domain: spi math

text:
C Lusso

X Chateau

Numerical Modeling of Disordered Foam in 3D: Effective Properties by Homogenization

We investigate the numerical modeling and simulation of mechanical effective properties of disordered foams in 3d. We consider the elasticity, the thermal conductivity as well as the permeability. We present a meshing procedure for the microstructure of randomly disordered wet foams. The material behavior is modeled by constitutive equations and a boundary value problem is set on the microstructure. We present and implement the homogenization approach for the determination of the effective mechanical properties. The local problems are discretized and solved with the finite element method. We investigate the monodisperse and disordered configuration of a wet foam by varying the volume fraction.

INTRODUCTION

The structure of foam materials involves several length scales from macroscopic scale to macroscopic scale. In the context, the modeling approach is an homogenization, or multi-scale approach [Allaire, 1989], consisting in three main stages: the representation, the localization and the homogenization. In the first step, the modeling of the microstructure is established through a representative section of the material microstructure, namely a Representative Volume Element (RVE). The formal definition of the RVE requires a condition of separation of scales (micro-macro) assumed all along this work. The wet foam structure is modeled as a continuous network of convex polyhedral surfaces and the meshing procedure is based on a Voronoi tessellation, which in turn is based on random packings algorithm [START_REF] Torquato | [END_REF]. The Voronoi tessellation then requires to evolve toward a physical structure of foam [Kraynik et al., 2003, Kraynik andReinelt, 1999], this process is called the relaxation. In the localization stage, at the microscopic scale, the local properties on the RVE are derived as the solutions of a boundary value problem set on the RVE. We consider the deformation of the solid Plateau border, modeled by an elastic rheology. We also determine the thermal conductivity, the heat transfer is modeled by the Fourier's law on the Plateau border. Morevover, we study the fluid flow within the porous part. To model the microscopic flow within the pores, we consider the Stokes equations for a viscous "Template" -2017/3/15 -8:40

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fluid. These boundary value problems are discretized and solved with the Finite Element Method. Finally, the macroscopic mechanical properties are predicted by mean of averaging the local fields within the RVE.

NUMERICAL MODELING OF THE FOAM STRUCTURE

The structure of a dry foam is modeled as a set of convex polyhedra packed to fill space. We adopt the Voronoi tessellation algorithm to generate a partition of a sphere into convex polyhedral cells. Then, the surface of the Voronoi tessellation is drived to evolve toward physically relevant geometry modeling the foam microstructure [Kraynik et al., 2003, Kraynik andReinelt, 1999]. This stage is called the relaxation [Kraynik et al., 2003, Kraynik andReinelt, 1999] and the algorithm provides a stable structure of foam. The surface is evolved toward some local minimum in surface energy by iterating Conjugate Gradient descent method of optimization, with the Surface Evolver software developed by Ken Brakke (1992) [Brakke, 1992, Phelan et al., 1995]. Besides, the cells are constrained to have equal volumes, thus modeling a monodisperse structure of disordered foam. Furthermore, the mesh is regularized, the facets of the triangulation are shaped as equilateral as possible and each vertex is moved to the average position of its neighboring vertices. We iterate this algorithm until the number of nonminimal features (in Plateau's law [START_REF] Weaire | [END_REF]) tends to 0. A wet foam structure can then be inferred starting from this relaxed structure. The extension to a wet foam structure with arbitrary volume fraction is achieved through the addition of the Plateau border. Subsequently the Plateau border's volume is gradually increased and adjusted. The volume meshing procedure is achieved with Gmsh [GMSH]. In the framework of the homogenization approach, the generated microstructure models the Representative Volume Element (RVE).

The figure 1 below depicted the mesh of a disordered microstructure with N = 128 cells and a volume fraction φ = 10%. We represent both the Plateau border (left) and the porous part (right).

STATEMENT OF THE LOCAL PROBLEMS

The elasticity problem

The elasticity model is adopted to describe the small deformation of the solid Plateau border. The domain Ω s ⊂ R 3 standing for the Plateau border is filled by a solid material. The rheology is defined by the linear Hooke's law and the constitutive equation reads:

σ = 2µε(u) + λ trace ε(u) in Ω s , (1)
where σ denotes the total stress tensor, ε the linearized strain tensor, λ and µ stand for the Lam coefficients. We consider the equilibrium state of the material, the momentum "Template" -2017/3/15 -8:40 balance law with no body force reads

div σ = 0 in Ω s . (2)
The boundary Γ is divided into disjoint parts Γ = Γ u ∪ Γ σ denoting the external surface and the internal surface respectively, corresponding to the boundary where the displacement u is imposed and to the boundary where the traction σN is imposed (nostress). In order to implement the homogenization approach, we apply an homogeneous loading E corresponding to uniform boundary conditions [Dormieux et al., 2002]. To achieve this we enforce

u = E • x on Γ u , (3) σN = 0 on Γ σ , (4)
where N stands for the outward unit normal on Γ and E is an order 2 tensor.

The thermal conductivity problem

The heat transfer model on the solid Plateau border is employed in order to prescribe the thermal conductivity. The behavior is defined by the linear Fourier's law and the constitutive equation reads

q(x) = -κ∇T (x) in Ω s , (5)
where T (x) ∈ R denotes the temperature within the material, q(x) ∈ R 3 is the heat flux and κ is the conductivity coefficient. We consider the thermal equilibrium state of "Template" -2017/3/15 -8:40

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the material, the thermal balance law is expressed as

div q(x) = 0 in Ω s . (6)
The boundary Γ is divided into disjoint parts Γ = Γ T ∪ Γ q . In order to implement the homogenization approach, we apply an homogeneous loading A corresponding to uniform boundary conditions. To achieve this, we enforce

T = A • x on Γ T , (7) q • N = 0 on Γ q , ( 8
)
where A is an order 1 tensor. The solution represents the local flux q and the local temperature T at the microscopic scale, induced by the macroscopic loading A.

The Stokes problem

At the microscopic scale, the flow of a viscous fluid is governed by the Stokes equations [Allaire, 1989]. The domain Ω f ⊂ R 3 standing for the porous part saturated by an incompressible Newtonian fluid. The constitutive equation is written

σ = 2µD(u) -p in Ω f , (9)
where u ∈ R 3 denotes the velocity, p ∈ R is the pressure field and µ > 0 is the viscosity. We consider the equilibrium state of the material and the fluid is assumed to be incompressible.

-µ div D(u)

+ ∇p = 0 in Ω f , div u = 0 in Ω f .
The boundary of the domain is disjoint into Γ = Γ sf ∪Γ e , where Γ sf stands for the solidfluid interface and Γ e is the external surface. We assume a no-slip boundary condition at the solid-fluid interface and the homogenization approach is implemented by applying an homogeneous loading α on the RVE.

u = 0 on Γ sf , σN = -(α • x) • N on Γ e .

NUMERICAL RESULTS

The problems , and are written in a variational form and discretized using the Galerkin method together with the finite element method. The resolution is achieved throught the FreeFem++ langage and the numerical method is wether the conjugate gradient or the GMRES solver for larger systems such as the N = 512 cells mesh.

Elasticity

We have determined the macroscopic properties by averaging the stress field on the whole domain. As the structure exhibits isotropy, the discrete macroscopic elastic tensor C h is reduced to 3 coefficients λ h + 2µ h , λ h and µ h where λ h and µ h are the discretized Lam coefficients. In the figure 2 below, we represent µ h and λ h with respect to the volume fraction φ. We have compared the result for different number of cells N , on both a r 0 (no refinement) mesh and a r 2 (2 refinements) mesh.

The macroscopic properties linearly increase with the volume fraction. The size of the RVE affects the results mainly for the elastic modulus. For the Lam coefficient, on the r 2 mesh, the converged homogenized solution is reached even with N = 32 cells, whereas for the elastic modulus, the converged homogenized solution is not reached even with N = 512 cells. In other words, with two refinements, the size of the RVE is large enough to obtain the converged homogenized solution λ h . In addition, by studying the periodic case of a Kelvin structure, we have inferred that our result with N = 512 are about 20% different from the converged homogenized solution.

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Thermal conductivity

We have determined the macroscopic thermal conductivity by averaging the flux field on the whole domain. As the structure exhibits isotropy, the macroscopic thermal conductivity tensor K K K h is reduced to κ h , where κ h is the discretized conductivity. In the figure 3 below, we represent κ h with respect to the volume fraction on both the r 0 mesh and the r 2 mesh.

The macroscopic thermal conductivity linearly increases with the volume fraction. The size of the RVE slightly influences the results since we obtain almost superimposed results with respect to the number of cell N , and the results are enhanced with two refinements (our approximation provides an upper bound of the homogenized solution). Meaning that the numerical error of approximation dominates the error of the homogenization approach. In addition, by studying the periodic case of a Kelvin structure, we have inferred that our result with N = 512 are about 5% different from the converged homogenized solution.

Permeability

The resolution is achieved through a MPI (Message Passing Interface) formulation in FreeFem++ langage with a MPI resolution on n p = 4 processors. The numerical method adopted to solve the linear system is the Mumps (MUltifrontal Massively Parallel Solver) which is a sparse direct solver. We determine the macroscopic permeability by averaging the velocity field on the whole domain. As the structure exhibits isotropy, the discrete macroscopic permeability tensor K h is reduced to K h = k h , where k h is the discretized permeability. In the monodisperse configuration, the average cell volume v B stands for the only characteristic length scale. In the figure 4 we represent the ratio k h r 2 , where r = 3v B 4π 1 3 , with respect to the volume fraction. We compare the result for different value of number of cell N .

The macroscopic permeability linearly decreases with the volume fraction. The size of the RVE significantly influences the results. More specifically, we that the number of cell has more influence on low volume fraction φ = 2, 5% on high volume fraction φ = 20%. This fact relies on the slightly higher number of elements for meshes with φ = 20%, leading to a more accurate approximation of the numerical solution in the finite element method.

CONCLUSION

First of all, we point out that an elaborate meshing procedure has to be carried out, which is time consuming and difficult to implement in the case of real microstructures. Indeed, the microstructure of foams presents irregularities such as singularities and requires to reach a stable state of equilibrium, modeling the physically relevant microstructure of foams. In this context, we manage to entirely automate the foam meshing procedure of disordered wet foams. Besides, our results are in good agreement with the well-known results on dependence of the mechanical properties of the foam with respect to the volume fraction (we have compared with estimates available in the literature data). As a consequence, we are confidant that a 512 bubbles RVE with two refinements mesh provides an accurate homogenized solution.

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Figure 1 .

 1 Figure 1. Meshed microstructure of a disordered foam with N = 128 cells and a volume fraction φ = 10%. Plateau border (left) and porous part (right).

"Figure 2 .

 2 Figure 2. Elastic modulus µ h normalized with µ = E 2(1+ν) = 1000 2,6 and Lamé coefficent normalized with K = λ + 2 3 = 1000 1,2 , with respect to the volume fraction φ. Left: no refinement (r 0 ) and right: 2 refinements (r 2 )

Figure 3 .

 3 Figure 3. Thermal conductivity κ h normalized with the conductivity coefficient κ = 100. Left: no refinement (r 0 ) and right: 2 refinements (r 2 ).

Figure 4 .

 4 Figure 4. Permeability k h r 2 with respect to the volume fraction φ.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from Agence Nationale de la Recherche (Grant no. ANR-13-RMNP-0003-01).


Tao Sun
email: tao@taosphysics.net

A More Complete Model for High-Temperature Superconductors

To date, the Hubbard model and its strong coupling limit, the t-J model, serve as the canonical model for strongly correlated electron systems in solids. Approximating the Coulomb interaction by only the onsite term (Hubbard U -term), however, may not be sufficient to describe the essential physics of interacting electron systems. We develop a more complete model in which all the next leading order terms besides the onsite term are retained. Moreover, we discuss how the inclusion of these neglected interaction terms in the Hubbard model changes the t-J model.

Introduction

It is generally believed [1] that some essential physics of strongly correlated electron systems in solids, such as high-temperature superconductors [2], can be described by the Hubbard model [3,4]

H = -t ij,σ γ ij c † iσ c jσ + U i n i↑ n i↓ , (1)
where c † iσ and c iσ are the creation and annihilation operators for an electron of spin σ at site i, and n iσ = c † iσ c iσ , the corresponding number operator of electrons. In Eq (1), γ ij = 1 for nearest-neighbor (n.n.) sites i, j, and 0 otherwise, which restricts the summation over the nearest-neighbor pairs. The t-term describes the kinetic energy due to hopping between adjacent sites, and the Uterm is the on-site approximation of the Coulomb interaction between electrons. Indeed, the Hubbard model offers the simplest way to describe both the band motion and atomic features of the interacting electron systems in solids.

