigenvectors). But in practice it is more delicate to compute eigenvectors than eigenvalues and in the numerical experiments we made (using Scilab), first just to check the Markov kernels M p for p P r1{2, 1q and M 0 (defined in Example 7) satisfy the hypergroup property, the algorithm based on Proposition 8 is more stable.

Thus we rather used the latter to proceed to the numerical experiments described below (the codes are available on request).

Let us recall the Achour-Trimèche's theorem [START_REF] Achour | Opérateurs de translation généralisée associés à un opérateur différentiel singulier sur un intervalle borné[END_REF]. Consider the differential operator L " B 2 ´U 1 B on r0, 1s with Neumann boundary conditions, where U : r0, 1s Ñ R is a smooth convex potential, which is assumed to be either non-increasing or symmetric with respect to the point 1{2. Then L satisfies the hypergroup property with respect to 0 (the finite sum in (4) has to be naturally extended into a denumerable sum, see for instance Bakry and Huet [START_REF] Bakry | The hypergroup property and representation of Markov kernels[END_REF]).

We would like to find an extension of this result to its discrete analogous setting of finite birth and death processes. Seeing B 2 ´U 1 B as a Metropolis modification of B 2 with respect to the probability measure admitting a density proportional to expp´U q (for this point of view, cf. e.g. [START_REF] Diaconis | On characterizations of Metropolis type algorithms in continuous time[END_REF]), a first guess is as follows. Let U be a convex and non-increasing function on r0, N s and consider the probability measure µ defined as

@ x P 0, N , µpxq ≔ 1 Z expp´U pxqq π 0 pxq (14)
where Z is the normalizing constant and π 0 is the invariant probability of M 0 , namely @ x P 0, N , π 0 pxq " " 1{N , if x P 1, N ´1 1{p2N q , if x P t0, N u

The usual choice for a Markov kernel admitting µ as reversible measure is the Metropolis perturbation of M 0 (initiated in Metropolis et al. [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]) given here by @ x " y P 0, N , P px, yq ≔ M 0 px, yq expp´pU pyq ´U pxqq `q " " M 0 px, yq , if x ă y M 0 px, yq expp´pU pyq ´U pxqqq , if x ą y Numerical experiments based on Proposition 8 show that the conjecture that P satisfy the hypergroup property with respect to 0 is wrong. We checked this assertion by taking N " 10 and by sampling the convex function U according to the following procedure: let pV pxqq xP 0,N ´1 be independent exponential random variables of parameter 1{N , we take @ x P 1, N , U px ´1q ´U pxq ≔ ÿ nP 0,N ´x

V pnq

Finally, we replaced in the above considerations the exploration kernel M 0 by M p , for p P r1{2, 1q. This should reinforce the log-concavity of the probability measure µ defined as in ( 14), where π 0 is replaced by the invariant probability measure π p of the Markov kernel M p . Nevertheless the conjecture still seems to be wrong (but less so when p becomes closer to 1).

Of course these experiments suggest that the right notion of log-concavity of a measure (or rather of a Markov kernel) has yet to be found in the discrete setting.

Aknowledgments:

The hypergroup property, the Carlen, Geronimo and Loss [START_REF] Carlen | On the Markov sequence problem for Jacobi polynomials[END_REF] method and the result of Achour and Trimèche [START_REF] Achour | Opérateurs de translation généralisée associés à un opérateur différentiel singulier sur un intervalle borné[END_REF] were explained to me by Dominique Bakry. He also transmitted to me the question of Persi Diaconis of knowing whether Eagleson's example of the biased Ehrenfest model could be recovered by the approach of Carlen, Geronimo and Loss [START_REF] Carlen | On the Markov sequence problem for Jacobi polynomials[END_REF]. Despite our conversations, he categorically refused to co-sign this paper. Maybe to leave me wish him here a happy 60th birthday ;-). I'm also grateful to Sylvain Ervedoza for helping with some preliminary numerical simulations and to the ANR STAB (Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations) for its support. miclo@math.univ-toulouse.fr Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex 9, France

apn 1 , n 1 q apn 1 , n

q 1 apn 2 , n 2 q ¨¨¨apn n´1 , n n q 1 apn n , n n q X nn

˚Partially supported by ANR-12-BS01-0019

 Proposition 8The Markov kernel P satisfies the hypergroup property with respect to 0 if and only if for all n P 0, N ´1 , the matrix pP ´θn,0 qpP ´θn,1 q ¨¨¨pP ´θn,n´1 q has non-negative entries.

