Thibault Bonnemain

Denis Ullmo

Mean Field Games in the weak noise limit : A WKB approach to the Fokker-Planck equation

Motivated by the study of a Mean Field Game toy model called the "seminar problem", we consider the Fokker-Planck equation in the small noise regime for a specific drift field. This gives us the opportunity to discuss the application to diffusion problem of the WKB approach "à la Maslov [8]", making it possible to solve directly the time dependent problem in an especially transparent way.

Introduction

Mean Field Games [START_REF] Lasry | Jeux à champ moyen. I -le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II -horizon fini et contrôle optimal[END_REF][START_REF] Huang | Large population stochastic dynamic games: closed-loop McKean-vlasov systems and the nash certainty equivalence principle[END_REF] are characterized by the coupling between a forward diffusion process for a density m(x, t) of agents with state variable x ∈ R n at time t, and a backward optimization process characterized by a value function u(x, t). In the simple case of quadratic mean field games [START_REF] Ullmo | Quadratic mean field games[END_REF] this takes the form of a system of coupled (forward) Fokker-Planck and (backward) Hamilton-Jacobi-Bellman equations ∂ t m(x, t) + ∇ ∇ ∇.(m(x, t)a a a(x, t)) -σ 2 2 ∆ m(x, t) = 0 FP , (1)

∂ t u(x, t) - 1 2µ ∇ ∇ ∇u(x, t) 2 + σ 2 2 ∆ u(x, t) = V [m t ](x) HJB , (2)
with initial and final conditions m(•, t = 0) = m 0 (•), u(•, T ) = c T (•). The coupling between the two PDE's is provided by the right hand side of Eq. ( 2) which involves the functional of the density m at time t, V [m t ](x), (which may also have an explicit dependence in x), and by the fact that the drift velocity in Eq. ( 1) is given in term of the gradient of the value function as a(x, t) = -1 µ ∇u(x, t). In the noiseless limit σ = 0, this system of equations reduces to a transport equation coupled to a Hamilton-Jacobi equation, both of which we associate with the classical dynamics of point particles. This limit is therefore rather intuitive, and in some respects simpler to analyze than the noisy regime. It turns out however that in many circumstances this limit is ill defined, which implies that it is mandatory to include a small but non zero noise. In that case, what one needs to analyze is the small (but non-zero) σ limit of the system Eqs. ( 1)- [START_REF] Cohen | A ray method for the asymptotic solution of the diffusion equation[END_REF], which quite naturally one would wish to study in terms of "classical trajectories" to make contact with the intuitive description one has in mind for the σ = 0 limit.

To avoid any misunderstanding, we stress right away that in this paper we will provide only a very modest step toward the solution of this general problem. To start with we will limit ourselves to the analysis of the particular case of a specific Mean Field Game toy model, the "seminar problem", introduced by Guéant and co worker in [START_REF] Guéant | Mean field games and applications[END_REF], and analyzed in some details in [START_REF] Swiecicki | phase diagram" of a mean field game[END_REF]. This Mean Field Game problem consists in finding the effective starting time of a seminar, fixed by a quorum condition, when all the participants try to optimize their behavior to avoid arriving too late or too early. The "state variable" x is therefore one dimensional, and corresponds simply to the physical space in which the motion of the agents takes place (the corridor leading to the seminar room) modeled as the negative real line x ∈] -∞, 0]. Absorbing boundary conditions {∀t, m(t, x = 0) = 0} are assumed since no one is expected to exit the seminar room before the beginning of the talk. Furthermore, the functional V [m t ](x) is taken uniformly zero, and the coupling between the HJB equation and the density of agents is just provided by a quorum condition on the number of agents in the room at the beginning of the seminar.

In the weak noise regime, it is shown in ref. [START_REF] Swiecicki | phase diagram" of a mean field game[END_REF] that this problem is associated with the drift field shown in Fig. 1, and reading to leading order

a(t, x) =          a (0) for x -a (0) (T -t) -x (T -t)
for -a (0) (T -t) x -a (2) (T -t) a (2) for -a (2) (T -t) x 0

, ( 3
)
where a (0) > a (2) are two constant drift velocities.

Figure 1: Regions of the (t, x) space, where T = 2 is the time when the seminar effectively begins, and their associated optimal drift a(t, x). In regions (0) and (2) the drift stays constant and is denoted respectively a (0) and a (2) (here 5 and 2). In region (1),the drift is linear in x.

The (admittedly limited) goal of this paper will therefore be to analyze the Fokker-Planck equation for this velocity field in the small σ regime, and to show that we can provide a very precise solution of this problem based just on the "classical trajectories" for a dynamics closely related to (but slightly different from) the σ = 0 limit of Eq. ( 1).

The fact that this can be achieved for the Fokker-Planck equation can be seen readily by multiplying Eq. ( 1) by σ 2 , and noting that it then has the structure of what Maslov [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF] has termed a "λ-pseudo differential operator", in the sense that each partial derivative is associated with a factor λ -1 ≡ σ 2 . This implies that a "semiclassical approximation" scheme can be applied to this equation in small σ 2 limit. This fact has of course been recognized for many years, and led to some publications [START_REF] Risken | The Fokker-Planck Equation[END_REF][START_REF] Shizgal | A uniform wkb approach to electron thermalization in gases[END_REF][START_REF] Stanislav | Fokker-planck equation approach to the description of soliton statistics in optical fiber transmission systems[END_REF]. Most of them, however, use a rather indirect approach, making use of transformation of variable to a form more directly related to the Schrödinger equation and through a normal mode decomposition (cf eg [START_REF] Caroli | Diffusion in a bistable potential: A systematic wkb treatment[END_REF] on the example of a diffusion in bi-stable potentials). We follow however here the philosophy of the ray method introduced in [START_REF] Cohen | A ray method for the asymptotic solution of the diffusion equation[END_REF].

Our goal will thus be to to show that a direct approach where the time-dependent WKB approximation is applied directly on Eq. ( 1) can be used effectively to obtain a extremely good approximation for the solution of the Fokker-Planck equation Eq. ( 1) with the drift field (3). We address thus here only the first (and simplest) step of the analysis of the coupled MFG equations system, and furthermore do this on a specific illustrative case. This gives us however the opportunity to discuss the application of the WKB approach in the perspective developed by Maslov [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF], in a way which is maybe a bit more transparent that what can be found in the literature [START_REF] Cohen | A ray method for the asymptotic solution of the diffusion equation[END_REF], and leads in our view to a rather intuitive interpretation.

The paper will be organized as follows. In section 2, we will give without justification the recipe for the construction of the WKB approximation. For the sake of clarity this will be done for a one dimensional problem, and we will assume that the initial density m 0 (x) is a Gaussian. Section 3 will then provide a derivation of these WKB expressions, together with a generalization to higher dimensionality and to a larger class of initial densities. Readers with little interest in these formal issues may skip that section and go directly to section 4 where the WKB approximation is applied to two simple examples where it turns out to provide the exact solution, as well as to the case corresponding to the drift field Eq. ( 3). Finally, we conclude in section 5, and, for self-containedness, briefly sketch two rather standard derivations in appendices AppendixB and AppendixC.

WKB approximation of a 1d Fokker-Planck equation

In this section, we provide, without any demonstration, the prescription for the construction of the WKB solution of the Fokker-Planck equation Eq. ( 1) in the small σ regime. We limit ourselves here to the one-dimensional case and to Gaussian initial densities

m 0 (x) = N exp - µ(x -x0 ) 2 2σ 2 , ( 4
)
where x0 is the center of the Gaussian and N = µ 2πσ 2 is a normalization factor. More general m 0 (x) could easily be considered (see section 3), but gaussians have an intrinsic interest, and, in addition, this also allows us to get the Green's function of the equation by reducing the width of the Gaussian to zero.

The semiclassical scheme follows three steps. The first one consists in constructing a Lagrangian symplectic manifold on which we can define an action. The second step uses this input to build the WKB approximation. Finally, we address how absorbing boundary conditions can be implemented in the semiclassical scheme.

