Laurent Miclo

On the Helmholtz decomposition for finite Markov processes

Keywords: Helmholtz decompositions, compact Riemannian manifolds, vector fields, gradient fields, Poisson equation, optimal transport, finite Markov processes, finite weighted graphs Mathematics Subject Classification (2020): 60J27, 60J60, 58J65, 46E27, 37C10, 58B20

Helmholtz decompositions break down any vector field into a sum of a gradient field and a divergence-free vector field. Such a result is extended to finite irreducible and reversible Markov processes, where vector fields correspond to anti-symmetric functions on the oriented edges of the underlying graph.

Introduction

The Helmholtz decomposition on a compact Riemannian manifold writes any vector field as a sum of a gradient field and a vector field whose divergence vanishes. In some sense, this decomposition corresponds to the Riemannian probability distribution and can be extended to any other probability measure admitting a positive and smooth density with respect to , with its corresponding skewed divergence, see Theorem 1 below. An interest of this extension is to explain why in the variational formulation of Benamou and Brenier [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF] of the Wasserstein distance, only gradient fields are needed.

Our main goal here is to transpose this decomposition to the finite Markov process setting, see Theorem 2 in Section 3, via some geometric definitions Laurent Miclo Toulouse School of Economics and Institut de Mathématiques de Toulouse, 1, Esplanade de l'université, 31080 Toulouse cedex 6, France, e-mail: miclo@math.cnrs.fr 1 inspired by Kenyon [8], interpreting 1-forms as anti-symmetric functions on the oriented edges of the underlying graph. The main difference with the continuous situation turns out to be that the two-sided flow of diffeomorphisms generated by a vector field has to be replaced by a one-sided flow on probability measures generated by some non-linear Markov generators. It will enable us to revisit the works of Maas [START_REF] Maas | Gradient flows of the entropy for finite Markov chains[END_REF] and Erbar and Maas [START_REF] Erbar | Ricci curvature of finite Markov chains via convexity of the entropy[END_REF] on optimal transport for finite Markov processes. Denote P `pV q the set of all probability measures on the finite state space V giving a positive weight to all its points. We will see furthermore that any smooth mapping P `pV q Q ρ Þ Ñ L ρ , where L ρ is an irreducible Markov generator admitting ρ as reversible measure, leads to a Riemannian structure on P `pV q which is natural in the context of finite optimal transport. But we will check that not all Riemannian structures on P `pV q are of this Markovian form, except when V has only two points.

The plan of this note is as follows. In the next section we recall in details the Helmholtz decompositions relative to "nice" probability measures on compact Riemannian manifolds, since we did not find a pedagogical exposition in the literature. This point of view is transferred to finite irreducible and reversible Markov processes in Section 3, which leads to the introduction of Markov-Riemannian metrics. The paper finishes with two appendices. One dealing, on finite state spaces, with a continuity property of the solutions to Poisson equations, which are at the heart of the Helmholtz decompositions. The second appendix computes a Metropolis-Riemann distance on the two point state space.

The compact Riemannian manifold setting

A reminder of the Helmholtz decomposition of vector fields is presented here, which will serve as a guide for the extension to the finite state space framework.

Consider M a compact and connected Riemannian manifold. Denote T M " Ů xPM T x M the corresponding tangent space, where T x M is the tangent space above x P M . Let ΣpT M q be the set of tangent vector fields on M , namely the smooth sections from M to T M .

Denote P `pM q the set of probability measures ρ on M admitting a smooth and positive density, still denoted ρ, with respect to the Riemannian probability measure . For any ρ P P `pM q, we see ΣpT M q as a tangent space above ρ and we endow it with the scalar product xx¨, ¨yy ρ given by @ b, b 1

P ΣpT M q, @@ b, b 1 DD ρ ż M @ b, b 1 D x ρpdxq
(where x¨, ¨yx in the scalar product in the tangent space T x M ). This introduces a notion of ρ-orthogonality on ΣpT M q.

Given a smooth function U : M Ñ R, the Riemannian gradient ∇U of U is an example of vector field. We denote G their set as U runs through all smooth functions. It appears that G is a linear sub-space of the space of sections ΣpT M q.

A vector field b P ΣpT M q is said to leave ρ P P `pM q invariant, if @ t ě 0, ϕ t pρq " ρ where pϕ t q tPR is the flow generated by b and where for any t ě 0, ϕ t pρq is the push-forward of ρ by ϕ t . Denote by Ipρq the set of b P ΣpT M q leaving ρ invariant.

Here is the statement of the Helmholtz ρ-decomposition of ΣpT M q:

Theorem 1 For any ρ P P `pM q, we have

ΣpT M q " G ' Ipρq
where the terms of the r.h.s. are ρ-orthogonal.

Before giving proof of this decomposition, let us recall a computational characterization of Ipρq. For any ρ P P `pM q, denote div ρ the ρ-skewed divergence defined by

@ b P ΣpT M q, div ρ pbq 1 ρ divpρbq
where divp¨q is the usual Riemannian divergence. The corresponding ρ-skewed Laplacian ρ is defined similarly @ f P C 8 pM q, ρ pf q div ρ p∇f q " pf q `x∇ lnpρq, ∇f y where is the Laplace-Beltrami operator. The second order operator ρ has several names, depending in the context in which it is used: the generalized Ornstein-Uhlenbeck diffusion generator in probability theory, the overdamped Langevin operator in analysis and the Witten Laplacian in geometry (especially concerning its extensions to differential forms).

The interest of ρ-skewed divergence is the well-known:

Lemma 1 For any fixed ρ P P `pM q, we have for any b P ΣpT M q, b P Ipρq ô div ρ pbq " 0 Proof This is a simple consequence of Stokes' theorem. Indeed, ρ P P `pM q and b P ΣpT M q being fixed, denote ρ t ϕ t pρq for any t ě 0. We have that ρ P Ipρq if and only if for any test function f P C 8 pM q, @ t ě 0, ρ t rf s " ρrf s or equivalently @ t ě 0, B t ρ t rf s " 0 It leads us to compute B t | t"0 ρ t rf s, since B t ρ t rf s can be computed similarly at any time t ě 0, replacing ρ by ρ t (which is well-known to belong to P `pM q too).

We have by definition of the flow,

B t | t"0 ρ t rf s " B t | t"0 ż f pϕ t pxqq ρpdxq " ż xb, ∇f y dρ " ż xρb, ∇f y d " ´ż divpρbqf d " ´ż div ρ pbqf dρ Since
the density ρ is positive, this expression vanishes for all f P C 8 pM q if and only div ρ pbq " 0, as desired.

We can now come to the Proof (of Theorem 1) The elements ρ P P `pM q and b P ΣpT M q being fixed, we consider the Poisson equation in U :

"

ρ pU q " div ρ pbq rU s " 0

This equation is well-known to admit a unique solution if and only if ρrdiv ρ pbqs " 0. We compute ρrdiv ρ pbqs "

ż div ρ pbq dρ " ż divpρbq d " 0
where Stokes' theorem has been invoked again. Consider U the unique solution of (1) and define β b ´∇U . We have div ρ pβq " div ρ pb ´∇U q " div ρ pbq ´ ρ pU q " 0 by construction of U . It remains to check that for any ∇U P G and any β P Ipρq are ρ-orthogonal. We compute

xx∇U, βyy ρ " ż x∇U, βy dρ " ż x∇U, ρβy d " ´ż divpρβqU d " ´ż div ρ pβqU dρ " 0 Remark 1 (a)
Endow C 8 pM q with the Fréchet structure of uniform convergence of functions and their derivatives. Corresponding semi-norms N L , for L P Z `, are constructed as follow. Let px pnq q nP N be a finite family of points from M and ą 0 be such that M is covered by the open balls Bpx pnq , q, n P N , and each of the balls Bpx pnq , 2 q is the basis of a system of coordinates px pnq k q kP m , where m is the dimension of M . For L P Z `, we define for any f P C 8 pM q, in Bpx pnq , q (and when l " 0, it is just the identity operator). This structure and Gateaux differentiability enable us to define the notion of a smooth curve r0, 1s Q t Þ Ñ f t P C 8 pM q, when the curve and all its tderivatives are continuous. A mapping U : C 8 pM q Ñ C 8 pM q is said to be regular if it transforms smooth curves into smooth curves, i.e. if r0, 1s Q t Þ Ñ Upf t q P C 8 pM q is a smooth curve when r0, 1s Q t Þ Ñ f t P C 8 pM q is a smooth curve. These notions can be extended to P `pM q (since it can be seen as a convex subset of C 8 pM q) and to ΣpT M q (e.g. by seeing it as a product of copies of C 8 pM q, via Whitney's embedding theorem).

N L pf q ÿ nP N ÿ lP 0,L max # sup xPBpx pnq , q |B k1,k2,...,k l f pxq| : k 1 , k
(b) Let us come back to the solution U of the Poisson equation (1), for given ρ P P `pM q and b P ΣpT M q. For x P M , consider pX ρ

x psqq sě0 a diffusion of generator L ρ starting from x. We have the probabilistic representation

U pxq " ´ż `8 0 Erdiv ρ pbqpX ρ x qpsqs ds (2)
where the integrant converges exponentially fast to 0 with respect to s, as the spectral gap of ρ can be bounded below uniformly in terms of the supremum norm of lnpρq (see also Appendix 3, where this property will be developed in the finite state space setting). In the sequel, a curve taking values in P `pM q or ΣpT M q will always implicitly assumed to be smooth. Girsanov formula relates explicitly the law of X ρ x r0, ss pX ρ x puqq uPr0,ss to that of the Riemannian Brownian motion X 1

x r0, ss, for any s ě 0, by stating that for any measurable bounded function F on the M -valued trajectories over the time interval r0, ss,

ErF pX ρ x r0, ssqs " E " F pX 1 x r0, ssq exp ´ż s 0 @ ∇ lnpρqpX 1 x pvqq, dX 1 x pvq D ´1 2
ż s 0 |∇ lnpρqpX 1 x pvqq| 2 dv ¯ı " E " F pX 1 x r0, ssq exp ´lnpρqpX 1 x psqq ´lnpρqpxq ´1 2 ż s 0 lnpρqpX 1 X pvqq `|∇ lnpρqpX 1 x pvqq| 2 dv
¯ı (where the scalar product and the norms are taken above X 1 X pvq). In conjunction with (2), it can be deduced that the mapping P `pM q ΣpT M q Q pρ, bq Þ Ñ U P C 8 pM q is regular in the sense described above in (a).

Let us give a classical consequence of the Helmholtz decomposition of Theorem 1 concerning the introduction of a (infinite-dimensional) Riemannianlike structure on P `pM q as in Ambrosio, Gigli and Savaré [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF].

A curve r0, 1s Q t Þ Ñ ρ t P P `pM q (recall it is assumed to be smooth according to Remark 1a) is said to be generated by a curve r0, 1s Q t Þ Ñ b t P ΣpT M q of vector fields when @ t P r0, 1s, ρ t " ϕ t pρ 0 q

where the time-inhomogenous flow pϕ t q tPr0,1s is defined through @ t P r0, 1s, @ x P M, d dt ϕ t pxq " b t pϕ t pxqq (starting with ϕ 0 being the identity mapping). When there is a curve r0, 1s Q t Þ Ñ U t P C 8 pM q of functions such that furthermore @ t P r0, 1s, b t " ∇U t we say pρ t q tPr0,1s is generated by a curve of gradient fields.