In the Hubbard model, there are four possible states on the lattice sites: empty state |0 , singly occupied states | ↑ (spin up) and | ↓ (spin down), and doubly occupied state | ↑↓ . Apparently, the double occupancy state | ↑↓ is not favorable when U is large. It is commonly believed that a model in which all the doubly occupied states are excluded can describe the low-energy physics of the system. Technically, such a singly occupied state system (with only empty and singly occupied sites) can be obtained by the Gutzwiller projection operator approach, which projects the Hubbard model to a subspace of the original Hilbert space. For large U , to the order of J = 2t 2 /U in the perturbation calculation, this projecting scheme leads to the well-known t-J model [START_REF] Spalek | t-J model then and now: A personal perspective from the pioneering times[END_REF][START_REF] Zhang | [END_REF]

H = -t ij,σ γ ij c † iσ cjσ + J ij γ ij (S i • S j - 1 4 ñi ñj ), (2)
where c † iσ = c † iσ (1 -n iσ ), ciσ = c iσ (1 -n iσ ), ñi = σ c † iσ ciσ , and S i is the spin operator at site i. Note that the projection operator also generates a so-called three-site term that has been neglected in the t-J model. That is, Eq (2) is in fact a model with only two-site terms. The t-J model can be viewed as a large U limit of the Hubbard model, and as a prototype to describe the low-energy physics of the interacting electron systems. Using the standard Green function technique, one can indeed show that the t-J model does exhibit a state with nonzero c i↓ c j↑ , signaling the occurrence of superconductivity. Nonetheless, it is still somewhat difficult to obtain a more complete understanding of hightemperature superconductors from the t-J model.

The Hubbard model is the simplest approximation of the general Hamiltonian of interacting electron systems, in which all the interaction terms except the on-site term (U -term) are neglected. It is possible that the neglected interaction terms may play important roles in the understanding of the physics of the strongly correlated electron systems. Some important physics may be omitted in the Hubbard model. As a result, the t-J model given in Eq (2), which is a projection to the subspace of singly occupied states of the Hubbard model, may not be sufficient to describe the essential physics of the interacting electrons.

In this paper, we will first describe the details of establishing a generalized Hubbard model, in which all of the two-site interaction terms between nearestneighbor sites are retained. We will then apply the Gutzwiller projection scheme to this more general model to obtain an effective Hamiltonian in the subspace of singly occupied states. Moreover, we will discuss how the inclusion of these neglected interaction terms in the Hubbard model modifies the t-J model.

The Model

We now present a systematic derivation of a more complete effective Hamiltonian with all two-site interaction terms between nearest-neighbor sites in the subspace of singly occupied states.

Bloch Representation

The general Hamiltonian describing the dynamics of the electrons in solids can be expressed as follows [3,[START_REF] Mahan | Many-Particle Physics[END_REF][START_REF] Li | Solid State Theory[END_REF] in Bloch representation

H = k,σ k c † kσ c kσ + 1 2 k1k2k 1 k 2 ,σσ k 1 k 2 | 1 r |k 1 k 2 c † k1σ c † k2σ c k 2 σ c k 1 σ , (3)
where c † k,σ and c k,σ are the creation and annihilation operators for electrons in the Bloch state with wave vector k and spin σ, k is the band energy, k runs over the first Brillouin zone, and the matrix element

k 1 k 2 | 1 r |k 1 k 2 = e 2 dxdx ψ * k1 (x)ψ k 1 (x)ψ * k2 (x )ψ k 2 (x ) |x -x | , (4)
where ψ k (x) and ψ * k (x) are the Bloch functions of the energy band. In Eq (3), the first term represents the kinetic energies of the electrons in the energy bands, and the second term is the Coulomb interaction energy of electrons.

Wannier Representation

The Hamiltonian in Eq (3) can be transformed to its Wannier representation form

H = ij,σ T ij c † iσ c jσ + 1 2 ijkl,σσ ij| 1 r |kl c † iσ c † jσ c lσ c kσ , (5)
where c † iσ and c iσ are the creation and annihilation operators for an electron with spin σ in a Wannier orbital localized at site i, T ij is the Fourier transform of the band energy k

T ij = 1 N k k e ik•(Ri-Rj ) , (6)
and the Wannier representation matrix element is given by

ij| 1 r |kl = e 2 dxdx φ * (x -R i )φ(x -R k )φ * (x -R j )φ(x -R l ) |x -x | , (7)
where φ(x -R i ) and φ * (x -R i ) are the Wannier functions localized around lattice site i.

In the most general cases where i = j = k = l, the Wannier matrix elements ij| 1 r |kl are the so-called four-center integrals, and the corresponding terms in the series of the Coulomb interaction in Eq (5) can be referred to as the four-site terms. Since the Wannier function φ(x -R i ) goes to zero rapidly when x is away from R i , the matrix element [START_REF] Mahan | Many-Particle Physics[END_REF] is not negligible only when the sites i, j, k, l are close enough so that the overlap between the Wannier functions is sufficiently large. Therefore, the Coulomb interaction energy can be conveniently approximated by a number of its leading terms of the series in the Wannier representation expression given in Eq [START_REF] Spalek | t-J model then and now: A personal perspective from the pioneering times[END_REF]. The biggest interaction term is the so-called on-site term (i = j = k = l)

1 2 U i,σ n iσ n iσ = U i n i↑ n i↓ , (8)
where U = ii| 1 r |ii . If only this on-site term is taken into account, the Hubbard model is obtained. The next leading order terms are the ones where the set {i, j, k, l} contains only two distinct nearest-neighbor sites. These terms may be referred to as the two-site interaction terms [9]. In this paper, we take one step further beyond the Hubbard model: In the approximation of the Coulomb interaction, we will retain all of these two-site interaction terms, in addition to the Hubbard on-site term.

Two-Site Interaction Approximation

We now describe a generalized Hubbard model with all of the two-site interaction terms between two nearest-neighbor sites. The two-side interaction terms are those where the set {i, j, k, l} consists of only one pair of nearest-neighbor sites. These terms fall under two categories:

1. Among the four sites i, j, k, l, three of them are in fact the same site, and the fourth one is a nearest neighbor of the other three. There are four such possibilities:

a. i = j = k, and l is the n.n. site of i:

1 2 ij,σ γ ij ii| 1 r |ij n iσ c † iσ c j σ = 1 2 ij,σ γ ij ii| 1 r |ij n iσ c † iσ c jσ (9)
b. i = j = l, and k is the n.n. site of i:

1 2 ij,σ γ ij ii| 1 r |ji n iσ c † iσ c jσ (10)
c. i = l = k, and j is the n.n. site of i:

1 2 ij,σ γ ij ij| 1 r |ii n iσ c † j σ c iσ = 1 2 ij,σ γ ij ji| 1 r |jj n j σ c † iσ c jσ (11)
d. j = l = k, and i is the n.n. site of j:

1 2 ij,σ γ ij ij| 1 r |jj n j σ c † iσ c jσ (12)
Eq (9) is true because the spin summation index σ can be replaced by σ in the expression. Similarly, the right-hand-side of Eq (11) results from interchanging the summation indexes i and j, in addition to the change of σ → σ. From Eq [START_REF] Mahan | Many-Particle Physics[END_REF], it can be verified that all the matrix elements in Eqs (9-12) are equal. Thus, all of the terms above can be combined into the following expression

X ij,σ γ ij c † iσ c jσ (n iσ + n j σ ), (13)
where the parameter X represents the matrix elements

X = ii| 1 r |ij = ii| 1 r |ji = ij| 1 r |ii = ji| 1 r |ii . (14)
Here, i and j represent the nearest-neighbor sites. 2. Among the sites i, j, k, l, there are two identical pairs, and moreover, the said two pairs are nearest neighbors. There are three such possibilities:

a. i = j, l = k, i and l are the n.n. sites:

1 2 ij,σ γ ij ii| 1 r |jj c † iσ c jσ c † iσ c j σ (15)
b. i = k, j = l, i and j are the n.n. sites:

1 2 ij,σσ γ ij ij| 1 r |ij n iσ n jσ (16)
c. i = l, j = k, i and j are the n.n. sites:

1 2 ij,σσ γ ij ij| 1 r |ji c † iσ c † jσ c iσ c jσ (17)
It has been argued that ii| 1 r |jj and ij| 1 r |ji are in the same order, but much smaller than ij| 1 r |ij [3,[START_REF] Strack | [END_REF]. Therefore, summing over all of the three terms in Eqs (15-17), we have the following terms in this category

1 2 V ij,σσ γ ij n iσ n jσ + 1 2 Y ij,σ γ ij (c † iσ c jσ c † iσ c j σ + σ c † iσ c † jσ c iσ c jσ ), ( 18
)
where the parameters V and Y represent the matrix elements

V = ij| 1 r |ij , Y = ii| 1 r |jj = ij| 1 r |ji . (19)
Eqs ( 13) and ( 18) contain all the two-site Coulomb interaction terms between two nearest neighbors.

Similarly, for the kinetic energy term, only the constant and nearest-neighbor hopping terms are considered for consistency

T ij = 1 N k k e ik•(Ri-Rj ) = T 0 δ ij + T 1 γ ij + ... , (20)
where

T 0 = T ii = 1 N k k , T 1 = T ij = 1 N k k e ik•(Ri-Rj ) , for n.n. sites ij , (21)
and (...) represents the hopping terms beyond nearest-neighbor sites. Since T 1 < 0, we will use t = -T 1 in the expressions hereafter. Substituting Eq (20) into Eq [START_REF] Spalek | t-J model then and now: A personal perspective from the pioneering times[END_REF], and approximating the Coulomb interaction by all the terms given in Eqs ( 8), [START_REF] Belitz | [END_REF], and (18), we have

H T 0 i,σ n iσ -t ij,σ γ ij c † iσ c jσ + U i n i↑ n i↓ + 1 2 V ij,σσ γ ij n iσ n jσ + X ij,σ γ ij c † iσ c jσ n iσ + n j σ + 1 2 Y ij,σ γ ij c † iσ c jσ c † iσ c j σ + σ c † iσ c † jσ c iσ c jσ . (22)
Eq ( 22) contains all the two-site interaction terms as well as the on-site term and can be considered the most complete generalization of the Hubbard model to date. The terms parametrized by X and V have been discussed in literature [START_REF] Strack | [END_REF]11]. The X-term is the so-called bond-change interaction, which describes the density-dependent nearest-neighbor hopping of electrons. The V -term is the so-called nearest-neighbor density interaction. The matrix element Y was discussed in reference [3]. When X = V = Y = 0, Eq (22) reduced to the Hubbard model. It is worth pointing out that, in contrast to the U -term, the terms parametrized by V , X, and Y retain the long-range nature of the Coulomb interaction. Mott argued that, based on the long-range nature of the Coulomb interaction, the correlated electron system undergoes a first-order metal-insulator transition triggered by electronic correlations [START_REF] Mott | Proc. Phys. Soc. Lond., A[END_REF]. The Hubbard model, however, only exhibits a continuous metal-insulator transition owing to the correlation effect of electrons [3]. The reason that the Hubbard model does not predict a discontinuity in the number of current carriers may be attributed to the lack of the long-range interactions [3,[START_REF] Mott | Proc. Phys. Soc. Lond., A[END_REF][START_REF] Belitz | [END_REF]. Therefore, a discontinuous metal-insulator transition may exist in the model given in Eq (22) [14].

Gutzwiller Projection Technique

Similar to the Hubbard model, in the model described by Eq (22), there are four possible states on the lattice sites: empty, singly occupied with spin up, singly occupied with spin down, and doubly occupied. The Hamiltonian describing the low-energy physics should not include the doubly occupied states. We need to project the Hamiltonian (22) to a subspace of the Hilbert space where only empty and singly occupied sites are allowed. This can be accomplished by the Gutzwiller projection approach. The Gutzwiller projection operators can be defined in the following form [4,[START_REF] Vollhardt | Proceedings of the International School of Physics "Enrico Fermi" Course CXXI[END_REF]

P s = i (1 -n i↑ n i↓ ), P d = I -P s , (23)
where n i↑ and n i↓ are the electron number operators at site i with spin up and down, respectively, i runs over all the lattice sites, and I is the identity operator. It can easily be verified that the projection operators P s and P d have the following properties (P s ) 2 = P s , (P s ) n = P s ,

(P d ) 2 = P d , (P d ) n = P d , P s P d = 0, ( 24
)
where n is any positive integer. The Gutzwiller projection operators can partition the original Hilbert space into two subspaces. The first one (denoted by S) contains the states with only empty and singly occupied sites, while the second one (denoted by D) contains the states with at least one doubly occupied site. The subscripts s and d denote the two subspaces S and D, respectively.