Symplectic manifold and classical action

The Fokker-Planck equation (1) can be written as Lm = 0 where we have introduced the

λ-pseudo differential operator L ≡ [λ -1 ∂ t • +λ -1 ∂ x (a•) -1 2 (λ -1 ∂ x ) 2 •] (with again λ ≡ σ -2 assumed large). Using the usual mapping λ -1 ∂ x → p, λ -1 ∂ t → E, L can be associated with the classical symbol L(x, t; p, E) = E + pa(x, t) -p 2 /2 , ( 5
)
which, if understood as a classical Hamiltonian leads to the canonical equations

ṫ = ∂ E L = 1 Ė = -∂ t L = -p∂ t a ẋ = ∂ p L = a(t, x) -p ṗ = -∂ x L = -p∂ x a . ( 6
)
Now, consider the initial Gaussian distribution Eq. ( 4) for x0 and µ given. It can be written in the semiclassical form m 0 (x 0 ) = N exp [λS 0 (x 0 )] with

S 0 (x 0 ) ≡ -µ (x 0 -x0 ) 2 2 . ( 7
)
At any point of space x 0 , one can therefore initiate a classical trajectory at t = t 0 with an initial momentum p 0 (x 0 ) = ∇S 0 (x 0 ) = -µ(x 0 -x0 ) , [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF] and fulfilling the "compatibility condition"

L(x, t; p, E) ≡ 0 . ( 9
)
The reunion of all these trajectories obtained from these initial conditions and the canonical equations ( 6) form a 2-dimensional manifold M = {(t, x(t, x 0 ), E(t, x 0 )p(t, x 0 )} where p, x and E respectively represent the value taken by p, x and E after evolving on this manifold from r 0 = (t 0 , x 0 ; E 0 (x 0 ), p 0 (x 0 )) for a time t -t 0 . An example of such a manifold is illustrated in Figure 2. To the manifold M, we can now associate a classical action

S(t, x) ≡

[L:r0→r]⊂M pdx + Edt [START_REF] Risken | The Fokker-Planck Equation[END_REF] where r0 = (t 0 , x0 ; E = 0, p = 0) is the point on M above X0 = (t 0 , x0 ), and r ∈ M is the point above X = (t, x). We stress that, since M is a Lagrangian manifold, the integral in Eq. ( 10) can be taken on any path on M joining r0 to r. For instance, the action S(t, x) can be computed either as

S 1 (t, x) = x0(t,x) x0 p 0 (x )dx S0(x0) + t t0 (p(s, x 0 ) ẋ(s, x 0 ) + E(s, x 0 ))ds ,
in which x 0 (t, x) is the initial position of the trajectory arriving at x at time t, or as

S 2 (t, x) = t t0 (p(s, x0 ) ẋ(s, x0 ) + E(s, x0 ))ds + x x(t,x0) p(x , t)dx ,
with p(x, t) the momentum coordinate of the point of M above (t, x). Both expressions lead to the same result (i.e. S 1 (t, x) = S 2 (t, x) = S(t, x)). This is illustrated on Fig. 3.

For the Gaussian initial density we consider, the definition of the initial momentum given by (8) and the compatibility conditions L ≡ 0 impose that p(t, x0 ) = 0 and E(t, x0 ) = 0 for all time, yielding

S(t, x) = S 2 (t, x) = x x(t,x0) p(t, x )dx , ( 11
)
where the path of integration on the manifold is taken at constant time t from the point above x ≡ x(t, x0 ) (evolution of the center of the distribution x0 ) to the point above x. As a final comment, it is worth mentioning that, for more general initial conditions, S 0 (x 0 ) can be non-zero and should be added to the right-hand side of (10).

Semiclassical approximation for m(t, x)

With this definition of the action, the WKB approximation for the density of probability is expressed as

m s.c. (t, x) = N ∂ x0 x(t, x 0 ) exp λS(t, x) - 1 2 t t0 (∂ x a)dτ , ( 12
)
where in the prefactor, x(t, x 0 ) is the position of a trajectory started at x 0 at time t = t 0 (with thus a momentum p 0 (x 0 ) given by Eq. ( 8)), and the integral in the exponential is taken along this trajectory.

Except for the fact that the exponent is real rather than complex, the only difference with respect to the traditional WKB expressions derived in optics or in the context of the Schrödinger equation is the extra term 1 2 t t0 (∂ x a)dτ in the exponent, which can be tracked back to the non-symmetric ordering of the operators p ≡ λ -1 ∂ x and x ≡ ×x in the Fokker-Planck equation which makes, in particular, the operator L non hermitian. Introducing L0 the hermitian approximation of L in which the term λ -1 ∂ x (a.) has been replaced by its symmetric counterpart

(1/2)[λ -1 ∂ x (a •) + aλ -1 ∂ x (•)],
and L1 = L -L0 , the additional term in Eq. ( 12) can be understood as arising from the perturbative effect of L1 on L0 (cf. Appendix AppendixA).

Absorbing boundary conditions

We shall illustrate below this WKB approach with the problem corresponding to the drift velocity field Eq. ( 3), problem for which we assume an absorbing boundary condition at x = 0. As such absorbing boundary conditions are rather common, we discuss now how to implement them in our semiclassical scheme.

Let us consider the semiclassical solution of the free problem (ie without the boundary condition)

m free (t, x) = N ∂ x0 x(t, x 0 ) exp λS(t, x) - 1 2 t t0 (∂ x a)dτ . ( 13
)
For sake of simplicity, we assume that (as will be the case in the examples we are going to consider), the trajectories on which S(t, x) is constructed are reaching x = 0 with positive velocity. Consider now the compatibility condition Eq. ( 9) at x = 0, for an arbitrary time t, and with the choice

E = ∂ t S L(0, t; p, ∂ t S) = 0 . It admits two solutions ẋ = a(x, t) -p = ± a 2 + 2∂ t S . ( 14
)
The one corresponding to a positive velocity is just p + (t) = ∂ x S(t, x = 0). We can however generate another set of trajectories initiated at time t at x = 0 with momentum p -(t) = a + a 2 (t, 0) + 2∂ t S(t, 0) and energy E(t) = ∂ t S(t, 0). These trajectories have negative velocities and thus "bounce" off the boundary point x = 0.

A "reflected" density

m ref (t, x) = N ∂ x0 x(t, x 0 ) exp λ S(t, x) - 1 2 t t0 (∂ x a)dτ , (15)
can therefore be constructed in exactly the same way as before using the reflected trajectories x and reflected action S. At x = 0, m ref (t, 0) = m free (t, 0) since ∂ t S(t, 0) = ∂ t S(t, 0) = E(t, 0) is the same for both (and thus one should just impose S(t 0 , 0) = S(t 0 , 0) for an arbitrary time t 0 ), and ∂ x0 x(t, x 0 ) = ∂ x0 x(t, x 0 ) since at x = 0 only the momentum has changed but not the position. Therefore, the total density

m tot (t, x) = m free (t, x) -m ref (t, x) (16)
is a semiclassical solution to the Fokker-Planck equation (1) which fulfills the absorbing boundary condition m tot (t, 0) = 0.

Derivation and generalization

We provide now a derivation (and some generalization) of Eq. ( 12). Our approach is very similar in spirit to the "ray method" developed by Cohen and Lewis [START_REF] Cohen | A ray method for the asymptotic solution of the diffusion equation[END_REF], but follow more closely the WKB formalism developed by Maslov [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF], that we feel might be easier to access for physicists.

We therefore want to describe the evolution of an initial density (at t = t 0 ) which is in the "semiclassical form"

m 0 (x) = φ 0 (x) exp [λS 0 (x)] , (17)
with x ∈ R d . Such form includes Gaussian densities such as Eq. ( 4), but are significantly more general.

By writing (σ -2 ≡ λ), the Fokker-Planck equation reads in the more general case,

0 = λ -1 ∂ t m + λ -1 ∇(a(t, x)m) - 1 2 λ -2 ∆m = Lm , ( 18
)
which up to the i factors, looks very much like a λ-pseudo differential Maslov operator of symbol

L(x, t; p, E) = E + a(x, t)•p -p 2 /2 . ( 19
)
Following Maslov's derivation [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF], let us consider the ansatz

m(t, x) = φ(t, x) exp [λS(t, x)] , (20)
with φ(t 0 , x) = φ 0 (x) and S(t 0 , x) = S 0 (x).

Writing X ≡ (t, x), P ≡ (E, p), Eq. ( 18) becomes

L φ(X)e λS(X) = 0 = e λS(X) R 0 φ(X) + λ -1 R 1 φ(X) + O(λ -2 ) (21) with R 0 = L(X; ∂ X S) , ( 22
)
R 1 = ∂ P L(X; ∂ X S), ∂ X φ + 1 2 Tr ∂ 2 PP L(X; ∂ X S)∂ 2 XX S + Tr ∂ 2 XP L(X; ∂ X S) φ . ( 23
)
Neglecting terms of order λ -2 and higher, solving Eq. ( 18) amounts to solving R 0 = 0 and R 1 = 0.

