Alix Deleporte

Quantum selection for spin systems

We report mathematical results on the process by which quantum order by disorder takes place for spin systems. The selection rules follow the influence of several competing contributions. Moreover there is no link between quantum selection and thermal selection. We present work in the general setting as well as toy models and examples for which quantum selection has an interesting behaviour.

I. INTRODUCTION A. Order by disorder

The understanding of low-energy states of nonintegrable quantum systems is a notoriously difficult task, with applications to the design of both quantum and regular computers, supraconductivity as well as superfluidity. In particular, the Anderson RVB model for high T c supraconductivity 2 has drawn attention to frustrated quantum spin systems.

In an effort to tackle this problem from a theoretical perspective, various approximation procedure are used, such as restriction to finite size systems 12,25,[START_REF] Waldtmann | The European Physical[END_REF] or generalizations to SU (N ) with N large 30 . In this paper we are interested in semiclassical methods 7,14,17 , which are inspired by Villain's "order by disorder" principle 31 .

This approach is motivated by the fact that, for frustrated spin systems, the classical minimal set does not consist of a single class of configurations given by a global symmetry. Spin ices 1,18,24,[START_REF] Matsuhira | [END_REF] feature a discrete set of classical minimal configurations, with extensive cardinality. For the Heisenberg AntiFerromagnetic model (HAF) on the Kagome lattice (see Figure 1), they form a continuous set which is not regular: the dimension of allowed infinitesimal moves is not constant on this set.

The idea behind "order by disorder" is that lowtemperature classical states, as well as quantum lowenergy eigenstates, are not exactly located on the classical minimal set but are spread out; in particular, their energies are shifted up by a factor depending on the behaviour of the classical energy near its minimal set. The flattest the classical energy landscape, the lowest the energy contribution. As a consequence, those states must concentrate only on the subset of the classical minimal set where the local energy landscape is the flattest. In short, the presence of thermal or quantum fluctuations actually restrict the possible locations of low-energy states.

At this point we already make an emphasis on the geometrical data needed to define what it means for the classical energy to be flatter near one minimal point than near another. As was already pointed out 14 , in the setting of classical low-temperature, the Gibbs measure depends on the classical energy itself and the volume element on the phase space. To the contrary, quantum states depend on the symplectic structure on the phase space, which is a finer geometrical notion: some phase space transformations preserve the volume form but not the symplectic structure. Thus, though thermal and quantum selection stem from the same intuition, the "flattest" classical points may not be the same in the two cases.

B. Results

In this article we clarify the process under which quantum selection takes place, and examine the links with thermal selection. A common heuristics states that quantum selection and thermal selection follow the same rules. This intuition, which leads to the claim that on the Kagome HAF low-energy states are coplanar, is sometimes misleading. Another claim states that quantum selection is determined by the classical frequencies in the linear spin-wave approximation. In fact there are additional terms, which do not play a role on antiferromagnetic systems but which appear in more general spin systems. We describe in detail those additional terms.

We report mathematical results, which define a function µ under which quantum selection takes place: as the spin grows, low-energy quantum states localize on the set of phase space on which both the classical energy and this function µ are minimal. We then analyse various model situations of irregular minimal classical sets in order to understand the link with thermal selection.

This article is organised as follows: Section II presents the general mathematical framework for the treatment of quantum selection in the context of spin systems. In Section III we use three toy models to illustrate the concepts and difficulties associated with quantum order by disorder. In Section IV we analyse practical examples such as the semiclassical HAF on the Kagome lattice. Section V presents a discussion of the consequences and applications of our work. The Appendix consists of exact computations which relate spin systems and Toeplitz quantization.

II. SPIN WAVE FREQUENCIES AND TOEPLITZ QUANTIZATION

In this section we expose the main mathematical ideas behind Toeplitz quantization, which allows to study spin operators in the large S limit from a rigorous point of view. We report our recent results on the topic and clarify the exact procedure under which quantum selection takes place.