We now derive the effective Hamiltonian in the subspace S using the Gutzwiller projection method. The specific treatment follows the formalism developed in Ref. [START_REF] Hubac | Brillouin-Wigner Methods for Many-Body Systems[END_REF]. The Schrödinger eigenvalue equation of the Hamiltonian H given in Eq (22) can be written as

H|Ψ = ε|Ψ , ( 25
)
where |Ψ is the wave vector function in Hilbert space and ε its corresponding energy eigenvalue. Since P s + P d = I, Eq (25) may be written as

H(P s + P d )|Ψ = ε(P s + P d )|Ψ . (26)
Operating P s and P d respectively to Eq (26) from the left, and taking the properties of P s and P d into account, we obtain

P s HP s -ε P s HP d P d HP s P d HP d -ε |Ψ s |Ψ d = 0,
where |Ψ s = P s |Ψ and |Ψ d = P d |Ψ , the wave vectors in the subspaces S and D, respectively. The projected Hamiltonian matrix above can be diagonalized as

H s -ε 0 0 H d -ε |Ψ s |Ψ d = 0,
where

H s = P s HP s + P s HP d (ε -P d HP d ) -1 P d HP s , (27)
H d = P d HP d + P d HP s (ε -P s HP s ) -1 P s HP d , (28)
are the projected Hamiltonians in the subspaces. In the following discussion, we are only interested in H s , the effective Hamiltonian in the singly occupied subspace.

Operating the projection operators P s and P d on H in Eq (22) from both the left and right, after some straightforward manipulations, we can obtain the following projected Hamiltonian matrix elements

P s HP s =T 0 i,σ ñiσ -t ij,σ γ ij c † iσ cjσ + 1 2 V ij,σσ γ ij ñiσ ñjσ + 1 2 Y ij,σ γ ij c † iσ c jσ c † j σ c iσ -ñiσ ñjσ , (29)
P s HP d = -(t -X) ij,σ γ ij c † iσ c jσ n j σ , (30)
P d HP s = -(t -X) ij,σ γ ij n iσ c † iσ cjσ , (31)
P d HP d = -(t -2X) ij,σ γ ij n iσ c † iσ c jσ n j σ + (U + 2T 0 ) i n i↑ n i↓ + (2V -Y ) ij γ ij n i↑ n i↓ n j↑ + n j↓ -n j↑ n j↓ + 1 2 Y ij,σ γ ij c † iσ c jσ c † iσ c j σ , (32)
where c † iσ = c † iσ (1-n iσ ), ciσ = c iσ (1-n iσ ), and ñi = σ c † iσ ciσ are the projected creation, annihilation, and number operators of electrons. Note that the Vand Y -terms do contribute to the zeroth-order of the effective Hamiltonian P s HP s . These are the terms neglected in the t-J model. Note also that the X and U -terms do not survive in the expression of P s HP s , indicating that they are not operators of the subspace S. As we will see below, however, they do contribute to the first-order terms in the perturbation expansion when the projected Hamiltonian matrix is diagonalized.

With the results above, we are now ready to obtain the effective Hamiltonian H s . For convenience, we rewrite

H s = H (0) s + H (1) s , (33)
H (0) s = P s HP s , (34)
H

s = P s HP d (ε -P d HP d ) -1 P d HP s . (1)
The zeroth-order term H (0) s has been obtained in Eq (29). In order to construct the effective Hamiltonian H s in the subspace S, the key operator to be determined is the inverse operator (ε -P d HP d ) -1 . Since this is an operator in the subspace D, using the completeness of the wave vectors |Ψ d , we have

(ε -P d HP d ) -1 = r,r |Ψ r d Ψ r d |(ε -P d HP d ) -1 |Ψ r d Ψ r d |, (36)
where the superscripts r and r are the labels of the states in the subspace D. Please note that these states in D must be virtual because this inverse operator is in the middle of H

s , which is an operator of S. As can be seen from Eq (32), the expression of P d HP d is complicated. Nevertheless, in the case where there is one virtual doubly occupied site [START_REF] Vollhardt | Proceedings of the International School of Physics "Enrico Fermi" Course CXXI[END_REF], the matrix element

Ψ r d |(ε -P d HP d ) -1 |Ψ r d = Ψ r d | ε -2T 0 -U + ij,σ γ ij c † iσ c jσ (t -2X)n iσ n j σ - 1 2 Y c † iσ c j σ -1 |Ψ r d , (37)
where

U = U + z (2V -Y ), 0 ≤ z ≤ z,
and z is the coordinate number. If we consider the parameters ε -2T 0 , t -2X, and Y to be small quantities compared with U , we can perform a perturbation expansion with respect to 1/U . To the leading term in 1/U , we have

(ε -P d HP d ) -1 = r,r |Ψ r d Ψ r d | - 1 U 1 + O ε -2T 0 U , t -2X U , Y U -1 |Ψ r d Ψ r d | = - 1 U 1 + O ε -2T 0 U , t -2X U , Y U -1 r,r |Ψ r d Ψ r d |δ r,r = - 1 U 1 + O ε -2T 0 U , t -2X U , Y U -1 = - 1 U + O ε -2T 0 U 2 , t -2X U 2 , Y U 2 . (38)
Substituting (38) into Eq (35) and using the expressions of P s HP d and P d HP s in Eqs (30) and Eqs (31), we obtain for the second term of H s in the form

H (1) s = -J 0 ij,σ i j ,σ γ ij γ i j c † iσ c jσ n j σ n i σ c † i σ cj σ , (39)
where J 0 = (t -X) 2 /U . In general, Eq (39) represents a term with four sites i, j, i , j involved. In the most general case where i = j = i = j , it is a four-site term and an operator of the subspace D. Therefore we examine its special cases where two of the four sites are in fact the same site. Since < ij > and < i j > are two nearest-neighbor pairs, there are only four such possibilities:

1. i and i are the same site 2. i and j are the same site 3. j and i are the same site 4. j and j are the same site Further examination reveals that cases 1 and 4 result in a zero operator, while case 2 is still an operator in the subspace D. Only case 3 with i = j provides a meaningful contribution to the operator in the subspace S H (1)

s = -J 0 ijk,σ γ ij γ jk c † iσ ñjσ ckσ + c jσ c † j σ ckσ . (40)
Please note that in Eq (40) the site j is the nearest neighbor of both sites i and k. Therefore, the two-site term can only result from the case of i = k in the summation. Writing the two-site and three-site terms separately, we have

H (1) s = (H (1) s ) 2site + (H (1) s ) 3site , (41)
where

(H (1) s ) 2site = -J 0 ij,σ γ ij ñiσ ñjσ + c † iσ c jσ c † j σ c iσ , (42)
(H (1) s ) 3site = -J 0 i =k,j,σ

γ ij γ jk c † iσ ñjσ ckσ + c jσ c † j σ ckσ . ( 43
)
Please note that the summation in Eq (43) excludes the case i = k. This threesite term is usually ignored in the discussion of the t-J model. We will also neglect this three-site term in the subsequent discussions. Substituting the expression of H (0) s in Eq (29) and H

s ) 2site in Eq (42) into Eq (33), we finally obtain, to the order of two-site interactions terms, the effective Hamiltonian H s

H s = T 0 iσ ñiσ + ij,σ γ ij -t c † iσ cjσ - 1 2 (2J 0 -Y ) c † iσ c † j σ c iσ c jσ + 1 2 (V -Y ) ñiσ ñjσ + 1 2 (V -2J 0 ) ñiσ ñjσ , (44)
which can be written as

H s = T 0 iσ ñiσ + ij,σ γ ij -t c † iσ cjσ - 1 2 J c † iσ c † j σ c iσ c jσ + 1 2 p ñiσ ñjσ + 1 2 q ñiσ ñjσ , (45)
with the coefficients given by the following equations

p = V -Y, q = V -2J 0 , J 0 = (t -X) 2 /U , U = U + z (2V -Y ), J = p -q, (46)
where 0 ≤ z ≤ z as discussed above. Eq (45) is the effective Hamiltonian in the subspace of singly occupied states. Since it is projected from a more general Hamiltonian retaining all terms up to the two-site interactions, we believe that Eq (45) can describe the low-energy physics of the interacting electron systems in solids more accurately and completely.

Please note that by rescaling H s with t, this model has only three dimensionless independent parameters T0 = T 0 /t, p = p/t, and q = q/t. One may expect that the system will exhibit ferromagnetism when q is sufficiently larger than p. On the other hand, the system will exhibit antiferromagnetism when p is sufficiently larger than q. Our studies demonstrate that this is true. More interestingly, our study also shows that this model exhibits superconductivity for some parameter range of p > q.

3 The t-J Model Apparently, Eq (45) reduces to the t-J model with J = 2J 0 in the case of X = Y = V = 0. We will show that when Y = V , Eq (45) will reduce to the t-J model as well (with J = 2J 0 -V in this case). Making use of the following operator identities

C † i↑ C i↓ = S + i = S x i + iS y i , (47)
C † i↓ C i↑ = S - i = S x i -iS y i , (48)
n i↑ -n i↓ = 2S z i , (49) we obtain
σ c † iσ c jσ c † j σ c iσ = -2 S x i S x j + S y i S y j , (50)
σ ñiσ ñjσ -ñiσ ñjσ = 4S z i S z j , (51)
where S α i , α = x, y, z, are the spin operators at site i. Furthermore, it is easy to verify that σ ñiσ ñjσ + ñiσ ñjσ = ñi ñj ,

where ñi = σ ñiσ . Eqs (50)-(52) immediately lead to the following well-known relation

σ c † iσ c jσ c † j σ c iσ + ñiσ ñjσ = -2(S i • S j - 1 4 ñi ñj ). ( 53
)
Writing the two terms in the summation over σ explicitly, the zeroth-order effective Hamiltonian Eq (29) can be written as

H (0)
q-terms is combined into the J-term to form a complete Heisenberg interaction term [17], the energies of the two spin configurations become the same, namely, J ñi ñj /4. Then the spin correlations can be attributed to the Heisenberg interaction.

Conclusion

Starting with a generalized Hubbard model with the next leading order Coulomb interaction terms, we have systematically developed a new model for the strongly correlated electrons in solids using the Gutzwiller projection approach. Our study demonstrates that the origin of the Heisenberg interaction term can be attributed to the competition between the energies of the two spin configurations of the nearest-neighbor electrons. A comparison of the conventional t-J model and the new model has been discussed. It seems that the coefficient of one of the interaction terms in the t-J model, i.e. ñi ñj , is not in the physical range, which may be the reason why the superconductivity dome has not been obtained from the t-J model.

We have studied the magnetism and superconductivity of this new model. The main results of this research have been summarized in a recent paper [START_REF] Sun | Green Function Theory of Strongly Correlated Electron Systems[END_REF]. Our study indicates that this model provides a more complete description of the physics of strongly correlated electron systems. The system is not necessarily in a ferromagnetic state as temperature T → 0 at any doping level δ ≥ 0. The system, however, must be in an antiferromagnetic state at the origin of the doping-temperature (δ-T ) plane (T → 0, δ = 0). Moreover, the model exhibits superconductivity in a doped region at sufficiently low temperatures. The details of the magnetic and superconducting properties of this model will be presented in forthcoming papers.

s=

  P s HP s = T 0 Y ) ñiσ ñjσ + ñiσ ñjσ .

Therefore, using Eqs (52) and (53), Eq (54) takes the form

We point out again that the terms parametrized by Y and (V -Y ) in Eq (55) are the neglected terms in the t-J model. Similarly, for the first-order term (H

s ) 2site given in Eq (42), we have

Summing up the terms in Eqs (55) and ( 56), the effective Hamiltonian can be expressed as in the t-J model form

The above equation shows that when V = Y , the model reduces to the t-J model with J = 2J 0 -V . However, it has been argued that V Y in the usual cases [3]. Also, since the value of V is usually large, the condition of J = 2J 0 -V > 0 (necessary for antiferromagnetic and superconducting states) is difficult to be realized. Therefore, the condition V = Y may be rarely physically satisfied. Finally, using the parameters p and q defined in Eq (46), the model can be written as

where J = p -q and J = p + q. When J = -J, which means p = 0 (i.e. the condition V = Y ), one has the t-J model. Our Green function analysis shows that a reasonable superconductivity dome appears at a finite value of p (p/q ∼ 2.5), which again seems to indicate that the coefficient of ñi ñj of the conventional t-J model, -J/4, is not in the physical range. Eqs (45) and (58) are two alternative expressions of the new model, which reveal the origin of the spin correlations from two different though equivalent perspectives. For two nonempty nearest-neighbor sites, there are two different spin configurations: the spin directions are either the same or the opposite. Eq (45) indicates that the energies for the two spin configurations are, in general, different (p = q), and that the relative direction of the spins on the nearestneighbor sites is determined by the competition between the energies of the two spin configurations. On the other hand, as shown in Eq (58), if the interaction terms are rearranged such that the amount of energy JS z i S z j from the p-and


Marc Arnaudon
email: marc.arnaudon@math.u-bordeaux.fr

Ana Bela Cruzeiro

Christian Léonard
email: christian.leonard@u-paris10.fr

Jean-Claude Zambrini
email: jczambrini@fc.ul.pt

An entropic interpolation problem for incompressible viscid fluids

Keywords: April 4, 2017 Incompressible viscid fluids, entropy minimization, diffusion processes, convex duality, stochastic velocities, Navier-Stokes equation

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In the article [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF], Arnold addressed a geodesic problem on the manifold of all volume preserving diffeomorphisms on the torus. The resulting geodesics offer us a natural description of the evolution of an incompressible perfect fluid. Unfortunately, very little is known on the global existence of these geodesics [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF] and Shnirelman proved in [START_REF] Shnirelman | The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid[END_REF][START_REF] Shnirelman | Generalized fluid flows, their approximation and applications[END_REF] that solutions do not exist in general. In the seminal article [START_REF] Brenier | The least action principle and the related concept of generalized flows for incompressible perfect fluids[END_REF], Brenier introduced a relaxation of Arnold's problem in terms of generalized flows. In this extended setting, global existence of generalized "minimizing flows" is much easier to obtain.