3.1. R 0 = 0, Hamilton-Jacobi equation
The equation (R 0 = 0) can be rewritten as an Hamilton-Jacobi equation on S L(X;

∂ X S) = ∂ t S + a(t, x)•∇S - 1 2 (∇S) 2 = 0 , ( 24
)
with an initial condition at

t = t 0 S(t 0 , x 0 ) = S 0 (x 0 ) . ( 25
)
Solution of this kind of equations is typically obtained through the method of characteristics. Here this amounts to build a one parameter family of rays (t, x; E, p) x0 (s) ≡ r(s, x 0 ), indexed by x 0 , which follow -for a fictitious time s -the Hamilton dynamics associated with L:

ṫ = ∂ E L = 1 Ė = ∂ t L = -p•∂ t a ẋ = ∂ p L = a(t, x) -p ṗ = -∂ x L = -p•∂ x a , ( 26
)
with initial the conditions r(0, x 0 ) = (E 0 , t 0 , p 0 (x 0 ), x 0 )

L(r(0, x 0 )) = 0 (27) corresponding to p 0 (x 0 ) = ∂ x0 S 0 (x 0 ) . ( 28
)
Eq. ( 27) fixes E 0 and it is clear from Eqs. (26) that we can take s ≡ t -t 0 .

As stressed in the previous section, the family of rays defined by Eqs. ( 26)-( 27) form a Lagrangian manifold, thus, according to the method of characteristics (cf appendix AppendixB), the solution of Eq. (24) reads

S(t, x) = X X0 Edt + p•dx , ( 29
)
where the integral is taken on any path on the manifold starting above the point X0 = (t 0 , x0 ) such that S 0 (x 0 ) = 0 and ending on the point above X = (t, x).

R 1 = 0, transport equation

We begin by focusing on the first term of R 1 that we rewrite more explicitly using the canonical Hamilton-Jacobi equations

∂ P L(X, ∂ X S), ∂ X φ = ∂ t φ + (a(t, x) -∂ x S)∂ x φ = ∂ t φ + ẋ∂ x φ = Dφ Dt , ( 30
)
where D Dt represents the time derivative along the flow. This allows us to write the equation (R 1 = 0) as a simple evolution equation

Dφ Dt = - 1 2 Tr ∂ 2 PP L(X, ∂ X S)∂ 2 XX S + Tr ∂ 2 XP L φ . ( 31
)
To solve this equation we will make use of Liouville's formula, which states that for a dynamical system

dx dt = f (x) , ( 32
)
and for any (d-1)-parameter family of trajectories x(t, α) indexed by α ∈ R (d-1) , the determinant

J(t, α) ≡ det ∂x(t,α) ∂(t,α) fulfills D ln J Dt = Tr df dx (x(t, α)) . ( 33
)
(Elements of a demonstration are given in appendix AppendixC for the sake of completeness.) Using the canonical equations we have

Ẋ = ∂ P L . ( 34
)
Noting that we can write X ≡ (t, x(t, x 0 )) and having J denote det[∂ t,x0 X], Liouville's formula reads

D ln(J) Dt = Tr ∂ X (∂ P L) = Tr[∂ 2 PX L + ∂ 2 PP L∂ 2 XX S] . ( 35
)
Hence Eq. ( 31) becomes

Dφ Dt + 1 2 D Dt (ln J)φ = - 1 2 Tr[∂ 2 XP L]φ , (36)
and, multiplying both sides by √ J,

D Dt √ Jφ = - 1 2 Tr ∂ 2 XP L √ Jφ . ( 37
)
Finally, we have

φ(x(t, x 0 )) = J(x(t 0 , x 0 )) J(x(t, x 0 )) φ(x(t 0 , x 0 )) exp - 1 2 t t0 Tr ∂ 2 XP L dτ ( 38
)
where J(x(t 0 , x 0 )) = 1 and, for L given by Eq. ( 19), Tr[∂ 2 XP ] = div a. In 1d J would simply become ∂ x0 x, yielding the prefactor in Eq. [START_REF] Smirnov | A Course of Higher Mathematics[END_REF].

It is also worth noting that Eq. ( 31) can be solved in multiple ways, another possibility would be

D Dt Jφ = + 1 2 Tr ∂ 2 P 2 L • ∂ 2 X 2 S Jφ , ( 39
) implying φ(x(t, x 0 )) = J(x(t 0 , x 0 )) J(x(t, x 0 )) φ(x(t 0 , x 0 )) exp + 1 2 t t0 Tr ∂ 2 P 2 L • ∂ 2 X 2 S dτ . ( 40
)
Here J serves only as a prefactor, and either expressions can be used.

Application to the seminar problem

For a 1d problem, and writing λ -1 ≡ σ 2 , the semiclassical expression for m reads

m(t, x) = N ∂ x0 x exp S(t, x) σ 2 - 1 2 t 0 (∂ x a)dτ . ( 41
)
We will use this expression to study the different drift regimes (cf Eq. ( 3)) presented by the seminar problem for Gaussian initial condition at t = 0

m 0 (x) = N exp - µ(x 0 -x0 ) 2 2σ 2 = N exp S 0 (x 0 ) σ 2 , ( 42
)
to which through Eq. ( 8) we associate the one-parameter family of initial points in phase space r(x 0 ) = (t = 0, x 0 , E 0 (x 0 ), p 0 (x 0 )) (43)

corresponding to p 0 (x 0 ) = ∂ x0 S 0 (x 0 ) = -µ(x 0 -x0 ) , E 0 (x 0 ) = p 2 0 (x 0 ) 2 -p 0 (x 0 )a(x 0 , t = 0) .

Constant drift

Let us start with the simple case of a constant drift a (this would correspond to regions (0) or (2) in Fig. 1). In order to obtain the density as expressed in Eq. ( 41) there are two terms we first need to compute, the prefactor ∂ x0 x(t, x 0 ) and the action S(t, x). To do so we start from the canonical equation of motion

ṗ = -∂ x L = -p∂ x a = 0 ẋ = ∂ p L = a -p ( = const. along a trajectory) . ( 44
)
For the one-parameter family of trajectories Eq. ( 43), this leads to

p(t, x 0 ) = µ(x 0 -x 0 ) x(t, x 0 ) = x 0 + [a -p(t, x 0 )]t = x 0 (1 + tµ) + t(a -µx 0 ) . ( 45
)
The prefactor is then readily obtained as

∂ x0 x(t, x 0 ) = 1 + tµ . ( 46
)
The action is computed noticing that, along the "center of mass" trajectory x(t, x0 ), the momentum p(t, x0 ) and energy E(t, x0 ) remain identically zero. Hence, M = {(t, x(t, x 0 ), E(t, x 0 )p(t, x 0 )} being Lagrangian,

S(t, x) = x x(t,x0) p(t, x )dx , (47)
p(t, x) being the momentum of the point above (x, t) on M. Denoting x 0 (t, x) the initial position of a trajectory arriving at x at time t (i.e. such that x = x(t, x 0 )), the second equation of (45) gives

x 0 (t, x) = x -at + µx 0 t 1 + µt , ( 48
)
and the first one

p(t, x) = - µ 1 + µt (x -(x 0 + at)) . ( 49
)
After integration, this last expression yields,

S(t, x) = - µt 1 + µt (x -x0 -at) 2 2t . ( 50
)
Finally, using Eq. (41) we have

m(t, x) = µ 2πσ 2 1 √ 1 + tµ exp - µt 1 + µt (x -x0 -at) 2 2tσ 2 , ( 51
)
which turns out to be the exact expression for the evolution of a initial Gaussian density in the case of a constant drift. This is actually expected since going back to the derivation of the semiclassical approximation, we see that the terms neglected contain only second (or higher) order spatial derivative of a which are identically zero in the case of a constant drift.

If µ → ∞, m(0, x) → δ(x -x0 ), and

m(t, x) → G(t, x, x0 ) = 1 2πtσ 2 exp - (x -x0 -at) 2 2tσ 2 , ( 52
)
which indeed is the exact Green function of the Fokker-Planck equation for a constant drift.

Absorbing boundary condition at x = 0

To implement the absorbing boundary condition at x = 0, we follow the procedure discussed earlier in section 2.3 and construct the "reflected" action

S(t, x) = S(t, 0) + x 0 p -(t, x )dx , ( 53
)
where p -(t, x) is the reflected momentum.