We excluded the case n " N , because the corresponding product matrix vanishes by the Hamilton-Cayley theorem.

Remark 9 Markov kernels usually refer to discrete time processes. Continuous time processes rather use Markov generators. A matrix L is called a Markov generator if its off-diagonal entries are non-negative and if the raw-sums all vanish. It is equivalent to the fact that we can find a positive number l ą 0 and a Markov kernel P such that L " lpP ´Iq, where I is the corresponding identity matrix. A technical advantage of Markov generators over Markov kernels is that it is straightforward to perturb them (in addition to the fact that continuous time is often easier to manage than discrete time). It is convenient for them to rewrite the above result under the following form.

Consider an irreducible birth and death Markov L generator on 0, N . For n P 0, N ´1 , denote by ´1 ă λ n,0 ă λ n,1 ă ¨¨¨ă λ n,n´1 ă 1 the eigenvalues of the minor of L corresponding to the rows and columns indexed by 0, n . The Markov generator L satisfies the hypergroup property (4) with respect to x 0 " 0 (and N replaced by N `1) if and only if for all n P 0, N ´1 , the matrix pL ´λn,0 qpL ´λn,1 q ¨¨¨pL ´λn,n´1 q has non-negative entries.

At the end of this section, we will explain how Proposition 8 enables to construct a relatively efficient algorithm to check the hypergroup property. First we prove Proposition 8 through a sequence of intermediate results.

We begin by showing that it is always possible to solve the commutation equation K µ 0 P " P K µ 0 explicitly in terms of P and µ 0 . It will remain to see if the obtained solution is non-negative, but the condition K µ 0 1 " 1 will be automatically satisfied. Indeed, for any matrix K commuting with P , we have K1 " KP 1 " P K1 so that K1 is an eigenfunction associated to the eigenvalue 1 and thus must be constant by irreducibility of P . The first component of K µ 0 1 is equal to δ 0 K µ 0 1 " µ 0 p1q " 1, so that we get K µ 0 1 " 1. Define @ n, m P 0, N , apn, mq ≔ P n p0, mq (note that apn, nq ą 0 for n P 0, N ) and for n P 0, N , the polynomial R n pXq given by R n pXq ≔ 1 apn, nq X n ´1 apn, nq ÿ

`1 apn, nq ÿ n 1 P 0,n´1 ÿ n 2 P 0,n 1 ´1 apn, n 1 q 1 apn 1 , n 1 q apn 1 , n 2 q 1 apn 2 , n 2 q X n 2 `¨¨p ´1q n 1 apn, nq


Maeva Ferrari

Karine Louati

Anne Miquel

Anthony Behin

Olivier Benveniste

Jérémie Sellam
email: jeremie.sellam@sat.aphp.fr

Prof Sellam Jérémie

Quickly progressive amyotrophy

Keywords: amyotrophy, myopathy, rapidly destructive osteoarthritis, femoro-patellar osteoarthritis, risk factor Correspondance to

While rapidly destructive OA is more recognized in hip, we report the case of a 50-year-old woman who presented a rapid chondrolysis in the patellofemoral joint in a context of rapid loss of muscular strength. She had arthralgia, myalgia and proximal muscular deficit of the limbs. Creatine phospho kinase level was elevated and electromyogram exam showed a myogenic syndrome. Neither immune

nor visceral disease was highlighted. As we suspected a polymyositis, we started corticosteroids and physiotherapy, then methotrexate and intravenous immunoglobulin. Concomitantly to the worsening of the muscular deficit and atrophy of hamstrings, she developed a persistent and disabling knee pain. Initial radiographs and magnetic resonance imaging (MRI) showed only a patellofemoral dysplasia and tiny cartilage damages. Because of aggravation of myalgia, we treated by mycophenolate mofetyl then rituximab. One year later, the knee remained painful and swollen. MRI showed signs of advanced osteoarthritis including an important loss of cartilage with an atrophy of hamstrings. Several articular corticosteroids injections were done. In the same time, the evolution of the muscular disease was unusual. Another histological analysis of muscle has highlighted a genetic myopathy due to mutation of calpain. Immunosuppressive treatments were stopped and a total joint replacement was performed.

We show for the first time a case of rapid chondrolysis of patellofemoral joint related to a severe genetic myopathy.

Reduced quadriceps strength or volume is associated with patello-femoral osteoarthritis (OA) [START_REF] Bennell | Update on the role of muscle in the genesis and management of knee osteoarthritis[END_REF][START_REF] Toumi | Association between individual quadriceps muscle volume/enthesis and patello femoral joint cartilage morphology[END_REF].