While both Arnold's and Brenier's problems are related to the evolution of inviscid fluids, usually described by the Euler equation (5), the present article introduces a stochastic analogue of Brenier's relaxed problem related to the description of the evolution of viscid fluids. This viscosity is usually modeled by adding an extra Laplacian term to the Euler equation, leading to the Navier-Stokes equation (9). Following Itô's stochastic description of parabolic diffusion equations, we shall introduce stochastic differential equations based on Brownian motion to take the viscosity into account.

Brenier's problem (8) amounts to minimize an average kinetic action, the averaging procedure being performed over the set of all absolutely continuous sample paths. This is no longer available when the sample paths are nowhere differentiable Brownian trajectories. However, it is still possible to consider some kinetic action in terms of Nelson's stochastic velocities. It happens that this stochastic action is a relative entropy (with respect to the Wiener measure) and that Brenier's minimization of an average kinetic action turns out to be an entropy minimization problem: the Bredinger problem (Bdg) stated below at page 8.

Literature. Brenier's relaxation of Arnold's geodesic problem was introduced in [START_REF] Brenier | A combinatorial algorithm for the Euler equations of incompressible flows[END_REF]. Its dual problem was established and investigated for the first time in [START_REF] Brenier | The dual least action problem for an ideal, incompressible fluid[END_REF] where it was emphasized that the pressure field is the natural Lagrange multiplier of the incompressibility condition. The connection between solutions to Brenier's problem and the notion of measure-valued solutions to the Euler equation in the sense of DiPerna and Majda was clarified in [START_REF] Brenier | Minimal geodesics on groups of measure-preserving maps and generalized solutions of the Euler equations[END_REF]. The regularity of the pressure field was explored in [START_REF] Brenier | The dual least action problem for an ideal, incompressible fluid[END_REF] and revisited in [START_REF] Brenier | Minimal geodesics on groups of measure-preserving maps and generalized solutions of the Euler equations[END_REF]. Further improvements about the dual problem and the regularity of the pressure were obtained later by Ambrosio and Figalli in [START_REF] Ambrosio | On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations[END_REF][START_REF] Ambrosio | Geodesics in the space of measure-preserving maps and plans[END_REF].

Considering viscid fluids (Navier-Stokes equation) instead of inviscid ones, we refer to [START_REF] Antoniouk | Generalized stochastic flows and applications to incompressible viscous fluids[END_REF][START_REF] Arnaudon | Generalized stochastic lagrangian paths for the Navier-Stokes equation[END_REF] for works where generalized flows are also considered. The present article is also about generalized flows in the setting of viscid fluids but with an alternate point of view. To our knowledge, it is the first attempt to extend Brenier's variational approach in this context.

Outline of the paper. More about the connection between Bredinger and Brenier problems and their relations with fluid dynamics is given at Section 1. Since Bredinger's problem enters the class of convex minimization problems, it admits a natural dual problem; this is exposed at Section 2. It is the place where the pressure enters the game. The general shape of the Radon-Nikodym density of the solution of Bredinger's problem with respect to the Wiener measure is described at Section 3. It is shown that, in general, this solution fails to be Markov but remains reciprocal. At Section 4, under the restrictive assumption that the pressure is a nice function, the kinematics of the solution is made explicit. This permits us to establish a connection between Bredinger's problem and the Navier-Stokes equation. Finally, the last Section 5 is devoted to a characterization of the existence of a solution to Bredinger problem when the state space is the torus T n .

In the present article, the difficult problem of the regularity of the pressure field is left apart, see Remark 4.2(b).

Notation. The sets of all Borel probability measures on a topological set Z is denoted by P(Z). For any probability measure m ∈ P(A) on the Borel set A, the push-forward of m by the Borel measurable mapping θ : A → B is denoted by θ # m ∈ P(B) and defined by θ # m(db) := m(θ -1 (db)), for any Borel subset db ⊂ B.

State space. In general the state space X for the fluid will be either the flat torus X = T n = R n /[0, 1] n or the whole space X = R n . We denote it by X when its specific structure does not matter.

Path space. We denote Ω := C([0, 1], X ) the set of all continuous paths from the unit time interval [0, 1] to X . As usual, the canonical process is defined by

X t (ω) = ω t ∈ X , t ∈ [0, 1], ω = (ω s ) 0≤s≤1 ∈ Ω
and Ω is equipped with the canonical σ-field σ(X t ; 0 ≤ t ≤ 1).

Marginal measures. For any Q ∈ P(Ω) and 0 ≤ s, t ≤ 1, we denote Q t = (X t ) # Q ∈ P(X ) and Q st = (X s , X t ) # Q ∈ P(X 2 ); they are respectively the laws of the position X t at time t and of the couple of positions (X s , X t ) when the law of the whole random trajectory is Q. In particular, taking s = 0 and t = 1, Q 01 := (X 0 , X 1 ) # Q ∈ P(X 2 ) is the endpoint projection of Q onto X 2 . If Q describes the random behavior of a particle, then Q t and Q 01 describe respectively the random behaviors of the particle at time t and the couple of endpoint positions (X 0 , X 1 ). We denote

Q x := Q(• | X 0 = x) ∈ P(Ω) and the bridge of Q between x and y ∈ X is Q xy (•) := Q(• | X 0 = x, X 1 = y) ∈ P(Ω).
In particular, as X is Polish, the following disintegration formulas

Q(•) = X Q x (•) Q 0 (dx) = X 2 Q xy (•) Q 01 (dxdy) ∈ P(Ω) are meaningful, i.e. x ∈ X → Q x ∈ P(Ω) and (x, y) ∈ X 2 → Q xy ∈ P(Ω) are measurable kernels.
Relative entropy. Let Y be a measurable space. We denote P(Y) and M(Y) respectively the sets of probability and positive measures on Y. The relative entropy of q ∈ M(Y) with respect to the reference measure r ∈ M(Y) is defined by

H(q|r) := Y log dq dr dq ∈ (-∞, ∞]
whenever the integral is meaningful, i.e. when q is absolutely continuous with respect to r and Y log -(dq/dr) dq < ∞. When r ∈ P(Y) is a probability measure, for all probability measures q ∈ P(Y), we have

H(q|r) = min H(•|r) = 0 ⇐⇒ q = r. (1)
A frequent use will be made of the additive decomposition formula

H(q|r) = H(f # q|f # r) + Z H(q f =z |r f =z ) (f # q)(dz) (2)
where f : Y → Z is any measurable mapping between the Polish spaces Y and Z equipped with their Borel σ-fields and q f =z is a regular version of the conditioned probability measure q(• | f = z).

• Note that as a definition q f =z is always a probability measure, even when q is not.

• It is necessary that f # q is σ-finite for the conditional probability measure q f to be defined properly. In particular, we see with (1) and (2) that

H(f # p|f # r) ≤ H(q|r), (3)
expressing the well-known property of decrease of the entropy by measurable push-forward.

Taking f = X 0 : Ω → X in (2) gives, for any Q, R ∈ M(Ω), H(Q|R) = H(Q 0 |R 0 ) + X H(Q x |R x ) Q 0 (dx)
whenever Q 0 and R 0 are σ-finite and H(Q 0 |R 0 ) makes sense in (-∞, ∞]. An interesting situation where unbounded path measures arise naturally is when the initial marginals

Q 0 = R 0 = vol
are prescribed to be the volume measure on X = R n . In this case,

H(Q|R) = X H(Q x |R x ) vol(dx).

Bredinger's problem

The main role of this article is played by the Bredinger problem, an entropy minimization problem stated below at (Bdg). The present section is devoted to a brief exposition of some relations between Bredinger's problem and the evolution of an incompressible viscid fluid. As our approach follows Brenier's one, we start with Brenier's problem and its relation with the evolution of an incompressible inviscid fluid.

Eulerian and Lagrangian coordinates. These coordinates correspond to two different descriptions of the same fluid flow through space and time. Let the state space X be a subset of R n .

(1) The Eulerian specification of the flow field in X is a vector field

(t, x) ∈ [0, 1] × X → v (t, x) ∈ R n
giving the velocity at position x and time t. One looks at the fluid motion focusing on specific locations in X .

(2) The Lagrangian specification of the flow field is a function

(t, x) ∈ [0, 1] × X → q(t, x) ∈ X
giving the position at time t of the parcel which was located at x at time t = 0. One looks at fluid motion following an individual particle. The labeling of the fluid particles allows keeping track of the changes of the shape of fluid parcels over time.

The two specifications are related by: v (t, q(t, x)) = ∂ t q(t, x). The total rate of change of a function or a vector field F (t, z) experienced by a specific flow parcel is

D t F (t, z) = d dt F (t, q(t, x)) x=q -1 t (z)
where z is fixed. This gives

D t F = (∂ t + v•∇)F, since d dt F (t, q(t, x)) x=q -1 t (z) = ∂ t F (t, q(t, x)) x=q -1 t (z) + ∂ t q(t, x)•∇F (t, q(t, x)) x=q -1 t (z) = ∂ t F (t, z) + v(t, z)•∇F (t, z).
This formula is meaningful if for each t, the map x → q(t, x) is injective. The operator

D t = ∂ t + v•∇ (4)
is sometimes called the convective derivative.

Euler equation. Let X be a bounded domain of R n . A fluid in X is said to be incompressible if the volume is preserved along the flow. This is equivalent to

∇•v = 0,
that is the divergence of the velocity field v vanishes everywhere. If the domain X has a boundary ∂X , the impermeability condition is

n•v = 0
where n is a normal vector to ∂X . From now on, we shall restrict our attention to domains X without boundary so that the impermeability condition is dropped down. The Euler equation is Newton's equation of motion

D t v = -∇p
where D t v is the convective acceleration and the scalar pressure field p : [0, 1] × X → R is part of the solution to be found out with v. The force -∇p is necessary for the volume to be preserved as time evolves. The fluid moves from high pressure to low pressure areas. Because of the expression (4) of the convective derivative, we obtain

   ∂ t v + v•∇v + ∇p = 0, t ≥ 0, x ∈ X ∇•v = 0, t ≥ 0, x ∈ X v(0, •) = v 0 , t = 0 (5)
which is the Euler equation of the unknown (v, p) for an incompressible fluid seen as a Cauchy problem with a given initial velocity field v 0 .

Arnold's flow of diffeomorphisms. The Cauchy problem (5) is notoriously difficult and there is some hope to understand it a little further by considering a variant which is closer to a variational approach of classical mechanics. Arnold [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] proposed to look at the following fixed endpoint version of the Cauchy problem (5):

   ∂ t v + v•∇v + ∇p = 0, 0 ≤ t ≤ 1, ∇•v = 0, 0 ≤ t ≤ 1, q 1 [v] = h, (6)
where X is a compact manifold with no boundary, typically X = T n ,

• q 1 [v] is defined by q 1 [v](x) := ω x 1 , x ∈ X , with ωx t = v(t, ω x t ), 0 ≤ t ≤ 1 ω x 0 = x, t = 0 ;
• h belongs to the group G vol := {g diffeo: det Dg = 1} of all volume and orientation preserving diffeomorphisms of X . This should be regarded informally since it is implicitly assumed in the definition of q 1 [v] that the field v admits a unique integral curve for each starting point x. In fact, the exact purpose of [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] is to describe the fluid evolution by means of pathlines (q t (•)) 0≤t≤1 which are seen as trajectories in G vol . One can prove that any solution (q t (•)) 0≤t≤1 of the action minimizing problem

[0,1]×X |∂ t q t (x)| 2 dxdt → min : q t (•) ∈ G vol , ∀0 ≤ t ≤ 1, q 0 (•) = Id, q 1 (•) = h (7)
where h is a prescribed element of G vol , is such that the velocity field v(t, z) = ∂ t q t (q -1 t (z)) is a solution of (6) for some pressure field p. This minimizer is nothing but a geodesic flow on G vol with prescribed endpoint positions Id and h. The pressure p disappears from the picture since -∇p can be seen as the force necessary to maintain the motion on the manifold G of all diffeomorphisms inside the submanifold G vol of volume preserving diffeomorphisms (the orientation is automatically preserved by continuity of the motion).