To compute this quantity, let us note

t abs = x 0 µ(x 0 -x 0 ) -a
the time at which the trajectory initiated at x 0 reaches 0 (and is thus "absorbed"). Since velocity is constant on a given trajectory, we can express the velocity before the bounce as ẋ+ (x 0 ) = -x 0 /t abs and thus just after the bounce as ẋ-(x 0 ) = +x 0 /t abs . Eqs. (44) then give

p -(t > t abs , x 0 ) = a - x 0 t abs = 2a -µ(x 0 -x 0 ) , (54)
x(t > t abs , x 0 ) = x 0 t abs (t -t abs ) = -x 0 (1 + µt) -at + µx 0 t . ( 55
)
Defining x0 (t, x) the initial position of a trajectory arriving at x(t, x 0 ) = x after reflection at x = 0, we thus have from Eq. ( 55)

x0 (t, x) = µtx 0 -at -x 1 + µt , ( 56
)
which inserted into Eq. (54) gives

p -(t, x) = 2a - µt 1 + µt x + x0 + at t . ( 57
)
Performing the integral in Eq. ( 53), and noting that the lower bound cancels the term S(t, x = 0), we thus have S(t, x) = 2ax -µt 1 + µt

(x + x0 + at) 2 2t , ( 58
)
giving for the total (incident plus reflected) density

m tot (t, x) = µ 2πσ 2 1 √ 1 + tµ exp - µt 1 + µt (x -x0 -at) 2 2tσ 2 -exp 2ax σ 2 exp - µt 1 + µt (x + x0 + at) 2 2tσ 2 . ( 59
)
This fulfills the absorbing boundary conditions m tot (t, 0) = 0 and, for the same reason as above is an exact solution of the Fokker-Planck equation. Note however that for finite µ, the Gaussian Eq. ( 42) fulfills the boundary condition m 0 (x = 0) up to exponentially small terms, and thus conversely Eq. ( 59), which satisfies the boundary condition exactly, satisfies the initial condition m tot (t, 0) = m 0 (x) only up to exponentially small terms. As µ → ∞, however, the boundary condition is exactly met by m 0 and Eq. (59) yields the exact Green function of the Fokker-Planck equation.

Linear drift

We will now consider a linear drift a(x, t) = x/(t -T ), with T > t the time at which the seminar begins, associated with region (1) in Fig. 1. The canonical equations become

     ẋ = ∂ p L = a -p = x t -T -p ṗ = -∂ x L = -p∂ x a = - p t -T , ( 60
) giving        p(t, x 0 ) = p 0 (x 0 ) T T -t = µT (x 0 -x 0 ) T -t x(t, x 0 ) = x 0 (T -t) T -µt(x 0 -x 0 ) . ( 61
)
We thus have ∂x/∂x 0 = (T -t + µtT )/T , which together with

t 0 (∂ x a)dτ = log[(T -t)/T ] yields for the prefactor to N ∂ x0 x exp - 1 2 t 0 (∂ x a)dτ = µ 2πσ 2 T 2 (µtT + T -t)(T -t) . ( 62
)
Turning now to the action, we have from the second equation of (61),

x 0 (t, x) = (x -µtx 0 ) T T -t + µT t , ( 63
)
which, inserted into the first equation of (61) gives for the momentum p(t, x),

p(t, x) = µ T (x 0 (T -t) -T x) (T -t)(T -t + µT t) , ( 64
)
leading by integration to

S(t, x) = x x(t,x0) p(t, x )dx = µ(x 0 (t -T ) -T x) 2 2(T -t)(T -t + µT t) . ( 65
)
Using Eq. ( 41), and computing the reflected action S(t, x) following the same procedure as in Section 4.1, giving

S(t, x) = µ(xT + (T -t)x 0 ) 2 2(t -T )(T -t + µT t) , ( 66
)
we get for the evolution of a Gaussian initial density with a linear drift velocity and absorbing boundary conditions at x = 0

m(t, x) = µ 2πσ 2 T 2 (µtT -T -t)(T -t) exp µ(xT -(T -t)x 0 ) 2 2σ 2 (t -T )(T -t + µT t) -exp µ(xT + (T -t)x 0 ) 2 2σ 2 (t -T )(T -t + µT t) . ( 67
)
As µ → ∞ we recover the Green function of the corresponding Fokker-Planck equation

G(t, x, x0 ) = T 2πσ 2 t(T -t) exp - T (x -T -t T x0 ) 2 2σ 2 t(T -t) -exp - T (x + T -t T x0 ) 2 2σ 2 t(T -t) . ( 68
)
Again, because the second x derivative of the drift is zero, expressions (67) and (68) are exact.

Coupling the two solutions

We now consider the full problem corresponding to the drift field Eq. ( 3), taking into account the possibility that agents beginning in region (0) or (2) (associated with constant drifts a (0) and a (2) ) may leak into region (1) (associated with a linear drift a(x, t) = x/(t -T )), and reciprocally1 . We focus here on times t ≤ T and on the configuration where the agents start their diffusion in region [START_REF] Caroli | Diffusion in a bistable potential: A systematic wkb treatment[END_REF], which is the one of interest from the point of view of mean field games. Corresponding expressions for a group of agents initially located in region (2) are given in appendix AppendixD.

We begin by defining x * (n) (x 0 ), p * (n) (x 0 ) and t * (n) (x 0 ), (n = 0, 2), the position, impulsion and time at which a trajectory initiated at r(x 0 ) (cf Eq. ( 43)) crosses the boundary between regions ( 1) and (n). Using Eq. ( 61) together with the fact that the boundary is the x = a (n) (t -T ) straight line, we may write

x * (n) (x 0 ) = a (n) (t * (n) -T ) = x 0 (T -t * (n) ) T -µ(x 0 -x 0 )t * (n) (x 0 ) . ( 69
)
We then compute t * (n) by inverting this last equation and obtain p * (n) inserting this newly found t * (n) expression in Eq. ( 61)

       t * (n) (x 0 ) = T 1 - µT (x 0 -x 0 ) a (n) T + x 0 + µT (x 0 -x 0 ) p * (n) (x 0 ) = a (n) T + µT (x 0 -x 0 ) + x 0 T . ( 70
)
Before the crossing (t < t * (n) ) the agents do not feel the effects of the drift change, and their trajectories remain the same as in Eq. ( 61). In region (1), (x * (0) < x < x * (2) ), the prefactor is thus obtained as Eq. ( 62) and the action as Eq. ( 65). We will now focus on the expression of the density after the crossing, the complete solution being simply obtained by patching the linear and the leaking densities imposing continuity of the solution.

Using the canonical equations in the region in which the agents are leaking, we have for t > t * (n) ,

   x (n) (t, x 0 ) = x * (n) + (a (n) -p * (n) )(t -t * (n) ) = x 0 T -t T -tµ(x 0 -x 0 ) p (n) (t, x 0 ) = p * (n) . ( 71
)
Let x 0 (t, x) be the initial position of a trajectory arriving at x at time t (thus x n (t, x 0 ) = x), t * (n) (t, x) the time at which this trajectory crosses the boundary between the two regions, and p * (n) (t, x) the momentum at the crossing

                 x 0 (t, x) = T (x -µtx 0 ) T -t -µT t t * (n) (t, x) = T 1 - µT x0 -x 0 (t, x) aT + x 0 (t, x) + µT (x 0 -x 0 (t, x)) p * (n) (t, x) = aT + µT x0 -x 0 (t, x) + x 0 (t, x) T . ( 72
)
We may now compute the prefactor

N ∂ x0 x n exp - 1 2 t * (t,x) 0 (∂ x a (1) )dτ = µ 2πσ 2 T T -t + µT t µT (x 0 -x) + x + a (n) (T -t + µT t) µT (x 0 -x) -µtx 0 , (73)
and the action

S (n) leak (t, x) = a (n) (t-T ) x p (1) (t, x )dx + x a (n) (t-T ) p * (n) (t, x )dx = -(a (n) ) 2 (t -T )(T -t + µT t) + 2a (n) (T -t + µT t)x + (1 -µT )x 2 + 2µT xx 0 + µ(t -T )x 2 0 2(T -t + µtT ) ,
(74) with p (1) given by Eq. ( 64). We note that if both x and x0 belong to the boundary between region (1) and region (n), the prefactor diverges because of diffraction effects that should be treated specifically.