Generation by vector fields is in fact equivalent to generation by gradient fields:

Proposition 1 Assume that pρ t q tPr0,1s is generated by the vector field curve pb t q tPr0,1s . Then it is also generated by the gradient field curve p∇U t q tPr0,1s , where for any t P r0, 1s, ∇U t comes from the Helmholtz decomposition of b t above ρ t : b t " ∇U t `βt with β t P Ipρ t q. Furthermore p∇U t q tPr0,1s is the only gradient field curve generating pρ t q tPr0,1s .

Proof Taking into account Remark 1, we get r0,

1s Q t Þ Ñ ∇U t is as smooth as r0, 1s Q t Þ Ñ b t .
Denote by pr ρ t q tPr0,1s the flow generated by the gradient fields p∇U t q tPr0,1s . Consider a test function f P C 8 pM q, we have for any t P r0, 1s,

B t ρ t rf s " B t ż f pϕ t pxqq ρpdxq " ż xb t , ∇f y ϕtpxq ρpdxq " ż xb t , ∇f y x ρ t pdxq
" ´ż f div ρt pb t q dρ t " ´ż f div ρt p∇U t q dρ t (4)

" B t r ρ t rf s
By integration, it follows that pρ t q tPr0,1s coincides with pr ρ t q tPr0,1s . Assume there is another gradient field curve p∇ r U t q tPr0,1s generating pρ t q tPr0,1s . According to (4), we have for any t P r0, 1s and any test function f P C 8 pM q, ż f div ρt p∇U t q dρ t "

ż f div ρt p∇ r U t q dρ t
It follows that for any given t P r0, 1s,

ż A ∇f, ∇pU t ´r U t q E dρ t " 0
so that taking f " U t ´r U t , we get ∇pU t ´r U t q " 0. Since this is true for any t P r0, 1s, it means the two gradient field curves p∇U t q tPr0,1s and p r ∇U t q tPr0,1s coincide. ˝ Remark 2 To make evolve the boundary BA of a smooth domain A in M , one usually resorts to a T pM q-valued vector field on BA and "pushes the boundary points" according to this vector field. It is well-known, see e.g. Mantegazza [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF], that it is sufficient to consider vector fields that are normal to BA, because their tangential components leave BA invariant (and just act as an infinitesimal reparametrization). Proposition 1 is a result analogous to this orthogonal decomposition of vector fields on BA into normal and tangential components. In fact we believe that both the Helmholtz decomposition and the normal and tangential decomposition are two instances of a more general result for measures on M (where BA would be identified with the Hausdorff trace of on BA) that is outside the scope of the present note.

To go in the direction of a Riemannian-type distance on P `pM q, we have to define the length of a curve. We start by associating to ρ 0 P P `pM q and to a curve of vector fields pb t q tPr0,1s the quantity Lpρ 0 , pb t q tPr0,1s q

d ż 1 0 }b t } 2 ρt dt (5)
where pρ t q tPr0,1s is deduced from ρ 0 and pb t q tPr0,1s via (3). Next, if pρ t q tPr0,1s is a curve in P `pM q, we define Lppρ t q tPr0,1s q inf Lpρ 0 , pb t q tPr0,1s q : pb t q tPr0,1s P Bppρ t q tPr0,1s q (

where Bppρ t q tPr0,1s q stands for the set of vector field curves generating pρ t q tPr0,1s .

Lemma 2 Let pρ t q tPr0,1s be a curve in P `pM q generated by a vector field curve. Consider p∇U t q tPr0,1s the unique gradient field curve generating pρ t q tPr0,1s according to Proposition 1. We have Lppρ t q tPr0,1s q " Lpρ 0 , p∇U t q tPr0,1s q

Proof Consider any vector field curve pb t q tPr0,1s generating pρ t q tPr0,1s and decompose it as in Proposition 1. The orthogonality property of Theorem 1 then shows that

@ t P r0, 1s, }b t } 2 ρt " }∇U t } 2 ρt `}β t } 2 ρt
implying that Lpρ 0 , pb t q tPr0,1s q ě Lpρ 0 , p∇U t q tPr0,1s q

This shows that the infimum of ( 6) is attained at the unique gradient field generating pρ t q tPr0,1s . The last step in constructing a Riemannian-like distance D on P `pM q is to define, for any ρ 0 , ρ 1 P P `pM q, Dpρ 0 , ρ 1 q inftLppρ t q tPr0,1s q : pρ t q tPr0,1s P Rpρ 0 , ρ 1 qu where Rpρ 0 , ρ 1 q is the set of curves in P `pM q, starting at ρ 0 and ending at ρ 1 , and generated by vector field curves. Lemma 2 shows that in this definition, it is sufficient to consider gradient field curves, thus enabling us to recover the variational formulation of Benamou and Brenier [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF] in terms of gradient field curves. It turns out that D is a Wasserstein distance, but we will not go further in this direction.

The finite Markov process setting

We propose here a version of Theorem 1 valid for finite Markov processes (as a first step toward a full Markov process extension?).

The Riemannian structure of M can be entirely encapsulated into the Laplace-Beltrami operator . By analogy, a finite framework consists in an irreducible Markov generator L pLpx, yqq x,yPV whose invariant probability π is supposed to be reversible. Denote V the underlying finite state space. It is endowed with a graph structure, by defining its edge set as

E tpx, yq P V 2 : Lpx, yq ą 0u (7)
Note that the edges are directed and that by reversibility we have @ x, y P V, px, yq P E ô py, xq P E Thus it could be tempting to consider the corresponding set of undirected edges. It is more convenient to choose an orientation for any undirected edge. More precisely we consider a set E Ă E, such that for any px, yq P E, either px, yq P E or py, xq P E, but not both px, yq and py, xq belong to E.

A function F : E Ñ R is said to be a vector field if and only if @ px, yq P E, F px, yq " ´F py, xq

We denote by ΣpEq the set of vector fields on the graph pV, Eq. Note that it is in natural bijection with R E the set of mapping from E to R. This definition is inspired by Kenyon [START_REF] Kenyon | Spanning forests and the vector bundle Laplacian[END_REF] (see also Remark 5 below). The latter paper also writes L in a form analogous to " div ˝∇ in the Riemannian setting. More precisely, given a function f : V Ñ R (whose space is denoted R V ), define the gradient of f as the vector field given by @ px, yq P E, ∇f px, yq f pyq ´f pxq

Endow V with the probability measure π and E with the measure µ defined by @ px, yq P E, µpx, yq πpxqLpx, yq

The operator ∇ can be seen as going from L 2 pπq to L 2 pµq. By definition, the divergence div is the opposite of the adjoint operator to ∇. Then we can write again:

L " div ˝∇ (10)
Denote G the set of vector fields which are gradients.

Pursuing our analogies with the Riemannian case, let P `pV q stands for all probability measures giving a positive weight to all points of V . Consider F P ΣpEq, to make it act on elements of P `pV q, consider the Markov generator L F associated to F via

@ x ‰ y P V, L F px, yq F `px, yqLpx, yq (11)
where we adopt the convention

@ px, yq R E, F px, yq " 0 (12)
and where F `px, yq stands for the non-negative part of F px, yq (the entries of the diagonal of L F are such that the row sums all vanish).

Remark 3

The Markov processes X F generated by L F mimick, as much as possible, dynamical systems generated by vector fields. In particular edges can be crossed only in one direction. Note that if F is a gradient ∇f , then the jumps of X F (strictly) increase f and it follows that X F converges in finite but random time toward a (random) local maxima of f . Recall that x P V is a local maxima for f (with respect to the neighborhood structure given by E) when for any px, yq P E, we have f pyq ď f pxq.

The Markov generator L F enables to associate to F the flow pϕ t q tě0 acting on P `pV q via @ ν P P `pV q, @ t ě 0, ϕ t pνq ν expptL F q (13) Contrary to the Riemannian case, we cannot consider negative times if we want to stay in P `pV q. The other major difference is that there is no underlying dynamical system acting on the points of V . Instead, it must be replaced by the Markov process generated by L F (whose time-marginal distribution are the pϕ t pνqq tě0 when the initial distribution is ν). We say that a vector field F P ΣpEq leaves our fixed reversible probability π invariant if and only if @ t ě 0, ϕ t pπq " π Denote again by Ipπq the set of F P ΣpEq leaving π invariant.

Let us define a scalar product on ΣpEq, seen as a tangent space to P `pV q above π P P `pV q, to get a notion of π-orthogonality. We define the scalar product xx¨, ¨yy π on ΣpEq through @ F, F 1 P ΣpEq, @@ F, F 1 DD π "

ÿ xPV @ F, F 1 D x πpxq
where

@ x P V, @ F, F 1 P ΣpEq, @ F, F 1 D x ÿ yPV ztxu F px, yqF 1 px, yqLpx, yq
Namely, we have

@ F, F 1 P ΣpEq, @@ F, F 1 DD π " ÿ x‰yPV F px, yqF 1 px, yqπpxqLpx, yq
(this quantity can also be identified with 2 xF, F 1 y L 2 pµq , where µ is defined in ( 9)).

We have the analogue of Theorem 1, but restricted to the probability π:

Theorem 2 We have ΣpEq " G ' Ipπq
where the terms of the r.h.s. are π-orthogonal.

Let us compute the divergence, as in Kenyon [START_REF] Kenyon | Spanning forests and the vector bundle Laplacian[END_REF], who used the following explicit form to deduce [START_REF] Mantegazza | Lecture notes on mean curvature flow[END_REF].

Lemma 3

We have for any F P ΣpEq:

@ x P V, divpF qpxq " ÿ yPV F px, yqLpx, yq (14)
Proof Consider a test function f P R V . We have We deduce the analogue of Lemma 1, at least for ρ " π:

µrF ∇f s " ÿ px,yqPE F px, yqpf pyq ´f pxqqπpxqLpx, yq " 1 2 ÿ px,yqPE F px, yqpf pyq ´f pxqqπpxqLpx, yq " 1 2 ÿ x,yPV F px, yqpf pyq ´f pxqqπpxqLpx, yq " 1 2 ÿ x,yPV F px, yqf pyqπpxqLpx, yq ´1
Lemma 4 We have for any F P ΣpEq, F P Ipπq ô divpF q " 0 Proof Consider a vector field F satisfying divpF q " 0. It is equivalent to the fact that for any f P R V , we have πrf divpF qs " 0. We compute, taking into account the convention (12), the reversibility of π for L and the antisymmetry of F , Now we have at our disposal all the ingredients necessary to the Proof (of Theorem 2) The arguments are exactly the same as those of the proof of Theorem 1. The vector field F P ΣpEq being fixed, we consider the Poisson equation in U :

πrf
" LpU q " divpF q πrU s " 0 ( 16
)
This equation is well-known to admit a unique solution if and only if πrdivpF qs " 0. From (15) and writing 1 for the constant function taking the value 1, we get πrdivpF qs " ´πrL F r1ss " 0 Consider U the unique solution of ( 16) and define G b ´∇U . We have divpGq " divpF ´∇U q " divpF q ´LpU q " 0 by construction of U .