A. Harmonic oscillators in Bargmann-Fock representation

The point of view of Bargmann on the quantum harmonic oscillator 5,[START_REF] Husimi | Proceedings of the Physico-Mathematical Society of Japan[END_REF] , is that quantum states should be seen as holomorphic functions on the complex space C d instead of the common choice L 2 (R d ). This idea can in fact be generalized to other phases spaces than C d , and allows to understand the large spin limit as a semiclassical limit from a rigorous point of view.

For positive k (which is seen as the inverse Planck constant), holomorphic functions on C d form a Hilbert space B k with the following scalar product

u|v B k = u(z)v(z) exp(-k|z| 2 )dz.
We naturally exclude from the space B k the functions with infinite norm. Example of functions in B k are the monomials z → z ν1 1 . . . z ν d d which, once normalized, form a Hilbert base of B k . Under this definition, B k naturally sits inside the space L k of all (not necessarily holomorphic) functions which are square-integrable with respect to the exponential weight above. The orthogonal projector Π k from L k to B k is used to define the quantum harmonic oscillator, which is the following operator on B k :

T k (|z| 2 )|u = Π k (z → |z| 2 u(z)).
This is a Toeplitz operator: the composition of a multiplication operator and a projection. The matrix elements are simply

u|T k (|z| 2 )|v B k = C d u(z)v(z)|z| 2 exp(-k|z| 2 )dz.
The monomials z ν are eigenfunctions of this operator, with eigenvalues k -1 (ν 1 + . . . + ν d + d). This contrasts with the L 2 (R d ) point of view on the harmonic oscillator, where the eigenvalues are the half-integers (ν 1 +. . .+ν d + d

2 ). This does not mean that this Toeplitz operator is not natural, or that other terms should be added; in experiments one can only measure gaps between eigenvalues, which coincide for the two settings.

The definition of the Toeplitz operator can be generalized. If H is any function on C d (which represents the classical energy on the phase space C d = R 2d ), the associated Toeplitz operator on B k is defined as

T k (H)|u = Π k (z → H(z)u(z)).
This defines a quantization:

T k (H 1 + H 2 ) = T k (H 1 ) + T k (H 2 )
, and in the large k limit, the commutator

[T k (H 1 ), T k (H 2 )] becomes close to -ik -1 T k ({H 1 , H 2 }).
The function H associated with the operator T k (H), which is unique, is called the symbol of T k (H). In the context of spin systems it coincides with the notion of upper symbol.

Toeplitz quantization follows the Wick order: if

H : z → z α z β , then T k (H) = k -αi ∂ α z β .
The Wick rule allows explicit computations for the Toeplitz quantization of any polynomial function in the coordinates.

Of great interest are Toeplitz operators associated with semipositive definite forms Q ≥ 0. As in the harmonic case, the infimum of the spectrum is linked with the classical frequencies, but is shifted with respect to the usual quantization procedure: if λ 1 , . . . , λ r are the non-zero classical frequencies for Q, then

µ(Q) := inf Spec(T k (Q)) = k -1 1 2 r i=1 λ i + 1 4 tr(Q) .
(1) The factor tr(Q) is specific to Bargmann quantization. In the Weyl representation, one has instead inf Spec(Op W (Q)) = 2 r i=1 λ i .

B. Toeplitz operators on spheres

Toeplitz quantization can be generalized from C d to other phase spaces, using tools of complex geometry 6 . In particular, this allows to define a quantization procedure on product of spheres: to any classical energy on a product of spheres, and any k, one can associate a quantum operator, acting on the tensor product of spaces C k+1 . Previously k was any positive real number, but now it needs to be an integer: the topology of the phase space only allows quantized values of the inverse Planck constant.