The reflected action is computed through the usual procedure, but, this time, taking into account that the reflected trajectory may also transit from a region to an other

Sleak (t, x) =S (n) leak (t, 0) + min[x;a (1) (t-T )] 0 p * (0) -(t, x )dx + min[max[x;a (1) (t-T )];a (2) (t-T )] a (1) (t-T ) p (1) -(t, x )dx + max[x,a (2) (t-T )] a (2) (t-T ) p * (2) -(t, x )dx , (75) with p * (n) -
the reflected leaking momentum in region (n) and p

(

-the reflected linear drift momentum. Complete, explicit, expressions are given in appendix AppendixD (cf Eqs. (D.2), (D.3) and (D.4)). However the contribution of reflected trajectories decay exponentially away from the absorbing boundary x = 0. Assuming t ≤ T as we do here, this implies that unless ≈ T , we can assume the contribution of reflected trajectories are important only when they are still in region (0), and the reflected action can be approximated as

Sleak (t, x) = 2a (0) x - 1 2(T -t + µtT ) -(a (0) ) 2 (t -T )(T -t + µT t) + 2a (0) (T -t + µT t)x + (1 -µT )x 2 + 2µT xx 0 + µ(t -T )x 2 0 . ( 76
)
We can show that, for this specific drift field, the reflected prefactor is the same as the direct one. Eventually, using Eq.(41), we have

m leak (t, x) = µ 2πσ 2 T T -t + µT t µT (x 0 -x) + x + a (n) (T -t + µT t) µT (x 0 -x) -µtx 0 exp S leak (t, x) σ 2 -exp Sleak (t, x) σ 2 . ( 77
)
Contrarily to constant and linear drifts which represent non-generic cases for which the WKB expression is exact, the above result is an approximation valid only in the semiclassical regime of small σ's. To be a bit more quantitative, we thus introduce the dimensionless parameter K defined as the ratio between the drift time τ drift = x(t, x0 )/a, the time needed to get from x = x(t, x0 ) to the location of the absorbing boundary condition x = 0 at speed a, and the diffusion time τ diffusion = x 2 (t, x0 )/σ 2 , time it would take to a purely diffusive process to spread the density from its center in x = x(t, x0 ) to x = 0. Thus

K = τ drift τ diffusion = σ 2 ax(t, x0 ) ∝ σ 2 . ( 78
)
The "small noise" [semiclassical] regime can be therefore characterized by K 1, and the large noise regime by K 1. Note that K usually depends on time. Fig. 4 shows a comparison between a numerical solution and the semiclassical approximation for different small values of K, fixing σ and varying t. As we can see the semiclassical approximation is almost indistinguishable from the numerical solution up to K = 0.33 and remains good for K slightly greater than one even if we can observe small discrepancies. Looking at larger values of σ (and hence K), cf. Fig. 5, we see that even for the largest value of K considered (K = 6.66), the agreement is still rather good although the difference with the exact result becomes more significant. The fact that the source of errors in the semiclassical treatment is generated only at the boundaries between the various regions explains the effectiveness of the approximation in this particular setup.

Conclusion

In this paper we proposed a new take on the WKB approximation scheme to study the Fokker-Planck equation. This approach, based on Maslov's geometric perspective, offers what we think to be a transparent way of tackling the Fokker-Planck equation, which we illustrated here on a problem motivated by a simple toy model of mean field games theory.

As stressed in the introduction, we have addressed here only a very small part of the program which would consist in providing a "ray theory" of mean field games in the small but non zero-noise limit. This program would involve a few steps (to start with a ray theory of the Hamilton-Jacobi-Bellman equation and then dealing with the coupling between the two) which are significantly more involved. We leave these for future research, but we are convinced that the WKB approach we propose provide a sound start for this program.

AppendixA. Non hermitian correction

As mentioned in section 2.2, the additional term 1 2 t t0 ∂ x a dτ in the expression [START_REF] Smirnov | A Course of Higher Mathematics[END_REF] can be seen as coming from the perturbation of the non hermitian operator L1 = L -L0 on the hermitian approximation

L0 = {λ -1 ∂ t • + λ -1 2 [∂ x (a•) + a ∂ x (•)] -1 2 (λ -1 ∂ x ) 2 •} of L.
To see this, we first need to apply L1 on m(t, x)

L1 m = λ -1 2 [∂ x (am) -a∂ x (m)] = λ -1 2 (∂ x a)m , (A.1)
and define its classical symbol L 1 = λ -1 2 ∂ x a. Using traditional first order perturbation theory for the action we thus obtain S → S 0 + δS with δS ≡ -

t t0 L 1 dτ = - λ -1 2 t t0 (∂ x a)dτ , (A .2)
(where the integral is taken on the unperturbed trajectory).

AppendixB. Method of characteristics

The method of characteristics is typically used to solve first-order partial differential equations. It aims to reduce a PDE to a family of ODEs that can be easily integrated. A rather complete discussion of this method can be found for instance in chapter II of [START_REF] Courant | Methods of Mathematical Physics[END_REF].

In the particular case of the Hamilton-Jacobi equation

∂ t S + a∂ x S - 1 2 (∂ x S) 2 = 0 , (B.1)
it is however extremely straightforward to check that the action defined by Eq. ( 10) is a solution. Indeed, using the least action principle, one has that for any X = (x, t), ∂ x S = p and ∂ t S = E, with p and E the momentum and energy of the trajectory reaching x at time t. Since all the trajectories involved have to fulfill the compatibility condition Eq. ( 9), this one reads L(x, t; ∂ x S, ∂ t S) = 0, which is precisely the Hamilton-Jacobi equation.

AppendixC. Liouville's formula

For completeness, in this appendix, we provide a brief derivation of the Liouville formula used in Section 3, as presented in [START_REF] Smirnov | A Course of Higher Mathematics[END_REF]. We consider a dynamic system described by

dx dt = f (x) (x ∈ R d ) , (C.1)
and consider a (d -1)-family of trajectories x(t, α) indexed by α ∈ R (d-1

AppendixD. Coupling the two solutions

This appendix aims at addressing what we left out of 4.3 for the sake of succinctness. We will first provide explicit expressions for the reflected action Eq. ( 75), then we will discuss the configuration where the agents begin in a constant drift region.

Explicit expression of the reflected action

Recalling Eq. ( 75)

Sleak (t, x) =S (n) leak (t, 0) + min[x;a (1) (t-T )] 0 p * (0) -(t, x )dx + min[max[x;a (1) (t-T )];a (2) (t-T )] a (1) (t-T ) p (1) -(t, x )dx + max[x,a (2) (t-T )] a (2) (t-T ) p * (2) -(t, x )dx , (D.1)
there are three domains in which Sleak (t, x) takes slightly different expressions. However, as mentioned in section 4.3, Eq. ( 75) can be approximated using only Eq. (D.2). This is shown in Fig. D.6 where the results of the two approximations, although obviously different for t ≈ T become more and more similar the smaller t gets.

• x < a (1) (t -T ) Sleak (t, x) = 2a (0) x - 1 2(T -t + µtT ) -(a (0) ) 2 (t -T )(T -t + µT t) + 2a (0) (T -t + µT t)x +(1 -µT )x 2 + 2µT xx 0 + µ(t -T )x 2 0 . (D.2) • a (1) (t -T ) < x < a (2) (t -T ) Sleak (t, x) = 1 2(T -t + µtT )(t -T ) µ(T 2 x 2 -2T (T -t)(2a (0) (T -t) + x)x 0 +(t -T ) 2 x 2 0 . (D.3) • x > a (2) (t -T ) Sleak (t, x) = 1 2(T -t + µtT ) (a (

Leak from a constant to a linear drift region

We begin, as in Section 4.3, by computing the position, time and momentum of the agents as they cross the boundary between an region of constant drift a (n) and region [START_REF] Caroli | Diffusion in a bistable potential: A systematic wkb treatment[END_REF]. Keeping the same notations and using the same method as earlier we have

         x * (n) (x 0 ) = a (n) (t * (n) -T ) = x 0 (1 + t * (n) µ) + t * (n) (a (n) -µx 0 ) t * (n) (x 0 ) = a (n) T + x 0 µ(x 0 -x 0 ) p * (n) (x 0 ) = µ(x 0 -x 0 ) . (D.5)
Using the canonical equations in region (1), we compute for t > t * (n) (x 0 )

           p (n) (t, x 0 ) = aT + x 0 -µT (x 0 -x 0 ) t -T x (n) (t, x 0 ) = at + x 0 -µt(x 0 -x 0 )
x 0 (t, x) =

x -at + µtx 0 (∂ x a (1) )dτ = µ 2πσ 2 (1 + tµ) µ(a (2) t -x + x0 ) a (0) -a (2) + a (0) µt + µ(x 0 -x) , (D.10)

and the new action

S (0) leak (t, x) = a (2) (t-T )
x(t,x0) p (n) (t, x )dx + a (0) (t-T ) a (2) (t-T ) p * (1) (t, x )dx + -2a (2) x + 2a (0) -a (2) (2 + µt)(t -T ) + x + µt(x -x0 ) -µ(x -x0 ) 2

.

(D.11) Finally the reflected action is computed as Sleak (t, x) =S -2a (2) x + 2a (0) -a (2) (2 + µt)(t -T ) + x + µt(x -x0 ) -µ(x -x0 ) 2 . (D.13)

Figure 2 :

 2 Figure 2: A typical manifold generated by the classical trajectories in region (1) of the drift field. In this case a = x t-T , µ = 1.5, σ = 0.4, T = 2 and x0 = -5. The dashed curves represent specific trajectories beginning at x 0 = -5.5, -5 and -4.5 from left to right.