Rapidly destructive OA is a recognized entity in hip but may occur in knee OA [START_REF] Burki | Rapidly destructive tibiofemoral knee osteoarthritis: clinicoradiological presentation and outcome after global medical treatment including nonarthroscopic joint lavage plus corticosteroid injection. A single center retrolective study[END_REF]. Here, we report a case of a rapid chondrolysis in the patellofemoral joint of a patient with rapid loss of muscular strength due to a severe myopathy.

A 50-year-old woman had myalgia, arthralgia, and proximal muscular deficit of the limbs from one year before the first visit. The rate of creatine phospho kinase (CPK) was 2856 UI/L (normal 10-160 UI/L). A myogenic syndrome was found at electromyographic exam. The antinuclear, antinuclear ribonucleoproteins, anti-synthetase (anti-J01, PL-7, PL-12, OJ, EJ), anti-SRP, anti-Mi2, anti-RNP, anti-Ro52, anti-PM-ScL, anti-Ku, and anti-3-hydroxy-3-methyl-glutaryl-CoA (HMGCoA) reductase antibodies were negative. The histology of deltoid displayed necrotized fibers, complement (C5b9) deposits on sarcolemma of some fibers and sparse inflammatory cells. No over-expression of HLA class I was observed and all the fibers were decorated in immuhistochemistry for the tested structural proteins, including dysferlin and sarcoglycans. Investigations did not identify any neoplasms nor any visceral lesions. Since a polymyositis was suspected, we started corticosteroids (1 mg/kg/day then tapered) and physiotherapy. Because of worsening of the muscle deficit and high CPK level (around 1400 UI/L), subcutaneous methotrexate until 20 mg/week and intravenous immunoglobulin were added. Concomitantly, corticosteroids dosage was increased at 1 mg/kg/day. Two years later, she developed pain in knees with effusions limitating walking. Clinical examination revealed an atrophy of hamstrings, adductors and gluteal deltoids. The analysis of the joint fluid showed <100 leucocytes/mm 3 in each knee. The initial radiographs showed a bilateral patellofemoral dysplasia without any joint space narrowing. Six months later, magnetic resonance imaging (MRI) examination of the right knee showed very tiny patellar cartilage damages corresponding to an early OA (Figure 1). Five corticosteroid injections in right knee and 3 in left knee (cortivazole and triamcinolone hexacetonide) were performed in the first 18 months with transient efficacy.

Fourteen months after methotrexate initiation, because of the aggravation of myalgia, muscular deficit and persistent increase of CPK level at 1615 UI/L, mycophenolate mofetyl (2 g/d) was added but rapidly stopped because of digestive intolerance, and replaced by rituximab (1000-mg infusion on days 1 and 15) without efficacy.

Nine months after the first knee symptoms, pain dramatically increased and the walk was possible only with a cane. One year after the first imaging evaluation, radiographs showed an almost complete joint space narrowing of patellofemoral joint. MRI showed ulcerations of patellar cartilage and a major atrophy of hamstringswith a normal quadriceps (Figure 2). Considering the functional impairment, one corticosteroid injection (triamcinolone hexacetonide) and one isotopic yttrium injection in each knee and a joint lavage of right knee were performed without efficacy.

Moreover, considering the clinical and biological unusual evolution with immunosuppressive drugs, new analysis on a second muscular biopsy was performed still showing isolated necrotized fibers, atrophic fibers and internalized nuclei without inflammation. Muscle western blot showed a decrease in intensity of the calpain band, exhibiting a genetic myopathy due to mutation of calpain. Then immunosuppressive drugs and corticosteroids were stopped, while intensive physiotherapy was maintained. As the different treatments for knee OA being ineffective, a total knee joint replacement was performed 2 years after the first symptoms.

Discussion

We report a case of a rapid chondrolysis of the patellofemoral joint related to a severe myopathy. Our hypothesis is that the uncontrolled myopathy associated with long-term corticosteroid use lead to a severe and rapid amyotrophy and fatty replacement of the hamstring muscles destabilizing the femoropatellar joint.