Brenier's generalized flow. Solving the geodesic problem (7) in G vol remains difficult. Indeed, the only known attempt is done in [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF] where a solution is proved to exist for h very close to the identity. Actually there are examples where such geodesics do not exist, see [START_REF] Shnirelman | The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid[END_REF][START_REF] Shnirelman | Generalized fluid flows, their approximation and applications[END_REF]. Therefore, Brenier [START_REF] Brenier | The least action principle and the related concept of generalized flows for incompressible perfect fluids[END_REF] relaxed (7) by introducing a probabilistic representation. Brenier's problem consists of minimizing an average kinetic action subject to incompressibility and endpoint constraints. It is

E Q [0,1] | Ẋt | 2 dt → min; Q ∈ P(Ω) : [Q t = vol, ∀0 ≤ t ≤ 1], Q 01 = π (8)
where π ∈ P(X 2 ) is a prescribed bistochastic probability measure, i.e. its marginals satisfy π 0 = π 1 = vol and Ẋt (ω) = ωt for any absolutely continuous path ω ∈ Ω with generalized time derivative ω. In the above action functional, it is understood that

[0,1] | ωt | 2 dt = ∞
whenever ω ∈ Ω is not absolutely continuous. Therefore any solution P of ( 8) is a path measure charging absolutely continuous paths. The constraint

(Q t = vol, ∀0 ≤ t ≤ 1)
reflects the volume preservation. The prescription that Q 01 = π varies among all the possible correlation structures between the initial and final positions with average profile

Q 0 = Q 1 = vol. It is a relaxation of q 1 (•) = h as can be seen by taking π(dxdy) = π h (dxdy) := vol(dx)δ h(x) (dy).
It is proved in [START_REF] Brenier | The least action principle and the related concept of generalized flows for incompressible perfect fluids[END_REF] that any P ∈ P(Ω) such that P t = vol, ∀t and P 01 = π Ẍt + ∇p(t, X t ) = 0, ∀t, P -a.e. ,

for some pressure field p, solves the geodesic problem (8). Keeping Arnold's point of view, we see that the P(X 2 )-valued flow (P 0t ) 0≤t≤1 is the generalized solution of Arnold's geodesic equation (7). In this approach one can recover the velocity field by defining a probability measure σ on

[0, 1] × X × R n , [0,1]×X ×R n f (t, x, v) σ(dtdxdv) := E P 1 0 f (t, X t , Ẋt ) dt
This measure can be considered as a generalized velocity that solves the Euler equation in the sense of DiPerna and Majda, see [START_REF] Brenier | Minimal geodesics on groups of measure-preserving maps and generalized solutions of the Euler equations[END_REF] for details.

Navier-Stokes equation. Navier-Stokes equation is a modification of the Euler equation where some viscosity term is added. Its Newtonian expression is

D t v = a∆v -∇p
where ∆v, the Laplace operator applied to v, represents a viscosity force, a > 0. This equation mixes the acceleration D t v which is easily expressed in Lagrangian coordinates and the viscosity term ∆v which is easily expressed in terms of Eulerian coordinates.

Rewriting everything in Eulerian terms leads to

   ∂ t v + v•∇v -a∆v + ∇p = 0, t ≥ 0, x ∈ X ∇•v = 0, t ≥ 0, x ∈ X v(0, •) = v 0 , t = 0. (9)
This is the Navier-Stokes equation of the unknown (v, p) for an incompressible fluid seen as a Cauchy problem with a given initial velocity field v 0 .

Introducing the Brownian motion. The presence of the Laplacian in (9) strongly suggests that considering Brownian paths instead of regular paths in (8) might lead us to an approach of the Navier-Stokes equation similar to Brenier's approach to the Euler equation. But one immediately faces the problem of defining the kinetic action [0,1] | Ẋt | 2 /2 dt since the Brownian sample paths are nowhere differentiable and any discrete approximation of the action diverges to infinity. One is forced to introduce an analogue of the average action

E Q [0,1] | Ẋt | 2 /2 dt that appeared in (8) by considering E Q [0,1] |v Q t (X)| 2 /2 dt (10)
with a relevant notion of stochastic velocity v Q t (X) introduced in place of the usual velocity v(t, X t ) = Ẋt , undefined in the present context where the path measure Q charges Brownian sample paths. A relevant notion of stochastic velocity was introduced by Nelson in [START_REF] Nelson | Dynamical theories of Brownian motion[END_REF]. The forward stochastic velocity is defined by

v Q t (X [0,t] ) := lim h→0 + 1 h E Q (X t+h -X t | X [0,t] ) (11)
and its backward counterpart by

v Q t (X [t,1] ) := lim h→0 + 1 h E Q (X t -X t-h | X [t,1] ), (12)
provided that X is a Q-integrable process and these limits exist in some sense. The stochastic action (10) computed with Nelson's stochastic velocity v Q = v Q can be expressed in terms of a relative entropy with respect to the reversible Brownian motion.

In the whole paper the reference path measure is the law R of the reversible Brownian motion on X = T n or X = R n with constant diffusion coefficient a > 0.

Definition 1.1. The reversible Brownian path measure R is defined by

R = X R x (•) vol(dx) ∈ Ω where for each x ∈ X , R x ∈ P(Ω) is the law of x + √ a B
where B is a standard Brownian motion on X starting from 0.

Roughly speaking, R is the Wiener measure with diffusion coefficient a on X starting from R 0 = vol ∈ P(X ). As R is a reversible Markov measure, its forward and backward velocities are opposite to each other: v R + v R = 0. Be aware that we have chosen Nelson's convention when defining the backward velocity

v Q t (X [t,1] ) = -[ v (X * ) # Q 1-t •X * ](X [t,1] ) where X * t := X 1-t , 0 ≤ t ≤ 1, is the time reversed canonical process. When X = R n , R ∈ M(Ω)
is an unbounded σ-finite measure and when X = T n , R ∈ P(Ω) is a probability measure. Note that in any case, the conditioned path measures R x ∈ P(Ω) are probability measures.

Girsanov's theory allows to show that for any Q ∈ M(Ω) with a finite relative entropy H(P |R) < ∞, there is some predictable vector field v Q (t, X [0,t] ) such that Q is the unique solution, among the path measures with the initial marginal Q 0 and which are absolutely continuous with respect to R, of the martingale problem associated with the family of second order differential operators defined for any twice differentiable function u and all 0 ≤ t ≤ 1, by

L t u = v Q (t, X [0,t] )•∇u + a∆u/2.
Moreover, we have

H(Q|R) = H(Q 0 |R 0 ) + 1 2a E Q [0,1] | v Q (t, X [0,t] )| 2 dt. (13)
For the details, see for instance [START_REF] Léonard | Girsanov theory under a finite entropy condition[END_REF]. These considerations were put forward a long time ago by Yasue in [START_REF] Yasue | A variational principle for the Navier-Stokes equation[END_REF] who introduced the stochastic action (10) but didn't take advantage of its representation (13) in terms of the relative entropy, although (13) is invoked in [START_REF] Yasue | A variational principle for the Navier-Stokes equation[END_REF]p. 135]. Since R is reversible with R 0 = R 1 = vol, we also obtain

H(Q|R) = H(Q 0 |vol) + 1 2a E Q [0,1] | v Q t | 2 dt = 1 2a E Q [0,1] | v Q t | 2 dt + H(Q 1 |vol)
where the stochastic drift fields v Q and v

Q

given by Girsanov's theory are precisely the forward and backward stochastic velocities of Q properly defined in some L 2 spaces. It is therefore natural to address the following entropy minimization problem

H(Q|R) → min; Q ∈ M(Ω) : [Q t = vol, ∀0 ≤ t ≤ 1], Q 01 = π (14)
in analogy with Brenier's problem (8). Of course, as Q 0 and Q 1 are prescribed to be the volume measure, we see that

H(Q|R) = 1 2a E Q [0,1] | v Q t | 2 dt = 1 2a E Q [0,1] | v Q t | 2 dt,
strengthening the analogy with (8).

Bredinger's problem. Recall that the dynamical version of the Schrödinger problem amounts to minimize the relative entropy

H(Q|R) → min; Q ∈ M(Ω) : Q 0 = µ 0 , Q 1 = µ 1 (15)
among all the path measures Q such that the initial and final marginals Q 0 and Q 1 are prescribed to be equal to given measures µ 0 and µ 1 ∈ M(X ). For more details on this convex optimization problem see [START_REF] Föllmer | Random fields and diffusion processes[END_REF][START_REF] Föllmer | Entropy minimization and Schrödinger processes in infinite dimensions[END_REF][START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF]. As Problem ( 14) is an hybrid of Brenier's problem (8) and Schrödinger's problem (15), we call it the Bredinger problem. We introduce the following extension of (15):

H(Q|R) → min; Q ∈ M(Ω) : (Q t = µ t , ∀t ∈ T ), Q 01 = π (Bdg)
with T a measurable subset of [0, 1] and (µ t ) t∈T a prescribed set of nonnegative measures on X . It is a slight extension of (14) where the state space X and the prescribed marginals (µ t ) t∈T are general. We still call the extension (Bdg) of ( 14) the Bredinger problem.

Of course, for this problem to admit a solution, it is necessary that H(µ t |vol) < ∞ for all t ∈ T , H(π|R 01 ) < ∞ (recall (3)) and

π 0 := π(• × X ) = µ 0 , π 1 := π(X × •) = µ 1 .
Proposition 1.2. Problem (Bdg) admits a solution if and only if there exists some Q ∈ M(Ω) such that Q t = µ t for all t ∈ T , Q 01 = π and H(Q|R) < ∞. In this case, the solution P is unique.

Remark 1.3. It can be checked without difficulty that this result is also valid when the state space X is a stochastically complete Riemannian manifold with smooth boundary, R is the reversible Brownian path measure with initial marginal R 0 = vol.

Fundamental example on the torus. As a basic important example, we take R ∈ P(Ω) the Wiener measure on the flat torus X = T n with initial marginal R 0 = vol : the normalized volume measure (so that R is reversible), µ t = vol for all t ∈ T = [0, 1] and π any bi-stochastic measure, i.e. such that π 0 = π 1 = vol. In this setting, (Bdg) becomes ( 14) which is as close as possible to Brenier's problem (8).

It is proved in Corollary 5.2 that in this precise setting, Bredinger's problem (14) admits a solution if and only if π is such that H(π|R 01 ) < ∞.

Remark 1.4. In the context of Remark 1.3, the fundamental example extends to a compact manifold X on which acts transitively a compact group of isometries with bi-invariant metric.

A simplified problem. It will be convenient to consider the easy version of the Bredinger problem (Bdg) with a finite set T = {t 1 , . . . , t K }:

H(Q|R) → min; Q ∈ M(Ω) : (Q t k = µ t k , 0 < t 1 < • • • < t K < 1), Q 01 = π (16)
where only a finite number of marginal constraints are prescribed.

Duality

The duality of Brenier's problem was studied in [START_REF] Brenier | The dual least action problem for an ideal, incompressible fluid[END_REF]. The present section is devoted to the dual problem of Bredinger's problem. In contrast with Brenier's problem which is affine, the strict convexity of Bredinger's problem allows for a rather standard treatment based on general convex analysis in infinite dimensional spaces stated below at Theorem 2.4. The main results of the section are the dual equality of Proposition 2.3 holding under weak hypotheses, and a characterization of the solution of Bredinger problem stated at Corollary 2.7, valid under restrictive regularity assumptions.

The dual problem of (Bdg) is stated at (19). Its unknown are a pressure scalar field p : [0, 1]×X → R in duality with the incompressibility constraint and a function η : X 2 → R in duality with the endpoint constraint π.

Dual equality. Let us denote for all

x ∈ X , R x := R(• | X 0 = x) ∈ P(Ω) and R µ 0 = X R x (•) µ 0 (dx) ∈ M(Ω).
They describe respectively the reference kinematics starting from x or from the initial distribution

R µ 0 0 = µ 0 . By (2) with f = X 0 , H(Q|R) = H(µ 0 |R 0 ) + H(Q|R µ 0 ) (17)
for all Q ∈ M(Ω) such that Q 0 = µ 0 . Therefore, as soon as H(µ 0 |R 0 ) < ∞ (this is verified when (Bdg) admits a solution), it is equivalent to solve the modified Bredinger problem

H(Q|R µ 0 ) → min; Q ∈ M(Ω) : Q t = µ t , t ∈ T , Q 01 = π, (Bdg µ 0 )
or (Bdg). The problems (Bdg) and (Bdg µ 0 ) share the same solution but their values differ from the quantity H(µ 0 |R 0 ) that only depends on the prescribed data R and µ. As far as duality is concerned, it will be a little bit more comfortable to consider (Bdg µ 0 ) rather than (Bdg). Take α a probability measure on T and consider the following weakening of (Bdg):

H(P |R) → min; P ∈ M(Ω) : (P t = µ t , for α-almost all t ∈ T ), Q 01 = π. ( 18
)
For instance, one may take α = K -1 1≤k≤K δ t k for the Bredinger problem (16). Choosing α = Leb [0,1] leads to Q t = µ t , for almost all t ∈ [0, 1] which is a slight weakening of Q t = µ t , ∀t ∈ [0, 1] in the original Bredinger problem (14). Nevertheless, the following result holds.