Figure 3 :

 3 Figure 3: Same manifold as in Figure 2 where are highlighted two paths with same beginning and end. Because of the Langrangian nature of the manifold we can write C 1 (E ṫ + p ẋ)dt = C 2 (E ṫ + p ẋ)dt.

Figure 4 :

 4 Figure 4: Spatial distribution of the agents at fixed time, dashed lines show the numerical solution while solid lines show the approximation. From left to right, K = 0.19 and t = 1.1, K = 0.24 and t = 1.3, K = 0.33 and t = 1.5, K = 0.56 and t = 1.7, K = 1.67 and t = 1.9. In this case T = 2, a (0) = 0.4, a (2) = 0.9, σ = 0.2, x0 = 1.2 and µ = 10 6 .

Figure 5 :

 5 Figure 5: Spatial distribution of the agents at fixed time, dashed lines show the numerical solution while solid lines show the approximation. From left to right, K = 0.74 and t = 1.1, K = 0.95 and t = 1.3, K = 1.33 and t = 1.5, K = 2.22 and t = 1.7, K = 6.66 and t = 1.9. In this case T = 2, a (0) = 0.4, a (2) = 0.9, σ = 0.4, x0 = 1.2 and µ = 10 6 .

2 )Figure D. 6 :

 26 Figure D.6: Spatial distribution of the agents, the thin line represents the numerical solution, the thick straight line the approximation using Eq. (75) and the thick dashed line the approximation using only Eq. (D.2). In this case T = 2, a (0) = 0.4, a(2) = 0.9, σ = 0.2, x0 = 1.2 and µ = 10 6 . From left to right, t = 1.7, t = 1.8, t = 1.9.

2 ( 3 +

 23 µt)(t -T ) -a (0)2 (T -t + µtT )

  (t, 0) + min[x;a (1) (t-T )] 0 p * (0) -(t, x )dx + min[max[x;a (1) (t-T )];a (2) (t-T )] a (1) (t-T ) p * (1) -(t, x )dx + max[x,a (2) (t-T )]

a ( 2 )-

 2 (t, x )dx , (D.12) that we approximate, as in Section 4.3, asSleak (t, x) = 2a (0) x + 1 2(1 + µt) a (2)2 (3 + µt)(t -T ) -a (0)2 (T -t + µtT )

  DerivationLet A a d × d matrix. We have det A = exp[Tr ln A], and thus

					d(ln det A) dt	=	d(Tr ln A) dt	.	(C.3)
	Now, for any function g of A, writing g(A) = n g n A n and using the cyclicity of the trace we have
				d(Trg(A)) dt	= Tr g (A)	dA dt	.	(C.4)
	Thus, if A ≡ ∂x ∂(t,α) and J(t, α) ≡ det A, we have
						d ln J dt	= TrA -1 dA dt	.	(C.5)
	Noting that here the total derivative d dt is the same as the partial derivative ∂ t taken at constant α,
	one furthermore has		dA dt	=	∂ 2 x(t, α) ∂t∂(t, α)	=	∂f (x(t, α)) ∂(t, α)	.	(C.6)
	Thus	d ln J dt	= Tr	∂(t, α) ∂x	∂f (x(t, α)) ∂(t, α)	= Tr	∂f ∂x	(x(t, α)) .	(C.7)
				d ln J dt	= Tr	∂f ∂x	(x(t, α)) .	(C.2)

) . Defining J(t, α) ≡ det ∂x(t,α) ∂(t,α) , the Liouville's formula states that :

  * (1) (t, x )dx , (D.[START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF] with p(n) the constant drift momentum of region (n) given by Eq. (49) and p *(1) the leaking momentum in region (1) obtained by inserting the third equation of Eqs. (D.6) into the second, yielding (t-T )(T -t + µtT ) + x 2 (1 + µT ) +2µ(t -T )xx 0 + µ(T -t)x 2 0 -2a (n) (t -T )(x + µtx 0 )In the case where agents begin in region (2), they may diffuse up to region (0), using, once again the same scheme, we compute the new prefactor

	and the action
	S leak (t, x) = (n)	a (n) (t-T ) x(t,x0)	p (n) (t, x )dx +	x a (n) (t-T )
	S leak (t, x) = (n)	1 2(1 + µt)(t -T )	-a (n)2 .	(D.9)
	N ∂ x0 x 0 (t, x 0 )	exp -	1 2
						,	(D.6)
						1 + µt
	from which we get the prefactor
	N ∂ x0 x n (t, x 0 )	exp -	1 2	,	(D.7)

t t * (n) (t,x) (∂ x a (1) )dτ = µ 2πσ 2 (1 + tµ) µ(t -T )(at -x + x0 ) a(T -t) + x + µT x -µ(T -t)x 0 p t * (2) (t,x) t * (1) (t,x)

Note that even if the derivative of the drift field a(t, x) is discontinuous at the boundary, p, E and L are continuous, and thus the Lagrangian character of our manifold M is trivially preserved.


Aurélie Van Hoye
email: aurelie.van-hoye@univ-lorraine.fr

Chantal Vandoorne

Gaëtan Absil

Flore Lecomte

Catherine Fallon

Pierre Lombrail

Anne Vuillemin

Van Hoye

Absil², Lecomte Vandoorne²

Lombrail

& Vuillemin

Health enhancing physical activity in all policies? Comparison of national public actors between France and Belgium

Keywords: physical activity, policy, actors, health in all policies

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The health benefits of physical activity have now been widely demonstrated, as well as the important role PA could play in the prevention of non-communicable disease and mortality [START_REF] Lee | Impact of Physical Inactivity on the World's Major Non-Communicable Diseases[END_REF][START_REF] Warburton | Health benefits of physical activity: the evidence[END_REF] . Despite these evidences, only 45% and 36% of the French and Belgian population do reach the WHO recommendations to preserve their health [START_REF] Commission | European Physical activity Factsheet[END_REF]. Moreover, numerous works [START_REF] Pratt | The cost of physical inactivity: moving into the 21st century[END_REF] have supported the large cost due to physical inactivity, with an estimate cost to the health care system internationally of nearly 54 billion worldwide in 2013 [START_REF] Ding | The economic burden of physical inactivity: a global analysis of major non-communicable diseases[END_REF]. Therefore, physical activity has not only beneficial effects on individual physical, mental and social health, but also an impact on communities and countries as a whole [START_REF] Daugbjerg | Promotion of physical activity in the European region: content analysis of 27 national policy documents[END_REF].

While sport promotion has a long history in some countries, like Belgium and France, the promotion of health enhancing physical activity (HEPA) is still under development [START_REF] Daugbjerg | Promotion of physical activity in the European region: content analysis of 27 national policy documents[END_REF]. This growing interest, potentially due to the increase of non-communicable disease and obesity [START_REF] Von Ruesten | Trend in obesity prevalence in European adult cohort populations during follow-up since 1996 and their predictions to 2015[END_REF], has encouraged other actors than the health sector to join their forces on strategies to promote PA [START_REF] Breda | Promoting health-enhancing physical activity in Europe: Current state of surveillance, policy development and implementation[END_REF]. This engagement is in line with the health promotion approach (9), encouraging PA to be supported as part of daily life, across all settings (work, home, school, transport, community) and the development of health in all policies (collaborative approach by incorporating health considerations into decision-making across sectors and policy areas [START_REF] Rudolph | Health in All Policies: improving health through intersectoral collaboration: Institute of Medicine of the National Academies[END_REF][START_REF] Ståhl | Health in all policies. Prospects and potentials Helsinki: Finnish Ministry of Social Affairs and Health[END_REF]). Policy may be defined as legislative or regulatory action taken by government at different levels through formal written codes or standards [START_REF] Schmid | A framework for physical activity policy research[END_REF]. In other words, PA policy is "a formal statement that defines PA as a priority area, states specific population targets and provides specific plan or framework for action" [START_REF] Bull | Developments in National Physical Activity Policy: an international review and recommendations towards better practice[END_REF]. Policy is rarely a single decision, but a set of decisions or even non-decision, created within a dynamic of negotiations between interested parties, recognized as more effective if interests of the involved stakeholders are defined and their potential influence analyzed [START_REF] Sallis | The role of built environments in physical activity, eating, and obesity in childhood[END_REF][START_REF] Sallis | Environmental and policy interventions to promote physical activitya[END_REF] . The objective of health in all policies is to get out of the 'silos' organization of the society and of policies [START_REF] De Leeuw | Nine questions to guide development and implementation of Health in All Policies[END_REF], by encouraging cross sectorial collaboration, such as for example between health, education and sport ministries, to answer to the necessity of dealing with health (especially PA) in comprehensive and inclusive ways, including making healthier choice easier choices [START_REF] Breda | Promoting health-enhancing physical activity in Europe: Current state of surveillance, policy development and implementation[END_REF][START_REF] De Leeuw | Nine questions to guide development and implementation of Health in All Policies[END_REF].