It U pxqGpx, yq πpxqLpx, yq " ´2πrU divpGqs " 0 Őf course, Theorem 2 can be extended to any ρ P P `pV q, it is sufficient to be given a corresponding irreducible generator L ρ admitting ρ as reversible probability measure and to consider all the above related notions. Let us assume that the corresponding graph pV, Eq remains the same, which amounts to assuming there is a function ρ : E Ñ p0, `8q such that @ px, yq P V 2 , x ‰ y, L ρ px, yq " ρpx, yqLpx, yq (when px, yq R E, x ‰ y, the values of ρpx, yq are irrelevant since Lpx, yq " 0, we can take for instance ρpx, yq " 0). The reversibility of ρ is equivalent to @ px, yq P E, δ ρ pxqρpx, yq " δ ρ pyqρpy, xq

where δ ρ is the density of ρ with respect to π:

@ x P V, δ ρ pxq ρpxq πpxq
Remark 4 Maas in [START_REF] Maas | Gradient flows of the entropy for finite Markov chains[END_REF] also "extends" the probability ρ into a function ρ of two variables of the state space (rather corresponding to the sides of (17)). He makes further assumptions, in particular that there exists a function θ : p0, `8q 2 Ñ p0, `8q such that ρpx, yq " θpρpxq, ρpyqq, which are not necessary for our purposes.

The definition of the gradient remains the same, but in (9) we have to consider @ px, yq P E, µ ρ px, yq ρpxqL ρ px, yq with the corresponding notion of divergence div ρ , so that L ρ " div ρ ˝∇. Definition [START_REF] Miclo | Processus de Markov inhomogènes finis[END_REF] has to be replaced by

@ x ‰ y P V, L ρ,F px, yq pF px, yqq `Lρ px, yq (18)
" pF px, yqq `ρpx, yqLpx, yq

The set Ipρq can be seen as the set of vector fields F leaving ρ invariant, with respect to the linear flow pϕ ρ,t q tě0 defined, as in (13), by @ ν P P `pV q, @ t ě 0, ϕ t pνq ν expptL ρ,F q but it seems more natural (cf. (24) below) to see Ipρq as the set of vector fields F leaving ρ invariant, with respect to the non-linear flow pψ t q tě0 described by its evolution:

@ ν P P `pV q, @ t ě 0, B t ψ t pνq " ψ t pνqL ψtpνq,F (19)
(and starting with ψ 0 pνq " ν). This non-linearity is a true discrepancy with respect to the Riemannian situation.

Remark 5 The fact that (8) does not depend on L nor ρ, suggests, as in Section 3.3 of Kenyon [START_REF] Kenyon | Spanning forests and the vector bundle Laplacian[END_REF] (see the definition of df peq there, with φ ye " φ xe " 1), that this definition corresponds more to an 1-form than to a (gradient) vector field, in analogy with differential geometry, where the differential of a function does not depend on the Riemannian structure, contrary to the gradient. Thus maybe we should have changed our terminology, replacing vector fields by 1-forms (and replace Helmholtz decompositions by Hodge decompositions, which is more traditional term for differential form fields, see for instance Section 3.44 of Aris [START_REF] Aris | Vectors, tensors and the basic equations of fluid mechanics[END_REF]). In the Riemannian setting, vector fields are important to describe corresponding dynamical systems, reason why we preferred to work with them. Nevertheless, in the finite Markov setting this link is distorted by the mapping F Þ Ñ L ρ,F , so it could be harmless to adopt the 1-form terminology.

Finally the scalar product xx¨, ¨yy ρ on ΣpEq should be defined through @ F, F 1 P ΣpEq, @@ F, F 1 DD ρ "

ÿ x‰yPV F px, yqF 1 px, yqρpxqL ρ px, yq (20)
leading to the notion of ρ-orthogonality on ΣpEq. With these definitions, Theorem 2 extends immediately into Theorem 3 For any ρ P P `pV q, we have

ΣpEq " G ' Ipρq
where the terms of the r.h.s. are ρ-orthogonal.

Another difference with the Riemannian case is the choice of the L ρ for ρ P P `pV q, role which previously were "naturally" played the ρ for ρ P P `pM q. A classical choice is to consider the Metropolis generators, which corresponds to @ ρ P P `pV q, @ px, yq P E, ρpx, yq " min ˆ1, ρpyqπpxq πpyqρpxq

˙(21)

" min ˆ1, δ ρ pyq δ ρ pxq ẇhich satisfies (17), since @ px, yq P E, δ ρ pxqρpx, yq " minpδ ρ pxq, δ ρ pyqq

(there is a relation between this construction of L ρ from L and ρ and that of ρ from and ρ, since they both correspond to the minimization of some trajectorial entropy, see [START_REF] Diaconis | On characterizations of Metropolis type algorithms in continuous time[END_REF]).

The favorite choice of Erbar and Maas [START_REF] Erbar | Ricci curvature of finite Markov chains via convexity of the entropy[END_REF], to extend the notions of Ricci lower bounds and their consequences for functional inequalities of finite Markov processes, is @ ρ P P `pV q, @ px, yq P E, δ ρ pxqρpx, yq " δ ρ pyq ´δρ pxq lnpδ ρ pyqq ´lnpδ ρ pxqq (23

)
Whatever the choice, some regularity of the mapping P `pV q Q ρ Þ Ñ L ρ P R V ˆV must be assumed for the considerations of Remark 1 to hold (for more details in this respect see the following appendix). For instance let us assume this mapping is locally Lipschitzian (seeing P `pV q as a subset of p0, 1q V ), condition which also insures the needed local existence and uniqueness of the solution to (19). Note that both ( 22) and ( 23) satisfy this condition. Proposition 1 and Lemma 2 are then also valid in the finite setting, via the same proofs. Indeed, it is sufficient to extend the notion of generation by vector or gradient field curves via: a continuous curve r0, 1s Q t Þ Ñ ρ t P P `pV q is said to be generated by a continuous curve r0, 1s Q t Þ Ñ F t P ΣpEq of vector fields when @ t P r0, 1s,

9 ρ t " ρ t L ρt,Ft (24)
We say furthermore that r0, 1s Q t Þ Ñ ρ t P P `pV q is generated by a continuous gradient field curve, when F t belongs to G for all t P r0, 1s. Then Proposition 1 and Lemma 2 extend literally to the finite Markov setting. More precisely, we are led to introduce as in ( 5), for any ρ 0 P P `pV q and any continuous curve r0, 1s Q t Þ Ñ F t P ΣpEq of vector fields, Lpρ 0 , pF t q tPr0,1s q

d ż 1 0 }F t } 2 ρt dt
where pρ t q tPr0,1s is the solution of (24). Next, as in ( 6), when pρ t q tPr0,1s is a continuous curve in P `pM q, we define Lppρ t q tPr0,1s q inf Lpρ 0 , pF t q tPr0,1s q : pF t q tPr0,1s P Bppρ t q tPr0,1s q ( where Bppρ t q tPr0,1s q stands for the set of continuous vector field curves generating pρ t q tPr0,1s . We get then that if Bppρ t q tPr0,1s q ‰ H, then Lppρ t q tPr0,1s q " Lpρ 0 , p∇U t q tPr0,1s q where p∇U t q tPr0,1s is the unique gradient field curve generating pρ t q tPr0,1s . By the usual convention, when Bppρ t q tPr0,1s q " H, we take Lppρ t q tPr0,1s q " `8. Following a traditional path, we construct a Riemannian-like distance D on P `pM q by defining, for any ρ 0 , ρ 1 P P `pM q, Dpρ 0 , ρ 1 q inftLppρ t q tPr0,1s q : pρ t q tPr0,1s P Rpρ 0 , ρ 1 qu (25

)
where Rpρ 0 , ρ 1 q is the set of continuous curves in P `pM q, starting at ρ 0 and ending at ρ 1 .

To check this distance takes finite values, we have to show that Rpρ 0 , ρ 1 q contains at least one curve generated by a continuous vector field. The following result is the main step in this direction.

Lemma 5 Let be given ρ P P `pV q and a signed measure η P HpV q with HpV q # η pηpxqq xPV P R V :

ÿ xPV ηpxq " 0 +
Then there exists a unique function U P R V with ř xPV U pxq " 0 and such that

η " ρL ρ,∇U (26)
Furthermore the mapping pρ, ηq Þ Ñ U is continuous.

Proof Equation (26) amounts to where the Markov generator r L ρ is defined via

@ x P V,
@ x ‰ y P V, r L ρ px, yq ρpyqL ρ py, xq (29)
(and the diagonal of r L ρ is such that all the row sums vanish). Note that r L ρ is a symmetric matrix so its reversible measure is the uniform distribution υ over V . Since r L ρ is irreducible, υ is also its unique invariant measure. These observations show that U is the unique solution of the Poisson equation " r L ρ pU q " η υrU s " 0 (30) The continuity of U in pρ, ηq is obtained as in the proof of Proposition 2 in Appendix 3.

The previous lemma proves that any C 1 curve pρ t q tPr0,1s in P `pV q is generated by the continuous gradient field p∇U t q tPr0,1s , where for any t P r0, 1s, ∇U t is obtained as in (26), where η is replaced by 9 ρ t and ρ by ρ t . In particular Rpρ 0 , ρ 1 q ‰ H, for any ρ 0 , ρ 1 P P `pV q.

The fact that in (26), η " 0 is equivalent to ∇U " 0 implies that D defined in (25) is a genuine Riemannian distance on P `pV q. Indeed, from the previous definitions, the scalar product above ρ P P `pV q of two tangent vectors η, η 1 P HpV q is given by

@@@ η, η 1 DDD ρ @@ ∇U, ∇U 1 DD ρ (31)
where ∇U and ∇U 1 are the respective unique solutions to (26) and to η 1 " ρL ρ,∇U 1 . The particular case where pL ρ q ρPP`pV q correspond to the Metropolis construction (starting from a given irreducible and fixed couple pπ, Lq) could be called the Metropolis-Riemann structure of P `pV q. In Appendix 3 we compute the associated distance D when V has two points and the jump rates of L are both equals to 1.

In general, let us compute more explicitly the scalar product given in (31):

Lemma 6 For any ρ P P `pV q and any η, η 1 P HpV q, we have

@@@ η, η 1 DDD ρ " ÿ x‰yPV r
A ρ px, yqpηpyq ´ηpxqqpη 1 pyq ´η1 pxqq with r A ρ p r A ρ px, yqq x,yPV the matrix given by

@ x, y P V, r A ρ px, yq 2 ż r0,`8q r P ρ,t px, yq ´1 |V | dt (32)
where @ t ě 0, r P ρ,t exppt r L ρ q (recall that the Markov generator r L ρ was defined in (29), that it is reversible with respect to the uniform distribution υ " 1 |V | and that the r.h.s. is absolutely convergent since the integrant is converging exponentially fast to zero).