Toeplitz operators on product of spheres include spin systems (with spin S = k 2 ). However the quantization procedure requires some care in the computations as can be seen on Table I.

classical quantum (S = k 2 ) z k k+2 Sz x k k+2 Sx z 2 k 2 (k,3) S 2 z + 1 k+3 zx k 2 2(k,3) (SxSz + SzSx) z 2 x k 3 (k,4) SzSxSz + 1 (k+3) Sx z 3 k 3 (k,4) S 3 z + k(3k+8) (k,4) Sz TABLE I.
Quantization of some symbols on the sphere.

The operator Sz has entries -1, -1

+ S -1 , . . . , 1 -S -1 , 1. We denote (k, j) = (k + 2)(k + 3) . . . (k + j).
The corrective terms of order k -1 are crucial for quantum order from disorder. The details for the computations in Table I are presented in the Appendix.

C. Quantum selection for Toeplitz operators

In a recent paper 9 , we developed mathematical tools in order to study quantum selection for general Toeplitz operators in the large k limit. We report that, in a general case (even if the set of minimal classical energy is irregular), quantum selection takes place for Toeplitz operators following a general criterion.

In order to apply our results to usual spin operators, as seen above, we need to consider Toeplitz operators with classical energy depending on k in the following way:

f = f 0 + k -1 f 1 + k -2 f 2 + . . . ,
where each term f j is a real function on the phase space. Indeed, the quantization of symbols which do not depend on k only yield a deformation of the usual spin operators. The Toeplitz operator T k (f ) is well-defined by linearity.

Quantum states with energy less than min(f 0 ) + Ck -1 are known to localize on Z = {f 0 is minimal} as k grows. In a neighbourhood of any point P 0 of Z, the function f 0 can be approximated by its quadratic Taylor estimate min(f 0 ) + Q, where Q is a semidefinite positive quadratic form which depends on P 0 .

The selection criterion is then

μ = µ(Q) + f 1 , in following sense: if (u k ) denotes a sequence of ground states of T k (f ), if a set V lies at positive distance from {x ∈ Z, μ(x) = min(μ)},
then for every j one has, as k → +∞,

V |u k (z)| 2 k -j .
The meaning of |u k (z)| 2 depends on the underlying manifold (for instance, on C d a factor exp(-k|z| 2 ) must be added), but as our quantum states are defined on the whole phase space, localisation properties can be formulated in a more elementary way than in the space representation.

The quantum ground state localizes, in the large k limit, only on the part of Z where μ is minimal; at any positive distance from this set, the ground state decays faster than any negative power of k. In fact, if E 0 is the energy of the ground state, then any quantum eigenstate with energy less than E 0 + k -1 for any localizes where μ is minimal. 28 In order to apply this result from a standard "operatorpresented" quantum spin Hamiltonian in the large spin limit, one needs first to compute, not only the associated classical energy at the main order, but also the socalled "subprincipal symbol" which contains the nextorder terms in the quantization procedure. For instance, starting with the operator S 2 z , the principal symbol is of course z 2 , and from Table I one can compute that a more accurate representation is

z 2 + k -1 (5z 2 + 1).
This subprincipal part, added to the trace and to the sum of symplectic eigenvalues of the quadratic part of the energy, yields the function µ which is the selection rule. In section III we apply this method to several models.

The physical interpretation of µ is the following: suppose that one wants to minimise the energy of a quantum state while constraining it to be localised at a precise point, where the classical energy has a local minimum. Then the energy of this minimal constrained state is naturally close to the classical energy, but is lifted up by quantum fluctuations. Indeed, quantum states have to spread out somewhat, and to reach parts of the phase space where the classical energy is not minimal. This energy lift is of the same order as the semiclassical parameter (here, k -1 ). Then µ, at this point, is the k -1 contribution to this energy lift.

In the context of spin systems, the selection rule is determined by the classical frequencies of the spin waves, and by non-trivial additional terms which must be taken care of. For the particular case of HAF systems, if each spin has the same number of neighbors, then the additional terms are constant, but on other systems on which quantum selection is studied, they can play an important role.

III. TOY MODELS

In order to understand quantum selection in the general case and in the particular case of spin systems, we first look at three simple toy models.