Despite the attention given to PA, previous work has shown that European policies were mostly prepared and implemented by a single actor, with health or sport ministry been the most implicated (around 55% of PA policy), but in other cases soliciting ministries of transport, environment or education [START_REF] Daugbjerg | Promotion of physical activity in the European region: content analysis of 27 national policy documents[END_REF]. Recent findings have underlined that almost all European countries had developed a national policy on one or more of the HEPA sectors and that only 59% of the European countries have a specific national coordinating mechanism on HEPA promotion [START_REF] Breda | Promoting health-enhancing physical activity in Europe: Current state of surveillance, policy development and implementation[END_REF]. Conceptually, countries have a unique influence on PA promotion, as this resulted from a socio-historical and cultural process. Previous work [START_REF] Daugbjerg | Promotion of physical activity in the European region: content analysis of 27 national policy documents[END_REF] has been centered on written documents or general indicators and has not taken the opportunity to compare internationally actors implicated in regard to the structure of the country and state organization, as well as their distinct dynamic in policy making process [START_REF] Daviter | Policy framing in the European Union[END_REF], especially about the type of PA targeted (e.g., sport, HEPA, active mobility…). To our knowledge, no previous study has identified mechanisms or instances enhancing 'physical activity in all policies' and inter-sectorial collaboration, due to the structure and policy system of the country (for example comparing a centralized state like France and a three levels state like Belgium).

France is a Republic, ruled by a president, elected by direct universal suffrage for a five-year term. The government, led by a prime minister nominated by the president, develops and guides policy implementation. The prime minister is accountable to parliament, which exercises legislative power and is made up of the National Assembly and the Senate. The state defines the competencies of each level of administration. France is a civil law country whose laws and regulations (acts, ministerial decrees, and administrative orders) are broken down into more than 60 codes by subject area (e.g. public health code, social security code, social action and family code). For major reforms and annual decisions, laws are enacted by legislation after discussion in parliament. Following enactment, decrees are issued by the prime minister. When specified in acts, some decrees must be assessed by the Council of State. Lower level regulations such as administrative orders are signed by the relevant minister.

Belgium is a constitutional monarchy, ruled by its seventh king, who does not wield power in the political sphere in its own, but acts in consultation with government ministers. Under the sixth reform of the state, the pyramid of the unitarian state made way for a more complex three-level structure. All three level are equal from the legal viewpoint, but can edict law only in regard to their competences: the Federal State (federal government and federal parliament), the Communities (Flemish, French, German-speaking) and the Regions (Walloon, Capital, Flemish). The Federal state include the entire Belgium territory, Walloon region entails part of the French speaking community and the whole German Community. Brussels Region entails the other part of French speaking community and part of Flemish community. Flemish region covers Flemish speaking community. Broadly speaking, the powers of the Federal State cover everything connected with the public interest (e.g., judicial system, army and federal police, social security, including health insurance, public debt, prices and incomes policy, etc.). Regions have powers in fields that are connected with their territory (e.g., economy, employment, agriculture, water policy, housing, public works, curative and preventive medicine, transport, environment, town and country planning). Since the speaking communities are based on the concept of "language" as a vehicle for culture, they have powers for culture, education, the use of languages, and assistance to individuals (protection of youth, social welfare, aid to families, immigrant assistance services, etc.).

To analyze the PA policy dynamic in Belgium and France, a network and policy making process [START_REF] Daviter | Policy framing in the European Union[END_REF] analysis will help to inform the key actors and their collaborations, questioning the potential "physical activity in all policies" [START_REF] Ståhl | Health in all policies. Prospects and potentials Helsinki: Finnish Ministry of Social Affairs and Health[END_REF]. Three main objectives were followed: 1/ to identify the competencies of ministries which play a role in HEPA and PA promotion, 2/ to question the presence of a leadership and coordination in HEPA policies implementation, 3/ to identify key public legal entities (e.g., agencies, administrations…) working on HEPA promotion and their relationship with ministries.

MATERIALS AND METHODS

Data collection

To collect and structure the data, the WHO HEPA PAT version 2 was used [START_REF] Organization | Healthenhancing physical activity (HEPA) policy audit tool (PAT). Version 2[END_REF]. The territory covered represents the whole country for France and the south part for Belgium (Federal, Wallonia, German and French Community, excluding Brussels and Flemish region). Reasons for not including the Flemish region was due to the fact that health and sport competencies have been attributed to the Region and Communities since the sixth reform of the Belgian state. Brussels Region was contacted, but no answers were obtained. The data collection started in France and Belgium in November 2014 and ended in January and April 2016 respectively. To have a common understanding and to able the comparison of the two countries, we use the concept competencies of ministries to analyze the data. In other words, the health and solidarity ministry is in charge of two competencies: health and solidarity).

Two complementary, concurrent and iterative data collections were conducted: a document analysis and a qualitative data collection. Internet based search was realized on the website of the different ministries and public legal entities. As the promotion of HEPA goes beyond the health sector alone, any relevant national policy document entailing as keyword "physical activity" or "walking", "cycling" were sought. The most recent version of the document was included in the inventory. Simultaneously, face-to-face or phone semi-structured interview (18 in France, 14 in Belgium) with experts in PA (recognized through their publications or membership in public legal entities) or representative of government were conducted (see Table 1 for details). The interviewees were selected taking their sectors as well as function into account. After the document and interview analysis, a final seminar was organized in each country to discuss and validate the findings.

RESULTS

Multi-sectoriality of HEPA

The ministries' competencies promoting HEPA in both country come principally from the same sectors : health, sport, environment and transport, as well as education, but with different degree of implication and a focus on different type of PA (see Figure 1). Justice is active on PA policy in France, but not in Belgium. In France, two ministries participate to HEPA promotion without specific policy: the ministry of higher education and research can potentially support HEPA through teaching and research; the ministry of work, employment, professional training and social dialog promote health, and potentially PA within the work sector and environment. In Belgium, the Public Health, Food Chain and Environment FPS has not been cited as an actor in HEPA, as it has no responsibilities in health promotion.

Figure 1. Type of PA promoted by ministries' competencies

Leadership and coordination in HEPA

In both countries, actors underline that there is no official instance or agency having a leadership on HEPA, but rather different actors that are implicated depending on the policy or the sector. "Does a coordination platform exists in regard to HEPA. Formally, no, we sometimes have questions, prevention counselling, that are across different matters, sporteducation, sport and health, but a general coordination, could such a thing exists? If it is to add a layer above the other, without a particular effect, this is just heavy." (General Administration of Sport, French speaking community).

In France, the National Sport-Well-being Plan in 2012, due to its dual piloting from health and sport ministries has placed these two ministries as non-official leaders recognized by nongovernmental actors. "There is no official leadership, but since the National Sport-Well-being Plan, an emerging leadership of Health and Sports ministry has been identified, which is stronger at regional level, than at national level" (Collective Seminar, France).

Specific permanent commission could be implicated in HEPA promotion: the general commission towards equity in territories (CGET), which implement the equity in territory policy and the inter-ministries coordination for the development of cycling use (CIDUV), the national center for sport development (CNDS) funding among other sport for health and the national council of sport (CNS; counselling on sport policies). In French-speaking Belgium, policies are rather centered on sport specifically or on PA among the multiple determinants of health, letting us though that there are no leadership in Belgium, but rather places where HEPA could be discussed, and potentially indirectly targeted: The Commission of Risk Prevention in Sport, the Superior Sport Council, the Regional Cycling Commission, the Superior Health Council. Working on HEPA in an intersectoral manner in these commission is restrained by the fact that "we (the commission of risk prevention in sport) can talk only about an activity that is ruled by the French-speaking community, like organized sport by sport federation, daily PA could not be regulated as easily, could you imagine a police officer asking a runner about its non-contraindication to PA practice?" (President of the commission of risk prevention in sport).

In France, scientific societies implicated in HEPA promotion are the French Society of Public Health (SFSP), the French Society of Exercice and Sport Medecine (SFMES), the Academy of Medecine, the French Society of Nutrition (SFN), the association of researcher in physical and sport activities (ACAPS), the French-speaking association in adapted physical activity (AFAPA). Belgian scientific societies could also be solicited on HEPA, but are at the moment not alerted on this aim, potentially explained by the size of the country: "In French-speaking community? No, there is not much, but there is no reason to have such, it's too small, Belgium is too restrictive" (President of the commission of risk prevention for health in sport)

Intersectorality: Ministries relationships

To question the intersectorality between ministries, we analyzed the different policies documents and interview trying to understand how they collaborated. To facilitate the understanding, we decided to select the competencies of ministries in France, and due to the three levels of complexity in Belgium, we selected federal public service, administrations or department. In France, the official relationship between the ministries 'competencies are based on different policies coming from the French Republic or the whole government (see Figure 2). "Some inter-ministries policies exist and contribute to PA, but the system need to be go out of silos and health objectives have to become more coherent" (Collective seminar, France).