Proof By definition, we have for any ρ P P `pV q, @@ ∇U, ∇U 1 DD ρ "

ÿ x‰yPV ρpxqL ρ px, yqpU pyq ´U pxqqpU 1 pyq ´U 1 pxqq " ÿ x‰yPV r L ρ px, yqpU pyq ´U pxqqpU 1 pyq ´U 1 pxqq " ÿ xPV r L ρ rpU ´U pxqqpU 1 ´U 1 pxqqspxq Note that r L ρ rpU ´U pxqqpU 1 ´U 1 pxqqs " r L ρ rU U 1 s ´U pxq r L ρ rU 1 s ´U 1 pxq r L ρ rU s so we deduce, taking into account the reversibility of υ for r L ρ xx∇U, ∇U 1 yy ρ |V | " υr r L ρ rU 2 s ´U r L ρ rU 1 s ´U 1 r L ρ rU ss " ´υrU r L ρ rU 1 ss ´υrU 1 r L ρ rU ss " ´2υrU r L ρ rU 1 ss " 2υrU η 1 s
according to (26), where η and U are replaced by η 1 and U 1 .

Recall that the solution of (30) is given by

@ x P V, U pxq " ż `8 0 Erηp r X ρ,x ptqqs dt
where r X ρ,x p r X ρ,x ptqq tě0 is a Markov process of generator r L ρ starting from x P V . Taking into account that η P HpV q, the r.h.s. can written

ż `8 0 Erηp r X ρ,x ptqqs ´υrηs dt " ż `8 0 r P ρ,t rηspxq ´1 |V | ÿ yPV ηpyq dt " ÿ yPV ηpyq ż `8 0 ˆr P ρ,t px, yq ´1 |V | ˙dt It follows that @@ ∇U, ∇U 1 DD ρ " 2 ÿ x,yPV η 1 pxqηpyq ż `8 0 ˆr P ρ,t px, yq ´1 |V | ˙dt " 2 ÿ x,yPV r A ρ px, yqη 1 pxqηpyq
Note that r A ρ p r A ρ px, yqq x,yPV is a symmetric matrix, since the same is true for r L ρ and thus for r P ρ,t for any t ě 0. Furthermore we have

@ x P V, ÿ yPV ˆr P ρ,t px, yq ´1 |V | ˙" 0
and by symmetry

@ y P V, ÿ xPV ˆr P ρ,t px, yq ´1 |V | ˙" 0 It follows that @ x P V, ÿ yPV A ρ px, yq " 0 " ÿ yPV A ρ py, xq and
the announced result follows.

Consider any scalar product, xxx¨, ¨yyy on HpV q. It can be extended into a semi-definite scalar product on R V , still written xxx¨, ¨yyy, by imposing that 1, the vector whose entries are all equal to 1, is orthogonal to HpV q and that xxx1, 1yyy " 0. Consider A pApx, yqq x,yPV the associated symmetric matrix. It is a non-negative matrix of rank |V | ´1 and all the row and column sums vanish and we can write @ η, η 1 P HpV q, @@@ η, η 1 DDD " ÿ x‰yPV Apx, yqpηpyq ´ηpxqqpη 1 pyq ´η1 pxqq

Any Riemannian structure on P `pV q is thus equivalent to the datum of a smooth mapping

P `pV q Q ρ Þ Ñ A ρ P A ( 33
)
where A is the space of all symmetric non-negative matrices of rank |V | ´1 whose row and column sums all vanish.

It is natural to wonder if all Riemannian structures on P `pV q come from a smooth family pL ρ q ρPP of irreducible Markov generators, respectively reversible with respect to the ρ. Or equivalently, if the mapping (33) is constructed, as in Lemma 6, from a family p r L ρ q ρPP of irreducible Markov generators reversible with respect to the uniform distribution υ. Let us call these metric structures Markov-Riemannian.

To investigate the existence of non-Markov-Riemannian structures, we consider a generic A P A and we wonder if we can find on V an irreducible Markov generator r L reversible with respect to υ such that A " r A p r Apx, yqq x,yPV with @ x, y P V, r Apx, yq 2

ż r0,`8q exppt r Lqpx, yq ´1 |V | dt
Consider a spectral decomposition of ´r L: p r λ n , r ϕ n q nP 0,|V |´1 , where p r ϕ n q nP 0,|V |´1 is an orthogonal basis of R V for its usual scalar product, where

@ n P 0, |V | ´1 , r Lr r ϕ n s " ´r λ n r ϕ n
and where r λ 0 " 0 and r ϕ 0 " 1.

It appears that a spectral decomposition of r

A is pλ n , r ϕ n q nP 0,|V |´1 , where

@ n P 0, |V | ´1 , λ n " 0 , if n " 0 2 r λn , otherwise
We are thus led to the following question. Consider an orthogonal basis pϕ n q nP 0,|V |´1 of R V , with ϕ 0 " 1. Let be given positive numbers pλ n q nP |V |´1 and take λ 0 " 0. The matrix of the operator A : R V Ñ R V described by @ n P 0, |V | ´1 , Arϕ n s " λ n ϕ n corresponds to a generic element of A. When is the operator r L defined by r Lrϕ 0 s " 0 and

@ n P |V | ´1 , r Lrϕ n s " ´2 λ n ϕ n (34)
a Markov generator? It amounts to check that the off-diagonal entries of r L are non-negative.

This problem is related to the determination of Markov sequences, see Definition 2.3 of Bakry and Huet [START_REF] Bakry | The hypergroup property and representation of Markov kernels[END_REF] and to the hypergroup property of the basis pϕ n q nP 0,|V |´1 .

When |V | " 2, the operator r L defined by ( 34) is always a Markov generator, since it is given by r L " ˆ´1{λ 1 1{λ 1 1{λ 1 ´1{λ 1

Ḃut

as soon as |V | ě 3, it is well-known that r L defined by (34) may not be a Markov generator.

Here is an example with V " t1, 2, 3u (which can be extended to any V with |V | ě 3). We take

ϕ 0 " ¨1 1 1 ' ϕ 1 " ¨1 ´2 1 ' ϕ 2 " ¨´1 0 1 ' Consider r
L described by (34), with λ 1 , λ 2 ą 0 to be chosen later. Introduce the function

f 2ϕ 0 `ϕ1 `3ϕ 2 " ¨0 0 6 ' If r
L was to be Markovian we would have

@ t ě 0, f t 2ϕ 0 `expp´2t{λ 1 qϕ 1 `3 expp´2t{λ 2 qϕ 2 ě 0
In particular we should have B t f t p1q| t"0 ě 0. But we compute that

B t | t"0 f t p1q " ´2 λ 1 `6 λ 2
quantity which is negative with λ 1 " 2 and λ 2 " 7.

On the finite set V , let RpV q be the set of couples pπ, Lq consisting of an irreducible Markov generator L reversible with respect to the probability measure π. Endow V with a total order ď and consider

Ý Ñ V tpx, yq P V 2 : x ă yu
The mapping

RpV q Q pπ, Lq Þ Ñ ppπpxqq xPV , pLpx, yqq px,yqP Ý Ñ V q P p0, 1q V ˆr0, `8q Ý Ñ V
is a bijection on its image and enables us to endow RpV q with a natural topology.

Let ΣpV 2 q be the set of general vector fields, which are the functions

F : V 2 Ñ R satisfying @ px, yq P V 2 , F px, yq " ´F py, xq (35)
(in particular F vanishes on the diagonal). Again, ΣpV 2 q is endowed with the topology inherited from R Ý Ñ

V . The divergence associated to L is the mapping transforming any vector field F P ΣpV 2 q into the function div L pF q defined on V via

@ x P V, div L pF qpxq " ÿ yPV F px, yqLpx, yq
For any vector field F P ΣpV 2 q and any pπ, Lq P RpV q, consider the Poisson equation in the unknown function U L,F described by " LpU L,F q " div L pF q πrU L,F s " 0

By reversibility of π for L and by the anti-symmetry property (35), we have πrdiv L pF qs " 0 so there is a unique solution U L,F to (36). Our main result here is:

Proposition 2 The mapping ΣpV 2 q ˆRpV q Q pF, π, Lq Þ Ñ U L,F P R V is continuous.
Proof Recall the probabilistic representation of U L,F :

@ x P V, U L,F pxq " ´ż `8 0 Erdiv L pF qpX x ptqqs dt (37)
where X x pX x ptqq tě0 ) stands for a left-limit and right-continuous Markov process starting from x and admitting L (resp. L) as generator.

To see the r.h.s. of (37) is well-defined, introduce the spectral gap of L: λpLq min " ´πrf Lrf ss πrf 2 s : f P R V zt0u, πrf s " 0 * which is the smallest non-zero eigenvalue of ´L (which is diagonalizable in R `by reversibility). We have ΛpLq ą 0 by irreducibility of L. For x P V and t ě 0, denote m x,t the law of X x ptq, seen as a row vector. Using matrix product, we have We deduce that for any test function f P R V , @ t ě 0, |m x,t rf s ´πrf s| " ˇˇπ "´m

@ t ě 0, m x,t " m x,
x,t π ´1¯f ıˇˇď d π " ´mx,t π ´1¯2  a πrf 2 s " d π " ´expptLq " m x,0 π ´1ı¯2  a πrf 2 s ď expp´λpLqtq d π " ´mx,0 π ´1¯2  a πrf 2 s " expp´λpLqtq g f f e π « ˆδx π ´1˙2 ff a πrf 2 s ď d πrf 2 s πpxq expp´λpLqtq
Applying this bound with f " div L pF q, we get @ x P V, @ t ě 0, |Erdiv L pF qpX x ptqqs| " |m x,t rdiv L pF qs ´πrdiv L pF qs| ď d πrdiv L pF q 2 s πpxq expp´λpLqtq which shows that the integral in (37) is absolutely converging. It is well-known, see for instance Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF], that the mapping RpV q Q pπ, Lq Þ Ñ λpLq P p0, `8q is continuous. More precisely, to come back to the symmetric setting of Chapter 2 Section 5 of Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF], note that λpLq is also the spectral gap of the symmetric matrix pLpx, yq a πpxq{πpyqq x,yPV . It follows we can apply the convergence under integral theorem to get the desired continuity from (37).

The previous result enables us to extend Proposition 1 to the finite setting, by showing that continuous curve r0, 1s Q t Þ Ñ ρ t P P `pV q generated (in the sense of (24)) by a continuous vector field curve is also generated by a continuous gradient vector field curve. Proposition 1 is even more than what we need. Indeed in Section 3, we fixed an element pπ, Lq of RpV q and only considered the other elements pr π, r

Lq P RpV q sharing with pπ, Lq the same oriented edge set E defined in [START_REF] Kato | Perturbation theory for linear operators[END_REF]. We end up with the previous set RpV q only when E coincides with V 2 minus its diagonal. Otherwise, the set of such elements pr π, r

Lq is a proper subset RpEq of RpV q. Of course by restriction, Proposition 2 is also valid if RpV q is replaced by RpEq.

The interest of RpEq is that a Girsanov formula holds between its elements, a finite setting analogue of Remark 2. More precisely, let pr π, r

Lq P RpEq and denote r X x p r X x ptqq tě0 be a left-limit and right-continuous Markov process starting from x and admitting r L as generator. Then for any T ě 0, the laws Lp r Xr0, T sq and LpXr0, T sq of r Xr0, T s and Xr0, T s are equivalent and the Radon-Nikodym derivative of Lp r Xr0, T sq with respect to LpXr0, T sq is given by dLp r Xr0, T sq dLpXr0, T sq " exp ¨ÿ px,yqPE ln ˜r Lpx, yq Lpx, yq ¸NT px, yq ´ż T 0 HpX t q dt ' where for any px, yq P E, N T px, yq is the number of jumps of Xr0, T s from x to y and @ x P V, Hpxq r Lpx, xq ´Lpx, xq

For a proof of this result, see e.g. the lecture notes [START_REF] Miclo | Processus de Markov inhomogènes finis[END_REF].