In the first toy model, which is the first historical example of quantum selection, thermal and quantum selection play the same role. In the second toy model, which has an irregular minimal set as does the HAF on the Kagome lattice, thermal selection is sharper than quantum selection: some classical configurations are equivalent from a quantum point of view (they share the same value of µ), but are discriminated by the Gibbs measure. Conversely, on the third toy model, there is no thermal selection, but quantum selection takes place.

A. Miniwells

The general study of quantum selection for the ground state of a Schrödinger operator was performed by Helffer and Sjöstrand 19 who exposed a WKB construction for a quasimode associated to the lowest energy. Quantum selection occurs when the potential V is minimal on a degenerate set Z. If Z is a smooth manifold on which V vanishes at order 2, the criterion for quantum selection is the trace of the square root of the Hessian matrix of V at the minimal points; in this Weyl setting, it corresponds exactly to the sum of the classical frequencies for the linearized system. Even in this case it does not correspond to the criterion for thermal selection (which is the product of these frequencies).

The simplest example is the operator P = -2 ∆ + V acting on L 2 (R 2 ), with V (x, y) = y 2 (1+x 2 ), vanishing at order two on the horizontal axis. It is already interesting to note that, though V itself is not a confining potential, P h only has discrete spectrum because of the quantum selection.

Around every point (x 0 , 0) of the horizontal axis, the quadratic terms in the potential are y 2 (1 + x 2 0 ). For this quadratic potential there is one non-zero classical quency, 1 + x 2 0 . This frequency is minimal at x 0 = 0, which is called the "miniwell" for this potential. Hence, the ground state of this operator concentrates on the point (0, 0), in the previous sense (for the Husimi transform).

In this setting, the value µ coincides with the effective potential given by the intuition of the adiabatic approximation 14 . At (x 0 , 0) one can approximate the behaviour of a low-energy state in the second variable as the ground state of the quadratic transverse operator -2 ∂ 2 y + (1 + x 2 0 )y 2 ; if e x0 (y) is the ground state of this operator, then the energy of a state of the form e x (y)f (x) is f, (-∂ 2

x + µ)f , so that µ acts as an effective potential (with new semiclassical parameter √ ).

B. Crossing points

With the Kagome lattice in mind, let us consider toy models where the minimal set of the classical energy is not a smooth manifold.

The first of this model is again a Schrödinger operator on R 2 , with potential V (x, y) = x 2 y 2 . The minimal set consists in the two axes, which meet at zero. On the horizontal branch (x 0 , 0), there is only one non-zero linear classical frequency, which is |x 0 |. This frequency is minimal at zero (note that this frequency is not smooth at zero). The same applies for the vertical axis. Once again, the operator P = -2 ∆ + V only has discrete spectrum and the first eigenfunction localizes at the origin, which is also the point of thermal selection (since the local dimension of the zero modes is maximal at this point). Because of the non-regularity of the classical frequency at the crossing point, for a potential W close to V which is also minimal on the two axes, the quantum system will still select the crossing point.

In this setting, the Born-Oppenheimer approximation fails at the crossing point, so that there is no simpler effective model. However µ still acts as an energy barrier, independently on the geometry.

A more general crossing is the Schrödinger operator on R 3 , with potential V (x, y, z) = x 2 y 2 z 2 . The minimal set is the union of the three planes {x = 0}, {y = 0}, {z = 0}, and the local zero dimension is maximal at the origin. However, on the plane {x = 0} the classical frequency is |yz|. All classical frequencies vanish identically on the three axes.

For this particular potential, there is a hierarchy of perturbations, and investigating the sub-sub-principal (order k -2 ) terms will lead to concentration at the origin and discrete spectrum. However, if non-degenerate transverse modes are added, they correspond (in the adiabatic regime) to a perturbation of order k -1 , in front of which the k -2 confinement at the origin is negligible; in the general setting, even for small perturbations, the quantum selected point can be any point on the three axes. This illustrates the discrepancy between quantum and thermal selection and shows that for the Kagome lattice, the points of quantum selection might not necessarily be the planar configurations, though those configurations have the maximal number of zero modes.