Lemma 2.1. Assume that α has a full support, i.e. supp α = T , and consider the following statements:

(i) (Bdg) admits a solution; (ii) t ∈ T → µ t ∈ P(X ) is weakly continuous on T ; (iii) (Bdg) is equivalent to (18). We have: (i) =⇒ (ii) =⇒ (iii).

Proof. Since for any P ∈ M(Ω), t ∈ T → P t ∈ P(X ) is weakly continuous (this follows from the continuity of the sample paths), it is necessary for (Bdg) to admit a solution such that t → µ t is also weakly continuous. In such a case, under the assumption that supp α = T , the constraint (P t = µ t , for α-almost all t ∈ T ) is equivalent to (P t = µ t , ∀t ∈ T ).

For the moment, it is assumed that the constraint µ = (µ t ) t∈T is a flow of probability measures and π ∈ P(X 2 ) is also a probability measure. In particular, this implies that (Bdg) is

H(Q|R) → min; Q ∈ P(Ω) : (Q t = µ t , ∀t ∈ T ), Q 01 = π
where Q lives in P(Ω) rather than in M(Ω). The aim of this section is to prove a dual equality for (Bdg).

Before stating it at Proposition 2.3, we must introduce some notions and notation. The value function of problem (Bdg µ 0 ) is denoted by

J(µ, π) := inf (Bdg µ 0 ) = inf {H(Q|R µ 0 ); Q ∈ P(Ω) : Q t = µ t , ∀t ∈ T , Q 01 = π} .
We denote respectively B(T ×X ) and B(X 2 ) the spaces of bounded measurable functions of T ×X and X 2 . Proposition 2.3. For any π ∈ P(X 2 ), µ ∈ P(X ) T and α ∈ P(T ) satisfying the Hypotheses 2.2 , we have

inf (Bdg) = H(µ 0 |R 0 ) + J(µ, π).
Moreover, for any class of functions A dense in B(T ×X ) × B(X 2 ) with respect to the pointwise convergence, the dual equality is

J(µ, π) = sup (p,η)∈A p, µ α + η, π - X log E R x exp T p(t, X t ) α(dt) + η(x, X 1 ) µ 0 (dx) , (19)
where we denote µ α (dtdx) := α(dt)µ t (dx).

Of course, the first identity is a direct consequence of (17). The remainder of the present section is devoted to the proof of the dual equality.

An abstract duality result. We begin stating an already known abstract result about convex duality. We shall apply it later to derive the dual problem of (Bdg) and some basic related relations. In these lines, we rely on general results about convex duality as presented for instance in the lecture notes [Léo]. Let U and V be two Hausdorff locally convex topological vector spaces with respective topological dual spaces U and V . Let us consider the following minimization problem

I( ) → min; ∈ U : T = k o (P)
where I is a convex (-∞, ∞]-valued function on U , T : U → V is a linear operator and k o ∈ V . We assume that the algebraic adjoint operator T * of T satisfies T * (V ) ⊂ U, so that one can write

[T * v]( ) = T * v, U,U = v, T V,V . It follows that the diagram U , U T *     T V , V is meaningful. The associated dual problem is v, k o -I * (T * v) → max; v ∈ V (D)
where

I * (u) := sup ∈U { , u -I( )} , u ∈ U
is the convex conjugate of I with respect to the duality U, U .

Theorem 2.4. Let us suppose that the following assumptions on I and T hold: (i) I is a convex σ(U , U )-lower semicontinuous function such that inf I > -∞;

(ii) there exists an open neighbourhood N of 0 in U such that sup u∈N I * (u) < +∞;

(iii) T * V ⊂ U. Then, the following assertions are verified.

(1) If inf(P) < +∞, the primal problem (P) admits at least a solution and if I is strictly convex, this solution is unique.

(2) The dual equality inf(P) = sup(D) holds. That is

inf{I( ); ∈ U : T = k o } = sup v∈V k o , v -I * (T * v) ∈ (-∞, +∞].
(3) The primal and dual problems are attained at ¯ and v respectively if and only if the following relations hold:

T ¯ = k o , ¯ ∈ ∂I * (T * v), where
∂I * (T * v) := ∈ U ; I * (T * v + w) ≥ I * (T * v) + , w U ,U , ∀w ∈ U denotes the subdifferential of I * at T * v.
Proof of Proposition 2.3. We apply Theorem 2.4 to the primal problem (Bdg). We first chose a relevant set of vector spaces and objective functions U, U * , I and I * . Then, we look at the constraint operators T and T * . Finally, Theorem 2.4 is applied in this setting.

The objective functions I and I * . Let us denote B(Ω) the space of all bounded measurable functions on Ω and equip it with the uniform norm u := sup ω∈Ω |u(ω)|, u ∈ B(Ω). Its topological dual space is denoted by B(Ω) . The convex function

Θ(u) := X log(E R x e u ) µ 0 (dx), u ∈ B(Ω) (20)
is well defined on B(Ω) because for any u ∈ B(Ω)

| log E R x e u | ≤ u , ∀x ∈ X , (21)
implying

|Θ(u)| ≤ u < ∞. (22)
Comparing (Bdg) with (P) and taking Lemma 2.5 below into account, we see that a good framework to work with is: (U, • ) = (B(Ω), • ) and

I(Q) := Θ * (Q) = sup u∈B(Ω) Q, u - X log(E R x e u ) µ 0 (dx) , Q ∈ B(Ω) , the convex conjugate of Θ. Lemma 2.5. For any Q ∈ B(Ω) , I(Q) = H(Q|R µ 0 ), if Q ∈ P(Ω) and Q 0 = µ 0 , +∞, otherwise.
Proof. This proof follows the line of the proof of [Léo12a, Lemma 4

.2]. Let Q ∈ B(Ω) be such that I(Q) < ∞. (a) Let us show that Q ≥ 0. Take u ∈ B(Ω), u ≥ 0.
As for all a ≤ 0, Θ(au) ≤ 0, we get

I(Q) ≥ sup a≤0 {a Q, u -Θ(au)} ≥ sup a≤0 a Q, u = 0, if Q, u ≥ 0 +∞, otherwise.
Hence, I(Q) < ∞ implies that Q, u ≥ 0, for all u ≥ 0, which is the desired result. (b) Let us show that Q is a positive measure. For a positive element of B(Ω) to be a measure it is sufficient (and necessary) that it is σ-additive. This means that for any decreasing sequence (u n ) n≥0 of measurable bounded functions such that lim n≥0 u n (ω) = 0 for all ω ∈ Ω, we have lim n→∞ Q, u n = 0.

Let (u n ) n≥0 be such a sequence. By dominated convergence, for all a ≥ 0, we obtain lim n→∞ Θ(au n ) = 0. Therefore,

I(Q) ≥ sup a≥0 lim sup n→∞ {a Q, u n -Θ(au n )} = sup a≥0 a lim sup n→∞ Q, u n
and I(Q) < ∞ implies that lim sup n→∞ Q, u n ≤ 0. Since, we already know that Q ≥ 0, this gives the desired result:

lim n→∞ Q, u n = 0. (c) Let Q ∈ M(Ω). Taking u = f (X 0 ) in sup u gives I(Q) ≥ sup f f, Q 0 -µ 0 = 0, if Q 0 = µ 0 , ∞, otherwise.
Hence, I(Q) < ∞ implies that Q 0 = µ 0 ∈ P(X ) which in turns implies that Q is also a probability measure. In this case, Q = X Q x (•) µ 0 (dx) and

I(Q) = sup u X (E Q x u -log E R x e u ) µ 0 (dx) (i) = sup k≥1 X sup u x :|u x |≤k E Q x u x -log E R x e u x µ 0 (dx) = sup k≥1 X sup v:|v|≤k (E Q x v -log E R x e v ) µ 0 (dx) (ii) = X H(Q x |R x ) µ 0 (dx) (17) = H(Q|R µ 0 )
which is the announced result. Let us give some precisions about this series of identities. At (i), we used the notation u x for the restriction of u to Ω x := {X 0 = x} ⊂ Ω.

Note that the inversion of sup u and X is valid since any function u ∈ B(Ω) can be identified with a measurable kernel (u x ∈ B(Ω x ), x ∈ X ) by u = u X 0 . Identity (ii) follows from a standard variational representation of the relative entropy of probability measures (see [START_REF] Gozlan | Transport inequalities. A survey[END_REF]Appendix] for instance) combined with Beppo-Levi's monotone convergence theorem.

This completes the proof of the lemma.

Let us compute I * = Θ * * defined by

I * (u) := sup Q∈B(Ω) { Q, u -I(Q)} , u ∈ B(Ω).
As Θ is convex (by Hölder's inequality) and lower σ(B(Ω), B(Ω) )-semicontinuous (by Fatou's lemma, it is lower • -semicontinuous and since it is convex, it turns out to be weakly semicontinuous), it is equal to its convex biconjugate. This means that

I * = Θ * * = Θ.
The constraint operators T and T * . For any

Q ∈ B(Ω) , we define Q ∈ B(T ×X ) by Q, p = T Q, p(t, X t ) α(dt), p ∈ B(T ×X ).
Clearly, when Q belongs to P(Ω), Q is the measure defined by Q(dtdx) = α(dt)Q t (dx). Hence, defining µ α (dtdx) := µ t (dx)α(dt), we see that Q = µ α is equivalent to Q t = µ t for α-almost all t ∈ T . For any Q ∈ B(Ω) , we define Q 01 ∈ B(X 2 ) by: Q 01 , η = Q, η(X 0 , X 1 ) , ∀η ∈ B(X 2 ). Putting everything together, the constraint operator is defined by

T Q := ( Q, Q 01 ) ∈ B(T ×X ) × B(X 2 ) , Q ∈ B(Ω)
and the full constraint of (18) writes as

T Q = (µ α , π), Q ∈ B(Ω) .

It is time to identify the topological space

V as V = B(T ×X ) × B(X 2 ) equipped with the uniform norm • T×X ⊕ • X 2 , so that its topological space V = B(T ×X ) × B(X 2 ) contains M b (T ×X ) × M b (X 2 ). Let us compute the adjoint T * of T . For any Q ∈ B(Ω) , p ∈ B(T ×X ) and η ∈ B(X 2 ), Q, T * (p, η) = T Q, (p, η) = Q, p + Q 1 , η = Q, T p(t, X t ) α(dt) + η(X 0 , X 1 ) .
Consequently, we see that

T * (p, η) = T p(t, X t ) α(dt) + η(X 0 , X 1 ). ( 23
)
The dual problem. We gathered all the ingredients to see that the dual problem of (18) is

T×X p dµ α + X 2 η dπ - X log E R x exp T p(t, X t ) α(dt) + η(x, X 1 ) µ 0 (dx) → max; p ∈ B(T ×X ), η ∈ B(X 2 ). ( 24
)
Proof of Proposition 2.3. It remains to verify the assumptions of Theorem 2.4 to obtain

J(µ, π) = sup (p,η)∈B(T×X )×B(X 2 ) p, µ α + η, π - X log E R x exp T p(t, X t ) α(dt) + η(x, X 1 ) µ 0 (dx) .
(i) Being a convex conjugate, I = Θ * is convex and lower semicontinuous with respect to σ(B(Ω) , B(Ω)). (ii) We see with ( 22), that the function I * = Θ is such that on the unit ball N = {u ∈ B(Ω); u ≤ 1}, we have:

sup u: u ≤1 Θ(u) < ∞. (iii) It is clear that for any p in B(T ×X ) and η in B(X 2 ), T * (p, η) = T p(t, X t ) α(dt) + η(X 0 , X 1 ) is in B(Ω)
. So far we have proved Proposition 2.3 in the special case where A = B(T ×X ) × B(X 2 ). The extension to the case where A is pointwise dense in B(T ×X ) × B(X 2 ) follows easily from an approximation argument combined with the dominated convergence theorem using (21). This completes the proof of Proposition 2.3.

Remark 2.6. We have chosen a strong topology on U = B(Ω) to insure the estimate sup u∈N Θ(u) < ∞ at (ii) above. This explains why Lemma 2.5 is needed.

Regular solutions. As a byproduct of this proof, Theorem 2.4 leaves us with the following

Corollary 2.7. Assume that Hypothesis 2.2 holds.

(1) If inf (Bdg) < +∞, the primal problem (Bdg) admits a unique solution.

(2) Let P ∈ P(Ω), p ∈ B(T ×X ) and η ∈ B(X 2 ). Both the primal problem (Bdg) and the dual problem (24) are attained respectively at P and (p, η) if and only if the constraints P t = µ t , ∀t ∈ T and P 01 = π are satisfied and

P = dµ 0 dR 0 (X 0 ) exp η(X 0 , X 1 ) -Q(p, η)(X 0 ) + T p(t, X t ) α(dt) R (25)
where

Q(p, η)(x) := log E R x exp T p(t, X t ) α(dt) + η(x, X 1 ) , x ∈ X , R 0 -a.s. Proof. Statement (1)
Θ (u) = e u X R x (•) µ 0 (dx) E R x e u = exp (u -log E R X 0 e u ) X R x (•) µ 0 (dx) = exp (u -log E R X 0 e u ) R µ 0 = dµ 0 dR 0 (X 0 ) exp (u -log E R X 0 e u ) R.
We conclude with (23), i.e. T * v = T * (p, η) = T p(t, X t ) α(dt) + η(X 0 , X 1 ).