Appendix 2: On the two point state space

The goal of this appendix is to compute a Metropolis distance on the set of positive probability measures on V t0, 1u, as an illustration of the constructions of Section 3.

As reference framework we choose L ˆ´1 1 1 ´1 ṡo π " p1{2, 1{2q and the corresponding edge set is E " tp0, 1q, p1, 0qu. Any ρ P P `pV q writes p1 ´ρp1q, ρp1qq, where ρp1q P p0, 1q. From now on, to simplify notations, ρ is identified with ρp1q which is denoted ρ P p0, 1q. We consider the Metropolis choice of generators described in (21): @ ρ P p0, 1q, L ρ " ˆ´ρp0, 1q ρp0, 1q ρp1, 0q ´ρp1, 0q ẇith ρp0, 1q " 1 ^ρ 1 ´ρ ρp1, 0q " 1 ^1 ´ρ ρ

Any vector field F P ΣpEq is determined by the value F p0, 1q P R which will simply be denoted F P R in the sequel. Note that in the present situation, any vector field is a gradient field. With this convention, the Markov generator (18) is given by @ ρ P p0, 1q, @ F P R, L ρ,F "

$ ' ' ' ' & ' ' ' ' % ˆ´F ρp0, 1q F ρp0, 1q 0 0 ˙, if F ě 0 ˆ0 0 ´F ρp1, 0q F ρp1, 0q ˙, if F ă 0
We also compute that, as in (20) and with the above notations, @ ρ P p0, 1q, @ F P R, }F } Let be given ρ 0 , ρ 1 P p0, 1q, the distance Dpρ 0 , ρ 1 q defined in (25) is described by D 2 pρ 0 , ρ 1 q " 2 min "ż 1 0 rρ t ^p1 ´ρt qsF 2 t dt : pF t q tPr0,1s P Dpρ 0 , ρ 1 q * (39)

where pρ t q tPr0,1s is the solution of (38) and where Dpρ 0 , ρ 1 q is the set of stepwise continuous mappings r0, 1s Q t Þ Ñ F t P R which are such that the corresponding pρ t q tPr0,1s does start at ρ 0 and end at ρ 1 . Indeed, our previous continuity assumption can be relaxed into stepwise continuity by classical arguments of regularization by convolution.

To compute Dpρ 0 , ρ 1 q, there is no loss of generality in assuming that ρ 0 ď ρ 1 and even ρ 0 ă ρ 1 , since Dpρ 0 , ρ 0 q " 0. Furthermore, we can restrict our attention to non-negative r0, 1s Q t Þ Ñ F t P R `. Indeed, if we have F t0 ă 0 for some t 0 P r0, 1s, then, since 9 ρ t has the same sign as F t for all t P r0, 1s, we can find t 1 ă t 2 P r0, 1s with t 0 P pt 1 , t 2 q such that ρ t1 " ρ t2 . It follows that in the minimization (39), it is advantageous to replace pF t q tPr0,1s by pG t q tPr0,1s defined by Let us consider the case where ρ 0 ě 1{2. The situation where ρ 1 ď 1{2 can be treated similarly and the case where ρ 0 ă 1{2 and ρ 1 ą 1{2 is deduced by writting Dpρ 0 , ρ 1 q " Dpρ 0 , 1{2q `Dp1{2, ρ 1 q.

@
Lemma 7 For 1{2 ď ρ 0 ă ρ 1 , we have Dpρ 0 , ρ 1 q " 4p a 1 ´ρ0 ´a1 ´ρ1 q

Proof Assuming, as we are allowed to, F t ě 0 for any t P r0, 1s, (38) reduces to @ t P r0, 1s, 9 ρ t " p1 ´ρt qF t since ρ t ě 1{2 for all t P r0, 1s. We deduce that @ t P r0, 1s, ρ t " 1 ´p1 ´ρ0 q expp´φ t q where @ t P r0, 1s, φ t "

ż t 0 F s ds
It appears that pF t q tPr0,1s belongs to Dpρ 0 , ρ 1 q if and only if

φ 1 " ln ˆ1 ´ρ0 1 ´ρ1
˙(40)

  x‰yPV πpxqLpx, yqF ´px, yqf pyq where F ´px, yq ´minpF px, yq, 0q is the non-positive part of F px, yq. By definition of L F , the first sum is equal to ÿ x‰yPV πpxqL F px, yqf pyq " πrL F rf ss ´ÿ xPV πpxqL F px, xqf pxq

		ÿ					ÿ
		xPV	πpxqL F px, xqf pxq "	´ÿ xPV	πpxqf pxq	y‰x	L F px, yq
							ÿ
					"	´ÿ xPV	πpxqf pxq	y‰x	F `px, yqLpx, yq
					"	´ÿ x‰yPV	πpxqLpx, yqF `px, yqf pxq
					"	´ÿ x‰yPV	πpyqLpy, xqF `py, xqf pyq
					"	´ÿ x‰yPV	πpxqLpx, yqF `py, xqf pyq
	Since F is a vector field, we have
					@ x, y P V,	F
		divpF qs
		ÿ
	"		πpxqLpx, yqF px, yqf pxq
		x,yPV
	"	´ÿ x,yPV	πpyqLpy, xqF py, xqf pxq
	"	´ÿ x,yPV	πpxqLpx, yqF px, yqf pyq
	"	´ÿ x‰yPV	πpxqLpx, yqF px, yqf pyq
	" `ÿ We compute ´ÿ x‰yPV πpxqLpx, yqF `px, yqf pyq

`py, xq " F ´px, yq Putting together the above computations, we end up with πrf divpF qs " ´πrL F rf ss (15) and the desired equivalence follows.

  remains to check that for any ∇U P G and any G P Ipπq are πorthogonal. By definition we have xx∇U, Gyy π "

		ÿ
			pU pyq ´U pxqqGpx, yq πpxqLpx, yq
		x,yPV
	By symmetry, we have
	ÿ
	x,yPV	U pyqGpx, yq πpxqLpx, yq "	´ÿ x,yPV	U pxqGpx, yq πpxqLpx, yq
	so that
		xx∇U, Gyy π "	´2 ÿ
			x,yPV

  "p1 ´ρt qF t ρ t p0, 1q , ifF t ě 0 ρ t F t ρ t p1, 0q , if F t ă 0

		ρ " 2	ÿ
	@ t P r0, 1s,	9 ρ t "
		" rρ ^p1 ´ρqsF t	(38)

x‰yPV F 2 px, yqρpxqL ρ px, yq

" 2rρ ^p1 ´ρqsF 2 Given a continuous curve r0, 1s Q t Þ Ñ F t P R, Equation (24) writes

  t P r0, 1s, G t " 0 , if t P rt 1 , t 2 s F t , otherwisesince pG t q tPr0,1s P Dpρ 0 , ρ 1 q and

	ż 1	ż 1
	rρ t ^p1 ´ρt qsF 2 t dt ą	rρ t ^p1 ´ρt qsG 2 t dt
	0	0

Acknowledgements I would particularly like to thank Jérôme Bolte and Stéphane Villeneuve for the discussions we had about this paper, and Thăng Lê for finding mistakes in a first version. Fundings from the grants ANR-17-EURE-0010 and AFOSR-22IOE016 are acknowledged.

Appendix 1: Continuity of the solution to the Poisson equation

The purpose of this appendix is to show that the solution U of (16) is continuous in terms of F, L and π.

Introduce the mapping h defined on R `via @ u P R `, hpuq ż u 0 a p1 ´ρ0 qe ´q dq so that

The optimization problem (39) amounts to minimize twice the above r.h.s. under the condition

which is equivalent to (40). Writing for any t P r0, 1s, ϕ t h 1 pφ t q 9 φ t , we are led to the simple problem of minimizing ş 1 0 ϕ 2 t dt under the contraint ş 1 0 ϕ t dt " A. It is well-known that the minimizer r0, 1s Q t Þ Ñ ϕ t is constant and so we end up with A 2 for the minimal value.

Thus we have shown that

˙" ¨ż ln ´1´ρ 0 1´ρ 1 0 a p1 ´ρ0 qe ´q dq '2 " p1 ´ρ0 q ¨ż ln ´1´ρ 0 1´ρ 1 0 e ´q{2 dq '2 " 8p1 ´ρ0 q ˆ1 ´c 1 ´ρ1 1 ´ρ0

˙2

" 8p a 1 ´ρ0 ´a1 ´ρ1 q 2 which is the desired result.


Aishwarya Venkataramanan
email: aishwarya.venkataramanan@univ-lorraine.fr

Cédric Pradalier

Integrating Visual and Semantic Similarity Using Hierarchies for Image Retrieval

Keywords: Similarity Search, Visual Hierarchy, Content-based image retrieval

Most of the research in content-based image retrieval (CBIR) focus on developing robust feature representations that can effectively retrieve instances from a database of images that are visually similar to a query. However, the retrieved images sometimes contain results that are not semantically related to the query. To address this, we propose a method for CBIR that captures both visual and semantic similarity using a visual hierarchy. The hierarchy is constructed by merging classes with overlapping features in the latent space of a deep neural network trained for classification, assuming that overlapping classes share high visual and semantic similarities. Finally, the constructed hierarchy is integrated into the distance calculation metric for similarity search. Experiments on standard datasets: CUB-200-2011 and CIFAR100, and a real-life use case using diatom microscopy images show that our method achieves superior performance compared to the existing methods on image retrieval.

Introduction

In recent years, the exponential growth of image data has made effective management of such data increasingly important. One way to manage this data is through Content-based Image Retrieval (CBIR), where a query image is used to retrieve images from the database that are visually similar to it [START_REF] Chen | Deep learning for instance retrieval: A survey[END_REF]. The comparison is performed on the features extracted from the images, represented by a set of numerical descriptors or vectors. The search is performed on these descriptors using a similarity measure such as the Euclidean or cosine distance. The challenge is to extract robust feature representations, such that objects belonging to the same category are close by, despite the changes in view-point, background and illumination. Recent methods employing Convolutional Neural Network (CNN) for feature extraction have proven to be robust and effective over the classical hand-crafted feature extractors [START_REF] Wan | Deep learning for content-based image retrieval: A comprehensive study[END_REF]. A commonly adopted approach is to extract feature vectors from the latent space of a trained CNN Fig. 1. Examples of top-5 images retrieved using cosine distance and our method on CIFAR100. Our method incorporates hierarchy while ranking image similarity, with retrieves images that are more visually and semantically relevant.

classifier. Deep classifiers use cross-entropy loss that learns feature representations to group together objects belonging to the same category, while separating them from the rest. Thus, given a query image, the retrieved data from the local neighbourhood in the latent space should ideally consist of images belonging to the same classification category.