C. Cancelling terms

We propose an example which serves to illustrate the effects of the different terms in the process by which quantum selection takes place.

Let us consider the following one-spin Hamiltonian:

H = S 2 z + ∆S z S x S z .
The principal symbol of this operator is

h 0 = z 2 (1 + ∆x).
If 0 < ∆ < 1, then h 0 is minimal on {z = 0}. It looks like h 0 is smaller near (-1, 0, 0) than near any other point, so that, at first sight, quantum order from disorder seems to take place in this setting.

Let us look at the three terms appearing in quantum selection:

1. For any minimal point, the associated linear classical frequency is zero, since the linear classical approximation is that of a free massive particle in one-dimensional space.

2. The trace of the quadratic form near a minimal point is non-zero; this term has contribution µ = 1 + ∆x 3. The next-order term in the expansion is

h 1 = 5z 2 -1 + ∆(9z 2 x -x)
In particular, on the set where h 0 is minimal, one has

h 1 = -1 -∆x.
The set selected by quantum order by disorder is the set where h 1 + µ is minimal, but the two terms cancel out. Hence there is no quantum selection at this order of expansion.

It can readily be seen that, if the spin S is even, then S z |0 = 0, so that H|0 = 0 and |0 is the ground state of H. As expected, the magnetization of |0 along the x axis is exactly zero. Hence, there is no quantum selection in the large spin limit for this model.

More involved theoretical examples where the classical degeneracy is not lifted at any order of S -1 include spin textures 13 . In other situations, there could be no quantum selection at first order, but next-order terms could break the degeneracy. In practice, one expects additional terms (such as second nearest neighbours interactions) which will destroy exact degeneracies.

IV. EXAMPLES A. Kagome lattice

The quantum HAF on the Kagome lattice is the Toeplitz quantization of the classical energy i∼j e i • e j .

Here i ∼ j means that the two sites i and j are linked by an edge. The Toeplitz quantization of the symbol above is

S 2 (S + 1) 2 i∼j S i • S j ,
so that, up to a multiplicative factor, the quantization of the classical HAF is the quantum HAF. As we wish to study quantum selection, it is important that in this case h 1 = 0.

The low-temperature properties of the S = 1 2 HAF are still unknown. Various numerical Ansätze or exact diagonalizations 12,[21][22][23]25,[START_REF] Waldtmann | The European Physical[END_REF]34 predict a spin liquid phase with polynomial decay of correlations, which is consistent with experiments 33 . The large S analysis sheds some light on this behaviour as we will see.

If e i = (x i , y i , z i ), since sites in the Kagome lattice are connected in triangles, up to a constant the classical energy reads

triangles e i + e j + e k 2 .
The minimal classical set consists in configurations where, on each triangle of sites, spins form a great equilateral triangle on the sphere. This set has a highly nontrivial structure: the presence of loops of triangles makes it non-smooth. The classical minimal set for one hexagon of triangles already has a crossing point as one of the toy models, on which two smooth manifolds cross. An interesting subset of classical minimal configurations consists in planar configurations, which form a discrete set. It is believed that quantum order by disorder selects these configurations, thus reducing the semiclassical study to a 3 colours Potts model on the Kagome lattice (with Hamiltonian unknown so far). Some of those planar configurations have been proven 7 to be local minima for the function µ which is the criterion for quantum selection, but it is unknown whether these are the global minima for µ or not. The results in the S = 1 2 case are compatible with this approach as there is an extensive number of coplanar states, the majority of which having no long-range order; this contrasts with the SU (N )case 30 which would predict a unique, ordered, selected configuration.