Figure 2. Intersectoral policies in France

In Belgium, despite a will of the French-speaking community to encourage cross-sectoral collaborations in its declaration of intentions 2014-2019, only the Walloon Cycling Plan support a collaboration between infrastructures and mobility administrations. Given this low number of collaborations in Belgium, we centered our work on informal relationship (see Figure 1 for details). At the regional and speaking community level, participants answered that they did not collaborated with the federal level. In the French speaking Community, some relationships have existed between sport and education sector, but are frozen at the moment.

In the sport sector, region and speaking community collaborates punctually, sharing their expertise on infrastructure and organization of sport respectively for specific promotion actions. The education and mobility administration also collaborate on cycling actions in schools, but this is not mentioned in policies. In the German community, a good communication and collaboration between the Health, Sport and Pedagogy departments exists in the administration.

. In Belgium, at federal level, the Superior Council of Health formulates scientific advices for politics and health professionals, the Belgian health care knowledge center (KCE) act as an interface between ministry of social security, ministry of public health and ministry of social affairs to provide scientific analysis and research, to advice policy makers on decisions relating to health care and health insurance. In the Walloon Region, the Walloon institute of evaluation, prospective and statistics (IWEPS) is a public scientific institute to help authorities' decisions. The regional cycling commission, include different ministries from the Walloon region and other stakeholders to implement the Walloon cycling plan. In the French speaking community, within the sport sector, the superior council of sport gives an expertise on sport aims to the sport ministry, the commission of risk prevention for health within sport works on the decrease of risks within and from sport practice.

DISCUSSION

The present study has revealed that HEPA policies in France and in Belgium are at their early stage, as cross-sectoral collaboration between actors seems to be on ministries' agenda since 2012 in France, and only in declaration of intentions in Belgium. One of the criteria recommended for successful policy development is the involvement of different stakeholder (6) (ministries, private sector, public legal entities). In comparison to previous European study [START_REF] Daugbjerg | Promotion of physical activity in the European region: content analysis of 27 national policy documents[END_REF][START_REF] Breda | Promoting health-enhancing physical activity in Europe: Current state of surveillance, policy development and implementation[END_REF], we found a larger diversity of ministries' competency implicated in HEPA promotion, but not more than half of the policies being cross-sectoral. Nevertheless, ministries focused on the type of PA they have in responsibility (PE for education ministries' competencies), but rarely on different types of PA. The novelty in France relies on plans coming emanating from the Republic or the Government as a whole, and not only collaborations between one or two ministries. This advance seems to enhance the appeal of ministries or public legal entities which have not HEPA as primary aims, but it is too early to draw conclusions in regard to the cross-sectoral dynamic at the moment. France seems to have more cross-sectoral policies, but some current plans, like the inter-ministerial ageing well plan have failed to keep the intersectorality created in their previous version. Reasons might be the enactment of a law on adapting society to an ageing population in 2015. The presence of a centralized state seems to be a positive factor in regard to policies implicating the whole government, as the three level structure in Belgium, with different government at each level, complicate the presence of such policies.

Despite the diversity of ministries implicated in HEPA promotion, their collaborations are rarely formal, but rather punctual, without a structured agenda and common goals. No real mechanisms of consultation exist to create a policy, but rather some working group, that often disappear when the law is enacted or when the plan ended, or even started. In Belgium, the three levels structure complicate the collaboration at the different levels, where few links are made within each level, but even fewer between the different levels, despite complementarity in actions. Moreover, in the interviews, many actors outside the sport and health sectors stated that HEPA is not the central point of collaboration, but rather a small part of a large range of health determinants or other aims. Therefore, their priority relied first on the need of a common definition and lobbying within each sector, as only a small part of representative of ministries have been committed towards HEPA, in the same way than plans include only few minor points on HEPA promotion within larger objectives.

Previous work [START_REF] Van Herten | Rationalising chances of success in intersectoral health policy making[END_REF] has analyzed inter-sectoral health policy in regard to three criteria to question its feasibility: the availability of evidence, the degree of support and the availability of tools for implementation. Considering HEPA promotion in both countries, our results underline 1/ the lack of knowledge regarding evidences in HEPA promotion and the need of its importance in each sector, 2/ no problem of support or conflicts, but no real engagement towards HEPA, and no concrete prevalence objectives for the future. 3/ The absence of coordination, the lack of guidance or procedure in policy creation (with some working group created to propose objectives within short timeframe), 4/ the limited amount and the opacity of funding allocated to HEPA in each sector are barriers to effective inter-sectoral collaboration.

Moreover, inter-sectoral leadership is lacking, despite the presence of different key public legal entities, which could play an active role. For example, the inter-ministerial committee for health in France was never solicited on HEPA aims. In both countries, lobbying towards HEPA in these instances could help to enhance the coordination mechanisms, such as having common goals for HEPA, making lobbying for HEPA, creating a professional network. The high number of public legal entities working on specific parts of HEPA promotion (e.g., active mobility, sustainable development, cancer prevention and treatment) increase the difficulty of creating an inter-sectorial coordination, for different reasons: 1/ impossibility of having a common agenda, 2/ the lack of definition of common goals, 3/ the diversity of professional profile and unease with inter-sectoral approaches [START_REF] Milbourne | Collaborative solutions or new policy problems: exploring multi-agency partnerships in education and health work[END_REF] . France, but their plurality could slow down the collective process. Implications rely on the need to define coordination for HEPA in each country, to support lobbying for HEPA in each sector specifically and raise awareness about HEPA in the different commission that could support its development, as well as officially integrate the implication of the key public legal entities in HEPA delivery in policy document.

Figure 3 :

 3 Figure 3 : relationship between administrations in Belgium

CONCLUSIONS

  HEPA inter-sectoral policies are at their early stage in France and Belgium, with only a fiveyear history in France and political will in Belgium. A broad diversity of sectors was implicated in HEPA policies: sport, health, transport, environment, and education, but often with light implication on HEPA. Despite this large range of ministries, no leadership or coordination exist to implement the HEPA policies, although different commission could work on this aim in their duty. No standardized procedure or mechanisms of consultation exists to enact HEPA policies. Ministries relationships are principally coming from plans implicating the whole government or republic in France, where in Belgium relationship are rather punctual. Public legal entities have an important role of inter-sectorial development in

Table 1 :

 1 organization participating to interview, type of interview and country mention of health), and ministries' competencies' policy promoting policies that could support PA without mentioning PA. Leadership and coordination was discussed within the different interviews and validation seminar, identifying public legal entities or mechanisms. The intersectorality was assessed using explicit references in policies documents and interview extracts about collaboration between ministries, with different types of relationship: 1/ formal and official through policies document, 2/ in construction, as actors as starting working together without official mention, 3/ punctual collaboration on common aims. Furthermore, intersectorality was questioned in regard to public legal entities supported by ministries which could have a role in HEPA policies implementation or delivery.

	Organisation	Interview type	Country
	General Operational Direction of Roads and	Face to face	Belgium: Walloon Region
	Infrastructures (INFRASPORT)
	General Operational Direction of Mobility and Hydraulic	Face to face	Belgium: Walloon Region
	Gates (RAVEL)
	General Operational Direction of Mobility and Hydraulic	Face to face	Belgium: Walloon Region
	Gates : Mobility strategy
	Agency of Quality of Life	Participated to	Belgium: Walloon Region
		final seminar
	General Administration of Sport	Face to face	Belgium: French Community
	Youth aid : project and prevention service	Face to face	Belgium: French Community
	Youth service	Phone interview Belgium: French Community
	General Administration of compulsory Education	Face to face	Belgium: French Community
	Catholic health union	Face to face	Belgium
	Birth and Child Office	Face to face	Belgium: French Community
	President of the commission of risk prevention for health	Phone interview Belgium: French Community
	in sport
	Head of the faculty of motricity sciences, physical	Face to face	Belgium
	education and physiotherapy of the UCLouvain
	Brussel's federation of General Practitioners	Face to face
	Department of Sport, Medias and Tourism	Face to face	Belgium: German community
		with health and
		elederly
	Department of Health and Elderly	Face to face	Belgium: German community
		with sport
	National Institute of Health Prevention and education	Phone interview France
	French Society of Professional in Adapted Physical	Phone interview France
	Activity

differentiate their policies: ministries' competencies' policy promoting HEPA (explicitly targeting PA to support health for population), ministries' competencies' policy promoting PA (without