Despite the ability of the CNNs to learn discriminative representations, it has been observed that sometimes, images unrelated to the query are retrieved [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF][START_REF] Park | A ranking algorithm using dynamic clustering for content-based image retrieval[END_REF]. These images are typically visually similar to the query, but semantically very different. Fig. 1 illustrates this for the images retrieved using cosine distance on CIFAR100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. While the retrieved images have certain visual similarities with the query, some of them are semantically very different. One way to address this is by incorporating domain knowledge while training, so that the network learns feature representations that are both visually and semantically relevant [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF]. The semantic data is obtained from expert domain knowledge or lexical databases such as WordNet [START_REF] Miller | Wordnet: a lexical database for english[END_REF]. However, incorporating external domain knowledge during training can be challenging since it would require significant changes in the training procedure, and can be detrimental when there are inconsistencies between visual and semantic similarities [START_REF] Brust | Not just a matter of semantics: The relationship between visual and semantic similarity[END_REF]. Also, it assumes that the specialized domain knowledge is available, which can be hard to obtain or irrelevant in certain situations, such as, querying the images of people or vehicles. In this paper, we demonstrate that the visual hierarchies obtained from the representations learned by the classification network are rich in visual and semantic information, achieving superior retrieval performance. Contrary to other approaches, our method does not require specialized domain knowledge or modifications to the training procedure, making it easy to integrate into an off-the-shelf classification network.

Our method relies on two key aspects for the retrieval: construction of a hierarchy that is visually and semantically meaningful, and integration of hierarchy into the distance calculation metric for similarity search. Our method for hierarchy construction involves identifying the classes in the latent space whose feature representations overlap with one another. This is based on the assumption that the overlapping classes are those that are very similar to each other, both visually and semantically. The clusters of overlapping classes are then grouped together into the same class at a higher hierarchy level. This process is repeated at different levels of the hierarchy until there are no overlaps in the feature representations, resulting in a hierarchical structure that captures both visual and semantic similarities between images. Once the hierarchy is obtained, we combine the cosine distance and a hierarchy-based distance to obtain a metric for ranking the images for retrieval. Experimental results show that our method improves image retrieval performance on two standard benchmarks: CIFAR100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] and CUB-200-2011 [START_REF] Wah | The caltech-ucsd birds-200-2011 dataset[END_REF], and a real life diatom image retrieval use case. Additionally, we demonstrate the robustness of our method by evaluating on images subject to perturbations that mimic practical difficulties encountered during acquisition.

Our contributions are as follows: (i) we develop a method to construct a hierarchy that is visually and semantically relevant and show that incorporating this hierarchy in the distance metric enhances the image retrieval performance, (ii) we design the method such that it does not require any additional domain knowledge and introduces no modifications to the model architecture or training procedure. The code is available in https://github.com/vaishwarya96/ Hierarchy-image-retrieval.git

Related Works

The role of visual attributes and semantics is crucial in describing images for identification or retrieval. Hierarchies are an intuitive way to express this, and several methods have leveraged it for image categorization and retrieval [START_REF] Silla | A survey of hierarchical classification across different application domains[END_REF]. This is done by integrating hierarchy obtained from the biological taxonomy or from lexical databases to learn feature representations that capture semantic similarity between the categories. [START_REF] Brust | Integrating domain knowledge: using hierarchies to improve deep classifiers[END_REF] uses WordNet to construct hierarchies and incorporates the domain knowledge obtained from it into a CNN classifier to improve the network's performance. [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF] extends the traditional cross-entropy loss used in flat classification to work on hierarchies. To do so, they compute the conditional probability of the nodes at each layer of the hierarchy and calculate the cross-entropy loss at each layer. DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] transfers semantic knowledge from text to visual object recognition. It involves pre-training a language model to learn word embeddings and re-training a deep visual model using these embeddings for image label prediction. [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF] uses an embedding algorithm that maps images onto a hypersphere. The distances on this hypersphere represent similarities that are derived from the lowest common ancestor height in a given hierarchy tree. One limitation of these methods is their reliance on an exist-ing domain knowledge, which can be difficult to obtain or irrelevant in some contexts.

A line of work relies on the visual features learnt by the network. [START_REF] Yang | Retrieving and classifying affective images via deep metric learning[END_REF] uses metric loss while training the network for image retrieval. Metric loss tries to pull closer the representations of images belonging to the same class, while pushing apart the inter-class representations. While this reinforces the condition that the intra-class representations are clustered tightly, it does not impact the semantic relationship between the instances. [START_REF] Zhang | Image-level classification by hierarchical structure learning with visual and semantic similarities[END_REF] constructs a visual hierarchy from the distance values between the image features and uses it to improve classification. Our method is similar to [START_REF] Park | A ranking algorithm using dynamic clustering for content-based image retrieval[END_REF], which uses Ward clustering for constructing the hierarchy and uses it for re-ranking the images retrieved using a distance metric. A drawback of using Ward clustering is that they are sensitive to outliers and can sometimes group classes with low semantic similarities. In contrast, our approach groups classes only when their features overlap, which results in a more meaningful and semantically relevant hierarchy.

Method

This section explains the two main steps of our method: the construction of the hierarchy from the learned feature representations, and the calculation of the distance metric for ranking the images.

Notations: Consider a CNN classifier trained on K classes, each class represented as {C 1 , ..., C K }. The feature vectors are extracted from the penultimate layer of the CNN. We apply dimensionality reduction on these feature vectors using PCA to reduce the redundant features and speed up the computation [START_REF] Wu | Dimensionality reduction for image retrieval[END_REF]. We automatically determine the number of principal components required to explain 95% of the original data variance by examining the eigenvalues in decreasing order. Let the feature vectors corresponding to the training images be x train . Assume that the class features follow multivariate Gaussian distributions with mean µ c and covariance Σ c , where c = {1, .., K}. Note that this assumption can be broken when the intra-class variance is large. In such cases, one can constrain the latent space to be Gaussian using the methodology in [START_REF] Venkataramanan | Self-supervised gaussian regularization of deep classifiers for mahalanobis-distance-based uncertainty estimation[END_REF]. Given a query image q, its feature vector is denoted as x q . Suppose the database consists of N images, their corresponding feature vectors are denoted as {x 1 , .., x N }.

Hierarchy Construction

The hierarchy construction involves identifying the classes with overlapping distributions of feature vectors in x train . Inter-class overlapping typically occurs when there are visual and semantic similarities in the class images. To construct the hierarchy, a bottom-up approach is used, where the overlapping classes in the latent space are merged in the higher hierarchy level. The overlapping classes are identified using the Bhattacharya Coefficient (BC), which is a statistical metric to measure the amount of overlap between two statistical populations.

The Bhattacharya distance between two classes C i and C j is given by:

D B (C i , C j ) = 1 8 (µ i -µ j ) T Σ -1 (µ i -µ j ) + 1 2 ln det Σ det Σ i det Σ j (1)
where

Σ = Σi+Σj 2
The BC is calculated as:

BC(C i , C j ) = e -D B (Ci,Cj ) (2)
The value of BC ranges from 0 to 1 inclusive, where a value of 0 signifies no overlap and 1 means full overlap. If the value of the BC between two classes is greater than or equal to a specified threshold t, then they are considered to be overlapping. Every cluster of classes that overlap with each other are identified. All classes within each cluster are merged into a unified class at a higher level of the hierarchy. On the other hand, if the BC is less than t, then these classes are considered to be distinct and are kept separate in the higher hierarchy level. The t value is a hyperparameter that depends on the dataset. The process of BC estimation is repeated at each level on the new set of classes, and the overlapping classes are merged to create a hierarchy structure. This is performed until there are no overlaps between the classes, or all the classes have been merged. At the higher hierarchy levels, some classes include feature vectors from multiple small classes that were merged. At every level, the µ and Σ values are recalculated from the feature vectors of the merged classes.

Distance Calculation

After constructing the hierarchy, we use it to perform the similarity search for image retrieval. The distances are calculated between the query feature x q and the image features x i in the database. Our final distance metric for ranking the images is a weighted sum of two distance values: cosine distance and a hierarchy based distance.

Cosine distance:

The cosine distance is a similarity measure between two feature vectors and is popularly used in image retrieval tasks. The cosine distance between x q and x i is given by:

D C (x q , x i ) = 1 - x q • x i ∥x q ∥∥x i ∥ ( 3
)
where • is the dot product and ∥ ∥ is the L2 norm.

Hierarchical distance:

The hierarchy is a tree structure H = (V, E) where V represents the nodes and E, the edges. H consists of K leaf nodes. We want to identify the position of any database image in the hierarchy, which means finding the leaf closest to each image. To identify the leaf nodes, we calculate the Mahalanobis Distance (MD) between the feature vectors x i and the feature vectors x train corresponding to each leaf node. The MD between x i and node c, is given as:

M D c (x i ) = (x i -µ c ) T Σ c -1 (x i -µ c ) (4)
x i is assigned to the leaf node n i corresponding to the smallest MD value.

Similarly, the leaf node n q corresponding to x q is identified. The distance calculation uses the height of the lowest common ancestor (LCA), where the LCA of two nodes is the parent node that is an ancestor of both nodes and does not have any child that is also an ancestor of both nodes. The hierarchical distance [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF] between x q x i is calculated as:

D H (x q , x i ) = height(LCA(n i , n q )) height(H) (5)
The final distance is given as:

D = D C (x q , x i ) + αD H (x q , x i ) ( 6
)
where α is a hyperparameter. The images are sorted in ascending order of D to rank the images based on similarity and retrieve them.

Robustness Analysis

Robustness analysis demonstrates the ability of the image retrieval methods to maintain accurate and reliable performance even when the images are distorted or corrupted. The distortions introduced during the robustness analysis mimic real-world scenarios where images can be affected by factors like blurring, occlusions, and changes in lighting conditions.

We consider the following scenarios: (i) Blurring -The images are subject to Gaussian blurring, simulating the effect of an out-of-focus camera or a lowquality image (ii) Saturation change -The saturation values of the original images are modified to simulate variations in colour intensity. (iii) Occlusions -Random occlusions are introduced in different parts of the images, simulating objects or regions being partially or completely blocked.

Experiments

Baselines

We compare our method against the following approaches: (i) Image retrieval using Euclidean Distance, (ii) Image retrieval using Cosine Distance, (iii) Our method using hierarchical distance only, (iv) Semantic Embeddings [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF], (v) Our method applied on the features extracted from semantic embeddings in (iv), (vi) Distance calculation using the method in Sec. 3.2 on hierarchy constructed using Ward clustering [START_REF] Park | A ranking algorithm using dynamic clustering for content-based image retrieval[END_REF], (vii) Distance calculation on semantic hierarchy obtained using domain knowledge.

Experimental Setup

We perform our experiments on three datasets with fine-grained visual features:

CUB-200-2011 [START_REF] Wah | The caltech-ucsd birds-200-2011 dataset[END_REF]. This is a widely used dataset for image retrieval, consisting of 200 classes of birds and a total of 11,788 images. The train data consists of 5,994 images and the test data consists of 5,794 images. The images were resized to 224 × 224, and random horizontal flipping and rotation (±30 • ) were used for data augmentation. Following [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF], we used ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF] for training with an SGD optimizer. The network was trained for 100 epochs with a batch size of 12. The values of t = 0.30 and α = 3 are obtained using hyperparameter search.

CIFAR100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. It consists of 60,000 images, of size 32 × 32 belonging to 100 classes of wide range of objects. The training dataset consists of 50,000 images and the remaining are test data. Following [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF], we used ResNet-110 [START_REF] He | Deep residual learning for image recognition[END_REF] for training with an SGD optimizer. The network was trained for 400 epochs with a batch size of 24. The values of t = 0.20 and α = 5 are obtained using hyperparameter search.