Whereas rapid destructive hip OA is well described [START_REF] Batra | Rapidly destructive osteoarthritis of the hip joint: a case series[END_REF][START_REF] Lequesne | Rapid destructive coxarthrosis[END_REF], rapid destructive tibio-femoral OA is less known (3). To our knowledge, it is the first reported case of rapid destructive patello-femoral OA. In our patient, the rapid loss of cartilage (in 1 year) suggests a rapidly destructive knee OA according to Ayral's definiton (i.e., a loss of tibiofemoral cartilage ≥ 50% in 1 year) (3).

In our patient, the rapid functional impairment may have lead to the dramatic decrease of the thickness of the knee cartilage as shown in tetraplegic patients [START_REF] Vanwanseele | Longitudinal analysis of cartilage artrophy in the knees of patients with spinal cord injury[END_REF][START_REF] Conroy | Muscle strengh, mass, and quality in older men and women with knee osteoarthritis[END_REF]. Unexpectedly, we observed an involvement of the posterior muscular groups of the thigh while quadriceps remained normal, while femoro-patellar OA is actually associated to involvement of quadriceps. A possible explanation is that the balance between agonist-antagonist muscles protects joint against excessive constraints [START_REF] Schipplein | Interaction between active and passive knee stabilizers during level walking[END_REF][START_REF] Sharma | Quadriceps strength and osteoarthritis progression in malaligned and lax knees[END_REF] and can worse OA when it is inversed [START_REF] Sharma | Quadriceps strength and osteoarthritis progression in malaligned and lax knees[END_REF][START_REF] Sharma | Examination of exercise effect on knee osteoarthritis outcomes: why should the local mechanical environment be considered?[END_REF]. Our patient had a fatty replacement of hamstrings with quadriceps still tonics that could promote a biomechanical unbalance and consequently rapid femoro-patellar chondrolysis.

In conclusion, severe and rapid thigh amyotrophy should be considered as a cause a rapid chondrolysis of the patellofemoral joint emphasizing the crucial role of muscular strength in patello-femoral OA development.

Acknowledgment: the authors thank Francis Berenbaum (Saint-Antoine hospital, Paris, France) for his help for the diagnosis and the discussion about this case and Alain Sautet (Saint-Antoine hospital, Paris, France) for the surgical treatment and the discussion.

Conflicts of interest: none


Sahli Amor

Khonsari Rh
email: roman.khonsari@gmail.com

R H Khonsari

Magnetic resonance sialography in bilateral Stafne bone cavities

Introduction Stafne bone cavities (SBC) are mandibular radiolucent lesions with challenging differential diagnoses. Many radiological methods have been proposed to demonstrate the presence of salivary gland tissues inside the cavities.

Case report

We report a case of bilateral angular mandibular radiolucent cavities located below the mandibular nerve canal. Magnetic resonance imaging (MRI) showed continuity between the submandibular glands and the tissue herniated into the cavities. The ductal system of both submandibular glands was evaluated using magnetic resonance sialography (MR sialography). We report an opacification of the glandular ductal system from the contents of the cavities up to its primary branches, confirming the diagnosis of SBC.

Discussion MR sialography allows to image submandibular gland ducts and demonstrates the presence of salivary gland tissue inside mandibular cavities with precise topographical information. This technique does not involve canulation or injection of contrast agents, and does not expose patient to radiations. MR sialography is a valuable addition to the common imaging procedures used in the investigation of SBCs.

Keywords

Stafne bone cavity; magnetic resonance imaging; magnetic resonance sialography

Introduction

The diagnosis of Stafne bone cavities is challenging. According to many authors, the most reliable method for assessing this diagnosis is computed tomography sialography (CT sialography). [START_REF] Stafne | Bone cavities situated near the angle of the mandible[END_REF][START_REF] Philipsen | Lingual and buccal mandibular bone depressions : a review based on 583 cases from a world-wide literature survey including 69 new cases from Japan[END_REF][START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF] This invasive technique involves cannulation of the salivary ducts, local injection of contrast agent and exposure to radiations. This article reports a case of Stafne bone cavities diagnosed using magnetic resonance sialography (MR sialography) and discusses this less invasive technique as a valuable alternative to CT sialography.