General shape of the solution

Assuming the dual attainment p ∈ B(T ×X ) and η ∈ B(X 2 ) as in Corollary 2.7-(2) is very restrictive. In general, even if the solution P writes as (25), p and η might be unbounded or even might take the value -∞ at some places.

It is proved in [BL] that, provided that the reference path measure R is Markov, the solution P to Bredinger's problem is a reciprocal path measure. A representation of the

| | s u 1 0 | | A B C Figure 1.
Radon-Nikodym derivative dP/dR is also obtained. In order to state these results below at Theorem 3.7, it is necessary to recall the definitions of Markov and reciprocal measures and also of additive functional. First of all, we need to make precise what a conditionable path measure is.

Definitions 3.1 (Conditionable path measure).

(1) A positive measure Q ∈ M(Ω) is called a path measure.

(2) The path measure

Q ∈ M(Ω) is said to be conditionable if for all t ∈ [0, 1], Q t is a σ-finite measure on X .
It is shown in [START_REF] Léonard | Some properties of path measures[END_REF] that for any conditionable path measure

Q ∈ M(Ω), the condi- tional expectation E Q (• | X T ) is well-defined for any T ⊂ [0, 1]
. This is the reason for this definition.

Definition 3.2 (Markov measure). A path measure Q on Ω is said to be Markov if it is conditionable and if for any t ∈ [0, 1] and for any events

A ∈ σ(X [0,t] ), B ∈ σ(X [t,1] ) Q(A ∩ B | X t ) = Q(A | X t )Q(B | X t ), Q-a.e.
This means that, knowing the present state X t , the future and past informations σ(X [t,1] ) and σ(X [0,t] ), are Q-independent. Definition 3.3 (Reciprocal measure). A path measure Q on Ω is called reciprocal if it is conditionable and for any times 0 ≤ s ≤ u ≤ 1 and any events

A ∈ σ(X [0,s] ), B ∈ σ(X [s,u] ), C ∈ σ(X [u,1] ),
see Figure 1,

Q(A ∩ B ∩ C | X s , X u ) = Q(A ∩ C | X s , X u )Q(B | X s , X u ) Q-a.e.
This property states that under Q, given the knowledge of the canonical process at both times s and u, the events "inside" [s, u] and those "outside" (s, u) are conditionally independent. It is clearly time-symmetric.

Remarks 3.4. We recall basic relations between the Markov and reciprocal properties. (a) Any Markov measure is reciprocal, but there are reciprocal measures that are not Markov. (b) For any reciprocal measure Q, the conditional path measures

Q(• | X 0 ) and Q(• | X 1 )
are Markov, Q-a.e.

Definition 3.5 (Additive functional). A measurable function

A [0,1] : Ω → [-∞, ∞) is said to an R-additive functional if for any finite partition k I k = [0, 1] of the time interval [0, 1] with intervals I k , we have A [0,1] = k A I k , R-a.e.
where for all k, A

I k = A k (X I k ) is σ(X I k )-measurable.
With some abuse of notation, we shall write:

A [0,1] = A = A(X [0,1] ) = k A(X I k ).
We are now ready to state Theorem 3.7 and its hypotheses.

Assumption 3.6. (Strong irreducibility). The reference measure R is Markov and it admits a transition density r defined by for all 0 ≤ s < t ≤ 1 by

R(X t ∈ dy | X s = x) := r(s, x; t, y) R t (dy), ∀x ∈ X , R s -a.e.
which is positive in the sense that r(s, x; t, y) > 0, ∀(x, y), R s ⊗ R t -a.e., ∀0 ≤ s < t ≤ 1.

Theorem 3.7. Under the above Assumptions 3.6, the following assertions hold true. (a) The solution P of (Bdg), if it exists, is reciprocal and

P = exp(A(X T ) + η(X 0 , X 1 )) R (26)
for some [-∞, ∞)-valued σ(X T )-measurable additive functional A(X T ) and some measurable function η :

X 2 → [-∞, ∞), with the convention that exp(-∞) = 0. (b) We consider the prescribed marginals µ t k ∈ P(X ), 1 ≤ k ≤ K at times 0 ≤ t 1 < • • • t K ≤ 1
and the prescribed endpoint marginal π ∈ P(X 2 ). If the Bredinger problem H(P |R) → min; P ∈ P(Ω) :

P t k = µ t k , 1 ≤ k ≤ K, P 01 = π, (27)
is such that inf (27) < ∞, its unique solution P is reciprocal and writes as

P = exp 1≤k≤K θ t k (X t k ) + η(X 0 , X 1 ) R (28)
for some measurable functions

θ t k : X → [-∞, ∞), 1 ≤ k ≤ K and some measurable function η : X 2 → [-∞, ∞).
Proof. See [BL].

Remark 3.8. Following Remark 1.3, it can be proved that Theorem 3.7 extends to a stochastically complete Riemannian manifold X with R the reversible Brownian path measure having marginal R 0 = vol.

We see that (25) has the desired shape (26) with

A(X T ) = T p(t, X t ) α(dt) (29)
and (29) holds true for the solution P of Bredinger problem under the hypotheses of Theorem 3.7-(b) where only finitely many time marginal constraints are considered.

Kinematics of regular solutions

In this section, the reference path measure R is the law of the reversible Brownian motion with diffusion coefficient a > 0 on X = T n or X = R n , see Definition 1.1. Definition 4.1 (Regular solution of Bredinger's problem). In view of (25) in Corollary 2.7 and (28) in Theorem 3.7, we say that (Bdg) admits a regular solution if it can be written as

P = exp η(X 0 , X 1 ) + s∈S θ s (X s ) + T p(t, X t ) dt R (30)
for some nice enough functions η : X 2 → R, p : T ×X → R where T ⊂ [0, 1] is a finite union of intervals and θ s with s running through a finite subset S = {s k ; 1 ≤ k ≤ K} ⊂ (0, 1). By "nice enough" it is meant that all the objects and equations built upon R, η, p and θ to be encountered in this section are meaningful or admit solutions and that the functions η, θ and p are such that for any x ∈ X , the function ψ x : [0, 1]×X → R specified by (34) below is well-defined, C 2 in space and piecewise C 1 in time.

Remarks 4.2. (a) The path measure P defined by (30) is the solution of a Bredinger problem of the form

H(Q|R) → min; Q ∈ M(Ω) : (Q t = µ t , ∀t ∈ S ∪ T ), Q 01 = π. (31)
(b) In the important case where the Bredinger problem is (14), i.e. the marginal constraint is the incompressibility condition: P t = vol for all t ∈ [0, 1], we know by Theorem 3.7 that the solution has the form (26): P = exp(A(X) + η(X 0 , X 1 )) R. But we did not succeed in proving that the additive functional writes as A(X) = [0,1] p(t, X t ) dt for some function p. (c) The expression (30) corresponds to a measure α(dt) = 1 {t∈T } dt + s∈S δ s (dt) in (25).

The kinematics of P . While the dynamics of P is specified by formula (30) which is expressed in terms of potentials p and θ, its kinematic description is specified by the velocity vector field v P appearing in the stochastic differential equation

dX t = v P t dt + dM P t , P -a.s.
where M P is a local P -martingale. We are going to calculate v P in terms of the potentials θ and p. This will permit us to establish some connection between P and the Navier-Stokes equation (9) in the special case where R is the Brownian path measure with diffusion constant a > 0.

Lemma 4.3. The velocity vector field is of the form v P t = aβ P t , P -a.s.

with β P predictable satisfying P -almost surely

β P t •dX t -a/2 |β P t | 2 dt = 1 {t∈S} θ t + p t dt + dψ X 0 t (X t ), ∀0 ≤ t ≤ 1, (33)
where we have set for any 0 ≤ t ≤ 1 and x ∈ X ,

ψ x (t, z) := log E R exp η(x, X 1 ) + s∈S,s>t θ s (X s ) + T ∩(t,1] p(r, X r ) dr X t = z , R t -a.s. (34)
Note that for t = 1, we have

ψ x (1, •) = η(x, •).
The identity (33) is the keystone of the computation of v P . It relates kinetic terms on the left-hand side with dynamical terms on the right-hand side.

Proof. Our calculation of v P is done by confronting the Feynman-Kac type formula (30) with the expression

dP dR = dP 0 dR 0 (X 0 ) exp [0,1] β P t • dX t - a 2 [0,1] |β P t | 2 dt , P -a.s. ( 35
)
issued from Girsanov's theory, where β P is a predictable vector field. To perform this identification, two operations are required.

(i) We disintegrate R and P along their initial positions, i.e. R = R n R x (•) R 0 (dx) and P = R n P x (•) P 0 (dx), where for any x ∈ R n , we denote

Q x (•) = Q x (• | X 0 = x).
The main advantage of this disintegration is that it allows to work with the Markov measures P x , while P is only reciprocal, see [BL]. (ii) The expressions (30) and (35) of dP/dR are not enough. We shall need for each 0 ≤ t ≤ 1, formulas for the Radon-Nikodym density

dP x [0,t] dR x [0,t] = E R x dP x dR x | X [0,t] (36)
of the restrictions to the σ-field σ(X [0,t] ). As regards (i), denoting b x the drift field of the Markov measure P for some predictable process β x . On the other hand, with (30), (36) and the Markov property of R, we obtain dP

x [0,t] dR x [0,t] = exp s∈S,s≤t θ s (X s ) + T ∩[0,t] p(r, X r ) dr + ψ x t (X t ) , R x -a.s.
where ψ is defined at (34). Comparing the differentials of the two expressions of dP x [0,t] /dR x [0,t] , we arrive at

β x t • dX t -a/2 |β x t | 2 dt = 1 {t∈S} θ t + 1 {t∈T } p t dt + dψ x t (X t ), ∀0 ≤ t < 1, P x -a.s. (38) which is (33).
Theorem 4.4. Let us assume that the dual parameters η, θ and p are such that P is regular in the sense of Definition 4.1. Then,

v P t (X [0,t] ) = v P t (X 0 , X t ) = a∇ψ X 0 t (X t ), ∀0 ≤ t ≤ 1, P -a.s.
, where ∇ψ X 0 t (X t ) stands for ∇ z ψ x (t, z) |x=X 0 ,z=Xt and for any x ∈ X , ψ x is given by (34). Moreover, for any x ∈ X , ψ x is a classical solution of the second-order Hamilton-Jacobi equation   

(∂ t + a∆/2)ψ + a |∇ψ| 2 /2 + 1 {t∈T } p (t, z) = 0, 0 ≤ t < 1, t ∈ S, z ∈ X , ψ(t, •) -ψ(t -, •) = -θ(t, •), t ∈ S, ψ(1, •) = η(x, •), t = 1. (39)
Proof. Let us work P x -almost surely with X 0 = x fixed and by means of (37), rewrite (38) as

dψ x t (X t ) = β x t • dM P x t + (a |β x t | 2 /2 -1 {t∈T } p t )
dt -1 {t∈S} θ t , P x -a.s. where Girsanov's theory ensures that dM P x t = dX t -b x t dt is the increment of a local P x -martingale. We see that t → ψ x t (X t ) is a P x -semimartingale. Our regularity assumption ensures that ψ x defined at (34) verifies the following Itô formula

dψ x t (X t ) = [ψ x t -ψ x t -](X t ) + ∇ψ x t (X t ) • dX t + ∂ t + a∆ψ x t /2 (X t ) dt, R x -a.s. = [ψ x t -ψ x t -](X t ) + ∇ψ x t (X t ) • dM P x t + ∂ t + a∆ψ x t /2 (X t ) dt + a|∇ψ x t | 2 (X t ) dt, P x -a.
s. The uniqueness of the decomposition of a semimartingale (Doob-Meyer theorem) allows for the identification of the previous two P x -almost sure expressions of dψ x t (X t ) and gives

β x t = ∇ψ x t (X t ) -1 {t∈S} θ t (X t ) = [ψ x t -ψ x t -](X t ) a|β x t | 2 /2 -1 {t∈T } p t (X t ) = ∂ t + a∆ψ x t /2 (X t ) + a|∇ψ x t | 2 (X t )
where the first and third equalities hold dtdP x -almost everywhere. The second one is valid P x -a.s. and we implicitly identified the semimartingale ψ x t (X t ) with its càdlàg modification. As ψ x is assumed to be regular for any x, we obtain

β P t (ω) = ∇ψ ω 0 t (ω t ), t ∈ [0, 1], ω ∈ Ω [ψ x t -ψ x t -](z) = -θ(t, z), t ∈ S, x, z ∈ X , (∂ t + a∆/2)ψ x (t, z) + a|∇ψ x | 2 (t, z)/2 + 1 {t∈T } p(t, z) = 0, t ∈ [0, 1) \ S, x, z ∈ X .
This completes the proof of the theorem.