Diatom Dataset. The diatom dataset consists of 9895 individual diatom images belonging to 166 classes [START_REF] Venkataramanan | Usefulness of synthetic datasets for diatom automatic detection using a deep-learning approach[END_REF]. The network was trained using Efficient-Net [START_REF] Tan | Efficientnet: Rethinking model scaling for convolutional neural networks[END_REF]. An SGD optimizer with a learning rate of 0.0002 was used. The images were resized to 256 × 256. This dataset contains high intra-class variance with distinct feature representations per class, and hence the latent space is not Gaussian, making it incompatible for BC calculation. The network was trained using the method in [START_REF] Venkataramanan | Self-supervised gaussian regularization of deep classifiers for mahalanobis-distance-based uncertainty estimation[END_REF], which ensures a Gaussian latent space. The training was performed for 100 epochs with a batch size of 12. The values of t = 0.25 and α = 3 are obtained using hyperparameter search.

For the CIFAR100 and CUB-200-2011, 80% of the train data was used for training and 20% for validation. For the diatoms, 70% of the dataset was used for training, 10% for validation and 20% for testing. The networks were trained on GeForce RTX 3090 with 24 Gb of RAM. The semantic hierarchy of CUB-200-2011 and CIFAR100 for the baseline experiments were obtained from [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF]. The diatom hierarchy was obtained from the taxonomy of the species, consisting of two levels: 'genus' and 'species'. The same model architectures were used for comparing all the baselines.

Setup for Robustness Analysis. The analysis is performed on the CUB-200-2011 dataset with the following settings. (i) A Gaussian Blur with kernel size [START_REF] Silla | A survey of hierarchical classification across different application domains[END_REF][START_REF] Silla | A survey of hierarchical classification across different application domains[END_REF] was applied to all the images, (ii) The saturation of images were adjusted with randomly chosen values in the range [0,1,0.5,1,2,5] and (iii) A random crop of dimension (70,70) was replaced by black patch in every image. The networks were retrained on the modified images, with the same training parameters.

Evaluation Metrics

We use MAP@k values for evaluating the image retrieval performance. MAP@k measures the average precision (AP) of the top k retrieved images, and then takes the mean across all queries. A higher value indicates better performance.

Results

Retrieval Results

Fig. 2 shows the plots of mAP@k values for different values of k for the three datasets. The results suggest that combining the visual and semantic cues improves the retrieval. Overall, our method achieves the best performance over the other baselines.

Effect of distances:

The cosine distance retrieves images that are visually similar to the query, whereas, the hierarchical distance takes into account the semantic relationships between classes. Both these aspects are important when retrieving the images, and removing one of them deteriorates the performance. There is a significant drop in the MAP values in CIFAR100 when using only the hierarchical distance. One explanation for this is that CIFAR100 contains classes that are visually diverse, and hence, visual cues are more important over the semantic ones here. This could also explain the marginal improvement in the metrics when combining both the visual and semantic attributes than when using only the visual information for CIFAR100 over the other datasets.

Effect of hierarchy: In the hierarchy-based methods, our approach outperforms the baselines. This is closely followed by our approach of distance calculation using semantic hierarchy. While semantic embeddings [START_REF] Barz | Hierarchy-based image embeddings for semantic image retrieval[END_REF] also relies on visual and semantic features for the retrieval, its drop in performance could be attributed to the errors introduced due to disagreements between the visual and semantic relationships while training [START_REF] Brust | Not just a matter of semantics: The relationship between visual and semantic similarity[END_REF]. Our approach of hierarchy construction merges the classes only when there is a correlation between visual and semantic attributes, and thus it is robust against these errors. The Ward hierarchy approach may sometimes group classes that are semantically unrelated, resulting in less effective performance. Qualitative Analysis: Fig. 1 shows some images retrieved using cosine distance and our method. While cosine distance retrieves images that are visually similar, some of them are semantically very different. Whereas, our method retrieves images that are visually and semantically meaningful.

Robustness Analysis: Fig. 3 shows the MAP@K plots for the robustness analysis on CUB-200-2011. Compared to the plot in Fig. 2, one can observe a drop in the metrics. However, our method still achieves the best performance over the other baselines, indicating its robustness even in challenging conditions.

Discussion and Conclusion

In this paper, we presented a method to tackle the problem of semantically dissimilar images retrieved in CBIR. By leveraging the learned feature representations from a CNN classifier, we construct a meaningful hierarchy that captures both visual and semantic information. This hierarchy is then integrated into the distance metric for similarity search, resulting in superior image retrieval performance, as demonstrated on fine-grained visual datasets. One limitation of our method arises when classes exhibit high variance in their features, resulting in potential overlaps with several or all other classes. Consequently, the network may incorrectly group them all into a single class when constructing the hierarchy. To mitigate this, we only group classes when the degree of overlap is above a threshold. One could also train the network using a metric loss to alleviate this, which we leave for future work.

Fig. 2 .

 2 Fig. 2. Retrieval performance on CUB-200-2011, CIFAR100 and Diatom dataset. Our method achieves state-of-the-art performance. The results suggest that incorporating hierarchy into content-based image retrieval is effective in improving the retrieval performance.

Fig. 3 .

 3 Fig. 3. Robustness analysis on the CUB-200-2011 dataset. Blurring, saturation change and occlusion were applied to test the impact on the retrieval. Results show that our method still achieves the best performance.


W Oukil

Non trivial zeros of the Zeta function using the differential equations

Keywords: Zeta function, Non trivial zeros, Riemann functional equation, Bernoulli numbers, Differential equations. AMS subject classifications: 00A05 1 Main results

Using the differential equations, we obtain a more flexible representation of the Zeta function in the critical strip. Thanks to the Riemann functional equation, we obtain an approach to prove that the Zeta function does not vanish at any point s of the critical strip such that 1 -2ℜ(s) = 0.

Theorem 1. Consider the Zeta function given by the Equation [START_REF] Titchmarsh | The Theory of the Riemann Zeta-Function[END_REF]. For every real sequence (t n ) n ⊂ R + and for every s ∈ B we have

lim n→+∞ t n n (n -1)! V s [q(n, u)](t n ) + ζ(s) s + 1 1 -s V s [u n ](t n ) = 0,
where q(n, u) := -

n+1 k=1 B k k! n!u n-k+2 (n + 2 -k)! ∀n ≥ 1, ∀u ∈ R,
and where for every n ≥ 1 and every real polynomial u ∈ R → g n (u) of order less then n + 1, the complex function t ∈ R + → V s [g n (u)](t), is defined as

V s [g n (u)](t) := +∞ t (t + 1) 1+s (u + 1) n+2+s - t 1+s u n+2+s g n (u)du -t 1+s +∞ t (u + 1) n+s u n+2+s +∞ u g n-1 (v) (v + 1) n+1+s dvdu.
For the particular sequence (t n = n) n∈N * , the representation of the Zeta function given by the previous Theorem, can be written as

lim n→+∞ n n+1 η s (n) n! ζ(s) s + 1 1 -s + ηs (n) = 0,
where η s (n) := n j=1 n j (-1) j (n + 1) -j j + 1 + s

- n-1 j=0 n -1 j (-1) j j + 1 + s n-1-j k=0 n -1 -j k n -j-k j + k + 2 + s , ηs (n) = - n+1 k=1 B k k! n (n + 2 -k)! n-k+2 j=0 n -k + 2 j (-1) j (n + 1) 2-k-j k + j -1 + s + n+1 k=1 B k k! n 2-k (n + 2 -k)!(k -1 + s) + n k=1 B k k! 1 (n + 1 -k)! n-k+1 j=0 w n,k,j k + j + 1 + s n-k-j m=0 v n,k,j,m k + j + m + 1 + s , w n,k,j := n -k + 1 j (-1) j k + j + 1 , v n,k,j,m := n -k -j m n -k-j-m .
In the conclusion of this manuscript, we present the approach to prove that the Zeta function does not vanish at any point s ∈ B such that 1-2ℜ(s) = 0.

Main Proposition

For every s ∈ B, the Equation ( 1) is equivalent to,

ζ(s) s = - 1 1-s - +∞ 1 u -1-s {u}du.
The aim is to studies the differential equation of solutions the functions

t → ψ s (z, t) := t s z + t 1 u -1-s {u}du , z ∈ C, t ≥ 1.
We focus only on the bounded solutions (there is a unique bounded solution.

All other solutions are oscillating and diverge to infinity in norm). More precisely, the strategy to prove the Theorem 1, is to find the initial condition

z s ∈ C such that sup t≥1 |ψ s (z s , t)| < +∞. Since sup t≥1 |ψ s (z, t)| < +∞ if and only if z = - +∞ 1 u -1-s {u}du, we obtain z s = - +∞ 1
u -1-s {u}du. For every s ∈ B we consider the following differential equation

d dt x = st -1 x + t -1 {t}, (2)
t ∈ [1, +∞)/N, x(1) = z, z ∈ C, x : [1, +∞) → C.
Lemma 2. Let be s ∈ B. For every z ∈ C there exists a unique continuous solution ψ s (z, t) : [1, +∞) → C of the differential equation (2). Further,

ψ s (z, t) = t s z + t 1 u -1-s {u}du , ∀t ≥ 1.
Proof. Let be s ∈ B fixed. The function

t → t s t 1 u -1-s {u}du, t ≥ 1, is C ∞ on [1, +∞)/N
ψ s (z, t) = t s z + t 1 u -1-s {u}du , ∀t ≥ 1.
Proposition 3. Let be s ∈ B and consider the continuous solution ψ s (z, t) : [1, +∞) → C of the differential equation (2). Then sup t≥1 |ψ s (z, t)| < +∞ if and only if for every real sequence (t n ) n ⊂ R + the initial condition z ∈ C satisfies the following equation

lim n→+∞ t n n (n -1)! V s [q(n, u)](t n ) + zV s [u n ](t n ) = 0,
where q(n, u) and V s are defined as in the Theorem 1.

Let us introduce the following notations,

p 2k+1 (t) := (-1) k j∈Z * 1 (j2π) 2k+3 sin(j2πt), p 2k (t) := (-1) k j∈Z * 1 (j2π) 2k+2 cos(j2πt) -1 , Lemma 6. We have t 0 {u} - 1 2 du = p0 (t) := j∈Z * 1 (j2π) 2 exp(ij2πt) -1 , ∀t ≥ 0,
and for every n ∈ N * we have

Φ n [p 0 ](t) = - 1 (n + 2)! n+1 k=2 B k n + 2 k t n-k+2 + p n (t), ∀t ≥ 0, ∀n ≥ 1.
Proof. Prove that

t 0 {u} - 1 2 du = p0 (t), ∀t ≥ 0,
The function u → {u} is 1-periodic, then there exists a continuous 1-periodic function p : R → R such that

t 0 {u}du = t 1 0 {u}du + p(t), ∀t ≥ 0. Since 1 0 {u}du = 1 0 u du = 1 2 , we get t 0 {u}du = t 1 2 + p(t), ∀t ≥ 0.
The function p is a piecewise C ∞ , continuous on R and 1-periodic. By Dirichlet Theorem, the Fourier series

n → n k=-n a k exp(ik2πt),
converge uniformly on R + to the function t → p(t), where (a k ) k ⊂ C are the Fourier coefficients of the function p.

t 0 {v} - 1 2 dv = j∈Z a j exp(ij2πt), ∀t ≥ 0.
By definition of the Fourier coefficients we have

a j = 1 0 exp(-ij2πu) u 0 {v} - 1 2 dv du = 1 2 1 0 exp(-ij2πu)u(u -1)du = 1 (2jπ) 2 , ∀j ∈ Z * ,
and

a 0 = 1 2 1 0 u(u -1)du = - 1 12 = - j∈Z * 1 (j2π) 2 .
The function p satisfies

p(t) = j∈Z * 1 (j2π) 2 exp(ij2πt) -1 = p0 (t), ∀t ≥ 0.