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3

Case report

An 80-year-old man was referred by the local neurology department for the screening of dental infections. He had no relevant medical history. Panoramic radiograph showed bilateral symmetric well-defined radiolucent lesions in the region of the mandibular third molars (Figure 1). Both lesions were located below the mandibular canal. The right radiolucent lesion was 1.5 x 1.2 cm in size, round, well-circumscribed and corticated. The left radiolucent lesion was 1.5 x 0.5 cm in size, ovoid, with no sclerotic borders. The patient had no history of trauma and the lesions were not tender. The main hypothesis was a bilateral posterior variant of SBC. Magnetic resonance imaging (MRI) showed that the bone cavities were filled with soft tissues, continuous with the submandibular glands. MR images were obtained on a 1,5 mm (SIEMENS Magnetom Aera 6448) device using both head and neck surface coils. Fast spin echo (FSE) T1 weighted axial images with the following parameters were acquired: Repetition Time (TR): 422 ms., Echo Time (TE): 9.5 ms., matrix: 320 x 248 mm, section thickness: 3 mm, examination period: 2 min 41 s. T2 'DIXON' 2D fast spin echo (FSE) multiecho with fat and water separation images and the following parameters were also obtained: TR: 5440 ms., TE: 98-110 ms., matrix 256 x 256mm, FOV: 210 cm 2 , section thickness 3.0 mm, examination period: 3 min 45 s. MR sialography involved different sequences: 3D Sialo MRI, fat-saturated isotropic Turbo spin echo (width: 1 mm, matrix: 192 x 135 mm, FOV: 180 cm 2 , examination period: 3 min 53 s.), with MIP (maximum intensity projection) and MP (multiplanar) reconstructions. The contents of the cavities emitted the same MR signal as the submandibular glands. MR sialography confirmed that the cavities were in continuity with the submandibular glands, with an opacification of the glandular ductal system up to its primary branches (Figure 3). A final diagnosis of bilateral SBC was retained. No further investigations or treatments were proposed.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

4 Discussion SBC was initially described by Stafne in 1942 as an asymptomatic well-defined radiolucent defect located on the medial face of the mandible. [START_REF] Stafne | Bone cavities situated near the angle of the mandible[END_REF] The radiological incidence of SBC is 0.08-0.48 %. [START_REF] Philipsen | Lingual and buccal mandibular bone depressions : a review based on 583 cases from a world-wide literature survey including 69 new cases from Japan[END_REF][START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF] SBC predominantly affects men aged 50-70. [START_REF] Philipsen | Lingual and buccal mandibular bone depressions : a review based on 583 cases from a world-wide literature survey including 69 new cases from Japan[END_REF][START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF] The usual localization of SBC (also called lingual posterior variant) is the submandibular region close to the inferior border of the mandible. The diagnostic criteria for SBC are: (1) homogenous round to ovoid radiolucency of the mandible, (2) sharp cortical limits with clear distinction from neighboring structures such as teeth, (3) ranging 1.0-3.0 cm in size, (4) typical localization below the inferior alveolar nerve canal, but limited by the mandibular angle and the lower borders of the horizontal branch and ramus. [START_REF] Philipsen | Lingual and buccal mandibular bone depressions : a review based on 583 cases from a world-wide literature survey including 69 new cases from Japan[END_REF][START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF] Variations in the presentation of SBC have been reported, such as multiple lesions, lesions above the mandibular canal and lesions without sclerotic margins. [START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF][START_REF] Branstetter | Imaging of a Stafne bone cavity: what MR adds and why a new name is needed[END_REF] Similar but more anterior defects have also been reported in the mandible near the apices of the premolars, associated with the sublingual glands (lingual anterior variant), and very rarely on the medial surface of the ramus, associated with the parotid gland (ramus variant). Usually, SBC is a unilateral condition. Bilateral occurrence is rare and multiple occurrences are even more uncommon. [START_REF] Philipsen | Lingual and buccal mandibular bone depressions : a review based on 583 cases from a world-wide literature survey including 69 new cases from Japan[END_REF] The origins and the mechanisms of disease are controversial. Several hypotheses have been proposed: incomplete calcification of Meckel's cartilage during ossification, trapping of the submandibular gland during ossification or trauma and compression from acquired vascular lesion of the facial artery. [START_REF] Minowa | Evaluation of static bone cavities with CT and MRI[END_REF] The rare occurrences of SBC at an early age are not in favor of a developmental disorder and most authors currently prefer theories involving the mechanical role of the submandibular glands in the formation of the cavities. [START_REF] Ariji | Stafne's bone cavity. Classification based on outline and content determined by computed tomography[END_REF][START_REF] Shimizu | CT analysis of the Stafne's bone defects of the mandible[END_REF] In fact, aging submandibular glands are progressively subjected to non-specific inflammatory infiltrations with subsequent fibrosis, hypertrophy and hyperplasia. These [START_REF] Schneider | Stafne bone cavities: systematic algorithm for diagnosis derived from retrospective data over a 5-year period[END_REF] proposed a diagnostic algorithm for bony lesions initially supected to be SBC on a panoramic radiograph, involving first Cone beam CT (CBCT) and then MR imaging to solve difficult cases. In fact, when standard radiographs are not sufficient, CT or CBCT can help to establish the differential diagnosis with traumatic bone cysts, ameloblastoma, fibro-osseous lesions, central giant cell lesions, salivary gland tumors, metastasis and vascular malformations. [START_REF] Kopp | Differential diagnosis of Stafne idiopathic bone cyst with volume tomography (DVT)[END_REF] Nevertheless, CT and CBCT do not provide optimal information on the contents of the cavities and MRI can help to characterize the soft tissues prolapsed into the defects. [START_REF] Probst | Imaging characteristic of Stafne bone cavity -panoramic radiography, computed tomography and magnetic resonance imaging[END_REF] The combination of sialography with CT or CBCT [START_REF] Li | Cone Beam CT sialography of Stafne bone cavity[END_REF] has been suggested to confirm the diagnosis of SBC, allowing the assessment of salivary gland tissue with an opacification of the salivary ducts inside the bony cavity. However, CT sialography has several disadvantages such as pain due to the injection of the contrast agent and exposure to radiation. Furthermore, radiologists generally have little experience in salivary gland cannulation and this technique has a high incidence of failure. Some authors have thus recommended MRI as the primary diagnostic imaging technique. [START_REF] Branstetter | Imaging of a Stafne bone cavity: what MR adds and why a new name is needed[END_REF][START_REF] Probst | Imaging characteristic of Stafne bone cavity -panoramic radiography, computed tomography and magnetic resonance imaging[END_REF] In the present case, MRI with conventional sequences showed that the mandibular bone cavities were filled with tissues similar to the submandibular gland. MR sialography allowed to obtain a hyperintense signal for fluid structures with removal of surrounding tissue on high T2-weighted sequences, and could thus highlight the salivary ductal system within the tissues herniated into the cavities. Using this technique, we could unequivocally confirm the diagnosis of SBC.