Fluid evolution. We wish to relate the stochastic velocity field v P with some evolution equation looking like Navier-Stokes equation (9). It is well known in classical mechanics that taking the gradient of Hamilton-Jacobi equation leads to the second Hamilton equation (Newton's equation). Let us do it with the second order Hamilton-Jacobi equation (39). For the PDE part, when t ∈ S is not an instant of shock, denoting for any x ∈ X the forward velocity of P (

• | X 0 = x) by x v := v P (•|X 0 =x) = a∇ψ x , we obtain 0 = ∇[∂ t ψ x + a/2 |∇ψ x | 2 + a/2 ∆ψ x + p] = ∂ t ∇ψ x + a∇ψ x • ∇(∇ψ x ) + a/2 ∆∇ψ x + ∇p = (∂ t + x v • ∇)∇ψ x + ∆( x v )/2 + ∇p
and multiplying by a, we see that

     (∂ t + x v • ∇)( x v ) = -a∆( x v )/2 -∇p , t < 1, t ∈ S, x v t - x v t -= -∇θ t , t ∈ S, x v 1 = ∇ y η(x, •), t = 1,
with p = ap. The left-hand side is the convective acceleration D t (

x v ) as in Navier-Stokes equation, but besides the gradient -∇p of a pressure in the right-hand side, we have -a/2 ∆ x v with the wrong sign. v P t (X t , X 1 ) only depend on the states X 0 , X t , X 1 and can be considered simultaneously in a sum or a difference without assuming the knowledge of the whole sample path. Let us emphasize for future use the identity

v P t (X t , X 1 ) = -[ v P * 1-t • X * ](X t , X 1 ), 0 ≤ t ≤ 1.
Since R is assumed to be reversible, i.e. R = R * , we see that dP * /dR = (dP/dR * ) • X * = (dP/dR) • X * and we obtain with (30) that P * = exp η * (X 0 , X 1 ) + Therefore, setting ϕ y (t, •) = -ξ y (1 -t, •), we obtain

X 1 v t (X t ) = a∇ϕ X 1 t (X t ) (44)
with ϕ y solution of (43). Taking the gradient of this equation and multiplying by a, we see that for any y ∈ X , y v solves (41).

Of course the marginal constraint cannot be verified by the velocities x v and y v , since they start or arrive at a Dirac mass. One must consider averages of these fields as in (40) to recover this constraint. For any smooth bounded functions, we have

In particular, taking α = 1/2 leads to (42) and completes the proof of the proposition.

Remark 4.6 (The pressure does not depend on the final position y). It is an important consequence of Theorem 3.7 that the pressure p and the potential θ do not depend on the final position X 1 . The only explicit appearance of X 1 is in the function η. Consequently, the pressure p in the Burgers equation (41) only depends on the actual position. This the constraints [Q t = vol, ∀0 ≤ t ≤ 1], Q 01 = π and such that H(Q|R) < ∞. The path measure of interest is

Q = X 3 R(• | X 0 = x, X 1/2 = z, X 1 = y) γ(dxdzdy)
with γ(dxdzdy) = π(dxdy)vol(dz) in P(X 3 ).

Proposition

R(• | X 0 = x, X 1/2 = z, X 1 = y) = R(X [0,1/2] ∈ • | X 0 = x, X 1/2 = z)R(X [1/2,1] ∈ • | X 1/2 = z, X 1 = y).
Let us check that Q satisfies the announced constraints.

We have Q 01 = π since for any measurable subsets A and B of X ,

Q 01 (A × B) = Q(X 0 ∈ A, X 1 ∈ B) = X 3 R(X 0 ∈ A | X 0 = x, X 1/2 = z)R(X 1 ∈ B | X 1/2 = z, X 1 = y) γ(dxdzdy) = X 3 1 x∈A 1 y∈B γ(dxdzdy) = γ(A × X × B) = vol(X )π(A × B) = π(A × B).
Let us show that for all 0 ≤ t ≤ 1, Q t = vol. Take 0 ≤ t ≤ 1/2 and denote R(X [0,1/2] ∈ • | X 0 = x, X 1/2 = z) = R x,z (•). Since π(• × X ) = π(X × •) = vol, we have γ(dxdz × X ) = π(dx × X )vol(dz) = vol(dx)vol(dz). Hence, for any measurable bounded function f on X , we have

X f dQ t = X 3 E R x,z [f (X t )] γ(dxdzdy) = X 2 E R x,z [f (X t )] vol(dx)vol(dz) (47) = X 2 E R 0,z-x [f (X t -x)] vol(dx)vol(dz) = X 2 E R 0,a [f (X t -x)] vol(dx)vol(da) = X E R 0,a X f (X t -x) vol(dx) vol(da) = X E R 0,a X f dvol vol(da) = X f dvol,
where the translation invariance of vol was used at the last but one equality. This shows that Q t = vol for all 0 ≤ t ≤ 1/2. A similar argument works for 1/2 ≤ t ≤ 1.

It remains to compute the entropy H(Q|R) to obtain a criterion of existence of a solution. Let us denote Q 0,1/2,1 (dxdzdy) := Q(X 0 ∈ dx, X 1/2 ∈ dz, X 1 ∈ dy) and Q xzy :=

  Hypotheses 2.2. (a) The constraint µ = (µ t ) t∈T is a flow of probability measures. (b) The mapping t ∈ T → µ t ∈ P(X ) is weakly continuous on T . (c) We choose α ∈ P(T ) such that supp α = T . It follows from Lemma 2.1 that we can assume Hypothesis 2.2-(b) almost without any loss of generality, so that (Bdg) and (18) are equivalent.

  is a direct application of Theorem 2.4-(1) and the only thing to be checked for (2) is the computation of the subdifferential ∂I * (T * v) of Theorem 2.4-(3) in the present setting. With I * = Θ given at (20), we obtain for any u ∈ B(Ω),

  x , since dX t = v P t dt + M P t , P -a.s. and for P 0 -almost all x in restriction to {X 0 = x} we have:dX t = b x t dt + dM P xThis shows that it is enough to obtain the drift b x of the Markov measure P x for any x.

				t	= v	P t dt + dM P t = v	P t dt + dM P x t , P x -a.s.,
	we see that v	P	has the special form
				v	P t = b X 0
	As regards (ii), Girsanov's representation (35) becomes for all 0 ≤ t ≤ 1,
			dP x [0,t] dR x [0,t]	= exp	0	t	β x r • dX r -	a 2	0	t	|β x r | 2 dr , P x -a.s.
	with
						b x t = aβ x t , P x -a.s.	(37)

t , for a.e. t, P -a.s.

  (t, X t ) dt R with η * (x, y) = η(y, x) for all x, y ∈ X , θ * s = θ 1-s for all s ∈ S * = {1 -s; s ∈ S} and p * (t, •) = p(1 -t, •) for all 0 ≤ t ≤ 1. Applying Theorem 4.4 to P * , we see thatv P * t (X [0,t] ) = a∇ξ X 0 t (X t ) with ξ y solution of    (∂ t + a∆/2)ξ + a|∇ξ| 2 /2 + p * = 0, t < 1, t ∈ S * , ξ(t, •) -ξ(t -, •) = -θ * (t, •), t ∈ S * , ξ(1, •) = η

	θ * s (X s ) +	p
	s∈S *	[0,1]

* * (y, •), t = 1.

  implying u, ∂ t µ = v t • ∇u + ∆u/2, µ t = v t • ∇u -∆u/2, µ t . It follows that for any 0 ≤ α ≤ 1, u, ∂ t µ t = v α

	R n	u d(P t -P 0 ) = E P	0	t	X 0 v s (X s ) • ∇u(X s ) +	1 2	∆u(X s ) ds
		= E P	0	t	v s (X s ) • ∇u(X s ) +	1 2	∆u(X s ) ds =	0	t	ds	X	[ v s • ∇u +	1 2	∆u](z) P s (dz)
	and
	R n	u d(P 1 -P t ) = E P	t	1	X 1 v s (X s ) • ∇u(X s ) -	1 2	∆u(X s ) ds
		= E P	t	1	v s (X s ) • ∇u(X s ) -	1 2	∆u(X s ) ds =	t	1	ds	X	[ v s • ∇u -	1 2	∆u](z) P s (dz)

t •∇u + a(1/2 -α)∆u, µ t which is equivalent to ∂ t µ + ∇ • (µv α ) = a(1/2 -α)∆µ.

  5.1. The path measure Q satisfies the constraints[Q t = vol, ∀0 ≤ t ≤ 1] and Q 01 = π. If H(π|R 01 ) < ∞, then H(Q|R) < ∞.Corollary 5.2. The entropy minimization problem (H T n ) admits a unique solution if and only if H(π|R 01 ) < ∞.Proof of Corollary 5.2. If H(π|R 01 ) < ∞, by Proposition 5.1 we have inf (Bdg) ≤ H(Q|R) < ∞ and we conclude with Corollary 2.7-(1) that (H T n ) admits a unique solution. Conversely, when (H T n ) admits a solution P , we haveH(π|R 01 ) = H((X 0 , X 1 ) # P |(X 0 , X 1 ) # R) ≤ H(P |R) < ∞.Proof of Proposition 5.1. As R is Markov, it satisfies

Acknowledgements

The second and the fourth authors were partially supported by the FCT Portuguese project PTDC/MAT-STA/0975/2014.

The forward velocity v P of P does not fulfill our hopes. But we are going to see that its backward velocity v P does. Recall (11) and (12) for the definitions of these velocities. Let us introduce

and for any 0 ≤ α ≤ 1, v α t (z) := E P (1 -α)

with

the average forward and backward velocities. In particular, α = 1/2 corresponds to the current velocity

. The reason for calling v α=1/2 the current velocity is that, among all the v α 's, it is the only one satisfying the continuity equation (42) below, see (45).

Theorem 4.5. For any y ∈ X , the backward velocity field

with p = ap and η y = η(•, y). On the other hand, the current velocity v cu satisfies the continuity equation

Moreover, y v t (z) = a∇ϕ y t (z), t ∈ S, where ϕ y solves the Hamilton-Jacobi-Bellman equation

The first equation of the system (41) is the desired Newton part of the Navier-Stokes equation (Burgers equation) with the right positive sign in front of the viscous force term: a∆ y v /2, see (9). The continuity equation ( 42) is the analogue of ∇•v = 0 in (9) which corresponds to the case µ ≡ 1.

Proof. Introducing the time-reversed

). On the other hand, as P is reciprocal, so is P * . Consequently, the forward and backward velocities v

means that all the fluid particles are submitted to the same pressure field ∇p regardless of their final positions y. A similar remark is valid for the shock potential θ.

Remark 4.7 (A mixture of flows tagged by their final positions). The solution P of Bredinger's problem is well described as the statistical mixture

where

. This velocity field is completely specified by (43) where the endpoint target y only occurs in the initial condition via the function -η(•, y). Formula (46) is a superposition principle. Each particle ending at y is subject to the gradient backward velocity field y v solving the Burgers equation (41) and the volume constraint P t = µ t , ∀t ∈ S ∪ T , (recall (31)) is recovered superposing all the flows tagged by their final positions, via formula (46). This superposition phenomenon is very reminiscent of the structure of the multiphase vortex sheets model encountered in [START_REF] Brenier | A homogenized model for vortex sheets[END_REF].

Remark 4.8 (The average velocity is not a gradient). The incompressibility constraint applied to a gradient velocity field v = ∇θ on the torus T n reads as 0 = ∇v = ∇•∇θ = ∆θ. But this implies that v vanishes everywhere. This is the reason why knowing that the average velocity is not a gradient leaves some room in our model. We know with (44) that y v t (z) = ∇ z ϕ y (z) is a gradient field. Consequently, the average backward velocity writes as

and we see that the dependence on z of P tz 1 := P (X 1 ∈ dy | X t = z) prevents us from identifying v t (z) with ∇ z [ ϕ y t (z) P tz 1 (dy)]. Introducing the average potential

we obtain

Existence of a solution on T n

We are going to prove a sufficient condition of existence of a solution of the Bredinger problem (14) in the special important case where the reference path measure R ∈ P(Ω) is the reversible Brownian motion on the the flat torus X = T n and µ t = vol, for all t. We refer to this problem as

It is an adaptation of a result in [START_REF] Brenier | The least action principle and the related concept of generalized flows for incompressible perfect fluids[END_REF] of existence of a generalized incompressible flow in T n . The specific property of the reversible Brownian motion R is the translation invariance

Combined with the translation invariance of vol (which is implied by (47)), this will lead us to the desired result. All we have to find is some path measure Q ∈ P(Ω) which satisfies

We have

= H(γ|R 0,1/2,1 )

The factorization property of the entropy is invoked at the equalities (i) and (iii). The identity (ii) is a consequence of is continuous in (x, y). As T n is compact, we have proved (48). This completes the proof of the proposition.