Integrate successively to obtain

Φ n [p 0 ](t) = n k=0 c k (n -k)! t n-k + p n (t), ∀t ≥ 0, ∀n ≥ 1,
where (c k ) k∈N defined as

c 2k+1 = 0 and c 2k = (-1) k j∈Z * 1 (j2π) 2(k+1) , ∀k ≥ 0.
By definition of the Bernoulli numbers (B k ) k∈N , we get

Φ n [p 0 ](t) = - 1 (n + 2)! n+1 k=2 B k n + 2 k t n-k+2 + p n (t), ∀t ≥ 0,
z s = - +∞ 1 u -1-s {u}du.
Proof. Let be s ∈ B fixed. There is a unique bounded solution. All other solutions are oscillating and diverge to infinity in norm. Let be z ∈ C and suppose that sup

t≥1 |ψ s (z, t)| < +∞.
Since ℜ(s) ∈ (0, 1), by the Lemma 2, we have

sup t≥1 |ψ s (z, t)| < +∞ =⇒ z + +∞ 1 u -1-s {u}du = 0, Then z s = - +∞ 1 u -1-s {u}du.
Prove that sup t≥1 |ψ s (z s , t)| < +∞. Since

z s + +∞ 1 u -1-s {u}du = 0,
by consequence,

z s + t 1 u -1-s {u}du = - +∞ t u -1-s {u}du, ∀t > 1.
By the Lemma 2, we get

ψ s (z s , t) = -t s +∞ t u -1-s {u}du, ∀t ≥ 1.
Then

sup t≥1 |ψ s (z s , t)| ≤ 1 ℜ(s) .
Proof of the Proposition 3. Let be s ∈ B and consider the bounded continuous solution ψ s (z s , t) : [1, +∞) → C defined in the Lemma 7. In order to simplify the notation, denote ψs (t) := ψ s (z s , t + 1), ∀t ≥ 0.

Using the fact t → ψ s (z s , t) is solution of the differential equation (2). By definition of Φ in the Notation 4, we get

(t + 1) d dt Φ n+1 [ ψs ](t) = (n + 1 + s)Φ n+1 [ ψs ](t) (3)
+ h s (n, t), ∀t ≥ 0, ∀n ≥ 1, Φ n+1 [ ψs ](0) = 0, ∀n ≥ 1, h s (n, t) := 1 2 t n+1 (n + 1)! + z s t n n! + Φ n [p 0 ](t), ∀t ≥ 0, ∀n ≥ 1,
where the function p0 is defined by the Lemma 6. The previous differential equation can be written as

d dt Φ n+1 [ ψs ](t) = (n + 1 + s)t -1 Φ n+1 [ ψs ](t) (4)
+ t -1 h s (n, t) - d dt Φ n+1 [ ψs ](t) , ∀t > 0, ∀n ≥ 1.
Integrate the differential equation (3), for every n ≥ 1 and t ≥ 0 we have

Φ n+1 [ ψs ](t) = (t + 1) n+1+s t 0 (u + 1) -n-2-s h s (n, u)du. ( 5
)
Integrate the differential equation (4), for every n ≥ 1 and t ≥ 1 we get

Φ n+1 [ ψs ](t) = t n+1+s Φ n+1 [ ψs ](1) (6)
+ t n+1+s t 1 u -n-2-s h s (n, u) - d dt Φ n+1 [ ψs ](u) du. Since d dt Φ n+1 [ ψs ](t) = Φ n [ ψs ](t), ∀n ≥ 1, ∀t > 0,
the Equation ( 6) is equivalent to

Φ n+1 [ ψs ](t) = t n+1+s Φ n+1 [ ψs ](1) (7)
+ t n+1+s t 1 u -n-2-s h s (n, u) -Φ n [ ψs ](u) du.
Since sup t≥0 | ψs (t)| = sup t≥1 |ψ s (z s , t)| < +∞, then for every fixed n ≥ 1 we have

Φ n+1 [ ψs ](t) ≤ Φ n+1 [1](t) sup t≥0 | ψs (t)| = t n+1 (n + 1)! sup t≥0 | ψs (t)|, ∀t ≥ 0, (8)
Since ℜ(s) ∈ (0, 1) the Equations ( 5) and (7) implies

+∞ 0 (u + 1) -n-2-s h s (n, u)du = 0, ∀n ≥ 1 Φ n+1 [ ψs ](1) + +∞ 1 u -n-2-s h s (n, u) -Φ n [ ψs ](u) du = 0, ∀n ≥ 1,
by consequence, for every n ≥ 1 and t ≥ 1, the function t → Φ n+1 [ ψs ](t) given by the Equations ( 5) and ( 7) can be written as

Φ n+1 [ ψs ](t) = -(t + 1) n+1+s +∞ t (u + 1) -n-2-s h s (n, u)du, (9)
Φ n+1 [ ψs ](t) = -t n+1+s +∞ t u -n-2-s h s (n, u) -Φ n [ ψs ](u) du. Implies ( t + 1 t ) 1+s +∞ t h s (n, u) (u + 1) n+2+s du = +∞ t h s (n, u) -Φ n [ ψs ](u) u n+2+s du.
Use the Equation (9), to obtain

( t + 1 t ) 1+s +∞ t h s (n, u) (u + 1) n+2+s du = +∞ t h s (n, u) u n+2+s du (10)
+ +∞ t (u + 1) n+s u n+2+s +∞ u h s (n -1, v) (v + 1) n+1+s dvdu.
By the notation of h s and by the Lemma 6, we have

h s (n, u) = p n (u) + 1 n! z s u n + q(n, u) . ( 11
)
where in order to simplify the notation, we denoted q(n, u) the following polynomial

q(n, u) := 1 2 n!u n+1 (n + 1)! - n! (n + 2)! n+1 k=2 B k n + 2 k u n-k+2 .
or since B 1 = -1 2 , the function q s can be written as

q(n, u) := - n! (n + 2)! n+1 k=1 B k n + 2 k u n-k+2 = - n+1 k=1 B k k! n!u n-k+2 (n + 2 -k)!
.

By the notation of the function h s and the Equation (11), for every real sequence (t n ) n ⊂ R + the equation ( 10) can be written as

(t n + 1) 1+s n! +∞ tn z s u n + q(n, u) (u + 1) n+2+s du = t 1+s n n! +∞ tn z s u n + q(n, u) u n+2+s du (12) + ǫ s (n) + t n 1+s (n -1)! +∞ tn (u + 1) n+s u n+2+s +∞ u z s v n-1 + q(n -1, v) (v + 1) n+1+s dvdu.
where in order to simplify the notation, for every n ≥ 1 we denoted

ǫ s (n) := -(t n + 1) 1+s +∞ tn (u + 1) -n-2-s p n (u)du + t n 1+s +∞ tn u -n-2-s p n (u)du + t n 1+s +∞ tn (u + 1) n+s u n+2+s +∞ u (v + 1) -n-1-s p n-1 (v)dvdu.
We have

|ǫ s (n)| ≤ (t n + 1) 1+ℜ(s) 2 + 1 n + ℜ(s) +∞ tn u -n-2-ℜ(s) du max v∈[0,1] |p n (v)| = t -n n n + 1 + ℜ(s) 1 + 1 n 1+ℜ(s) 2 + 1 n + ℜ(s) max v∈[0,1] |p n (v)|, ∀n ≥ 1.
By the Notation 5, we have lim n→+∞ max v∈[0,1] |p n (v)| = 0, we deduce that

lim n→+∞ |nt n n ǫ s (n)| = 0.
Using the last limit, the Equation (12) implies

lim n→+∞ t n n (n -1)! z s V s [u n ](t n ) + V s [q(n, u)](t n ) = 0,
where for every n ≥ 1 and every real polynomial u ∈ R → g n (u) of order less then n + 1, the function t → V s [g n (u)](t), is defined as

V s [g n (u)](t) := +∞ t (t + 1) 1+s (u + 1) n+2+s - t 1+s u n+2+s g n (u)du -t 1+s +∞ t (u + 1) n+s u n+2+s +∞ u g n-1 (v) (v + 1) n+1+s dvdu. Then V s [u n ](n) = n j=1 (-1) j n j (n + 1) -j j + 1 + s - n-1 j=0
(-1) j n -1 j

1 j + 1 + s n-1-j k=0 n -1 -j k n -j-k j + k + 2 + s
and by the notation of q(n, u) given by the Equation (13)

V s [q(n, u)](n) = - n+1 k=1 B k k! n (n + 2 -k)! n-k+2 j=0 (-1) j n -k + 2 j (n + 1) 2-k-j k + j -1 + s + n+1 k=1 B k k! n 2-k (n + 2 -k)!(k -1 + s) + n k=1 B k k! 1 (n + 1 -k)!

Conclusion

In this conclusion, we present the approach to prove that the Zeta function does not vanish at any point s ∈ B such that 1 -2ℜ(s) = 0. Let be s ∈ B and suppose that |ζ(s)| = 0. By Theorem 1, we have

Notation 4 .

 4 Let g : R + → C be a continuous function, we denote the function Φ[g] : R + → C as Φ[g](t) := t 0 g(u)du, ∀t ≥ 0, Notation 5. For every k ∈ N define the function p k : R → R as

  k,j,m k + j + m + 1 + s where w n,k,j := n -k + 1 j (-1) j k + j + 1 , v n,k,j,m := n -k -j m n -k-j-m .

lim n→+∞ n n+1 η s (n) n! + ( 1 -

 1 s)η s (n) = 0, Thanks to the Riemann functional equation, the point 1 -s is also a zero of the function ζ, thenlim n→+∞ n n+1 η 1-s (n) n! + sη 1-s (n) = 0,

  ∀n ≥ 1, Lemma 7. For every s ∈ B, there exists a unique z s ∈ C such that the solution ψ s (z s , t) : [1, +∞) → C of the differential equation (2) satisfies sup t≥1 |ψ s (z s , t)| < +∞. Further,

W. Oukil 3 Proof of the Theorem 1

Proof of the Theorem 1. Let be s ∈ B. By the Equation (1) we have,

Thanks to the Main Proposition 3 and the Lemma 7 we get

where we recall that

and where for every n ≥ 1 and every real polynomial

We can apply following integration for every arbitrary sequence (t n ) n ⊂ R + . Consider the particular case where t n = n for every n ≥ 1. By change of variable, we have

Thanks to the Binomial theorem, we have

For ℑ(s) = 0, by taking the imaginary part of the subtraction, can be written as

where by using the definition of the Bernoulli numbers, the quantity of f s (n) is defined as an alternating series. Using the monotonicity of the absolute value of the general term, we obtain lim n→+∞ n n+1 f s (n) = 0.