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  the mechanical properties of the glands and induce enough pressure on the lingual mandibular cortical bone to cause local bone resorption. The diagnosis of SBC is generally established on routine panoramic radiographs. Schneider et al. (2014)

6 MR

 6 sialography is a non-invasive technique and does not involve canulation, injections into the salivary ducts or exposure to radiations. MR sialography is based on the application of techniques initially developed for MR cholangio-pancreatography and urinary tract imaging to the head and neck region. The technical background of MR sialography is the concept of hydro-MR 11 : fluids such as gall, urine or saliva induce a hyper-intense signal when surrounding tissues are removed on high T2-weighted sequences. Saliva is thus used as an inner contrast agent. Some authors propose to increase the contrast of MR sialography images by stimulating salivary secretion using sialagogues, hyper-hydration and lemon juice or tartaric acid administration before image acquisition. 11 MR sialography tends to progressively replace sialography and has been validated as an alternative to conventional sialography in patients with sialolithiasis, radiation induced xerostomia and Sjögren syndrome. 11-13 MR sialography allows the study of the 4 major salivary glands (left and right submandibular and parotid glands) and of the surrounding soft-tissue structures in a single imaging session, while CT sialography would required several painful and technically challenging injections of contrast agent. 12 Different MR sequences and protocols are reported in the literature for the evaluation of the submandibular ductal system: 3D CISS, 3D FSE with MPR, MIP and VR (volume rendering) reconstructions and 2D Sialo MRI with Turbo Spin Echo Single Short.11- 13 In our experience, the most clinically relevant images were obtained using 3D Sialo MRI with Turbo Spin Echo isotropic fat-saturated and reconstructions using MIP (Maximum Intensity Projection) and MPR (Multi Planar Reconstruction).The disadvantages of MR sialography are similar to the disadvantages of standard MRI: availability, high cost, image acquisition length, counter-indication in patients with atrisk implantable devices (pacemakers) and in claustrophobic patients.We believe MR sialography technique is a valuable alternative to CT sialography in the diagnostic algorithm of SBCs.

Figure 1 .

 1 Figure 1. Panoramic radiograph. Well-defined bilateral radiolucent lesions in the right and

Figure 2 .

 2 Figure 2. MR imaging showing that the mandibular bone cavities seen in Figure 1 are filled

Figure 3 .

 3 Figure 3. Axial reconstruction ("MIP" sialo-MRI) demonstrating that the bony cavities