B. Simple models for the Kagome lattice

An easy case which allows to understand the large S behavior of the HAF on the Kagome lattice consists in a loop of four triangles. In this situation the classical minimal set is (once accounted for the global SO(3) action) the union of three circles C 1 , C 2 , C 3 , two of each crossing at exactly one point. The crossing points correspond to planar configurations. There is a symmetry exchanging C 2 and C 3 . In Figure 2 we plot the value of µ along C 1 and along C 2 , with parameter an angle which is 0 or π on the crossings; this confirms the general belief that µ is minimal on planar configurations.

The Husimi tree, proposed by Douçot and Simon 14 , also serves as a toy model for the study of the Kagome lattice. It is depicted on Figure 1.

The advantage of this model is that the classical minimal set is much simpler than on the Kagome lattice. Indeed, on the Husimi tree, once the three vectors on a parent triangle are chosen along a great equilateral triangle on the sphere, there is one degree of freedom in the choice of the spins for each child triangle. Thus the minimal set is a torus of dimension (triangles) -1, parametrised by the angles between the equilateral triangles at neighbouring sites.

Douçot and Simon 14 reported that the classical frequencies are not constant on the classical minimal set: in particular, in this situation there is quantum selection (the selected points are presumed to be coplanar configurations except for the spins at the leaves which are free), but there is no thermal selection since there are equivalent for a class of phase space transformations which preserve the volume.

C. Anisotropic XXZ chain

Let us take up from an example proposed by Douçot and Simon 14 and define the following Hamiltonian acting on a closed chain of N spins:

H = J i S i • S i+1 + i S z i (S z i+1 -S z i ).
The principal term in the classical energy is

h 0 = J i e i • e i+1 + i z i (z i -z i+1 ).
The next-order contribution is

h 1 = 2J i e i • e i+1 + i (-2z i z i+1 + 5z 2 i -1).
If J < 0, the minimum of h 0 is reached on ferromagnetic configurations {e i = e}, indexed by S 2 . Near any of these minimal configurations, the linear spin wave theory is the same, up to a factor -J + 1 -z 2 in the potential. Hence µ is minimal as z = 0.

On ferromagnetic ordered configurations, one has

h 1 = JN + 3N z 2 .
Again h 1 is smaller when z = 0. The sum µ+h 1 , which is the criterion for quantum selection, is minimal as z = 0, hence the ground state is located on this set. In this paper, we reported evidence that quantum order by disorder does not have the same rules as thermal order by disorder. In experimental settings of lowtemperature quantum systems, there is competition between quantum and thermal selection. We present an analysis of orders of magnitude.

On the system SrCrGaO which is an experimental realization of the Kagome lattice, the interaction strength J is presumed 17 to be of order

J k B 50K.
In experimental realizations, the spin S cannot be very large so that the order of magnitude of the contribution µ + h 1 is also of order 10K × k B . This means that, below these temperatures, quantum selection predominates over thermal selection, since the magnitude of the quantum fluctuations is much greater. On the Kagome lattice there is no competition presumed between quantum and thermal selection, but in other cases it could even lead to a phase transition from thermal order (at medium temperatures) to quantum order (at very low temperatures). The temperatures involved in this analysis can be reached for large systems by modern experimental methods.

In our recent paper 9 we also computed the relative contributions at low temperature on systems for which the quantum selection criterion µ + h 1 is minimal at two points, one of which is a regular "miniwell" point, the other a crossing point. In this situation, if the temperature is such that thermal effects are of the same order as quantum effects, then the crossing point will be selected (the quantum fluctuations do not see the difference between the two points, and the thermal fluctuations select the one with maximal local zero dimension). However at lower temperatures, the regular point will be selected. The interpretation is that µ + h 1 acts as an effective Hamiltonian, which is smooth on the miniwell, but which is typically non-regular at the crossing point (see Figure 2). This confinement leads to an increased quantum energy (this shift is of order S -4/3 ). Hence there are more low-energy quantum states near the miniwell than near the crossing point. This is a theoretical instance of a phase transition, which is of course very peculiar (since µ + h 1 reaches the same value at two very different points).

B. Selection on the Kagome lattice

The actual computation of µ on examples, even as simple as a chain of triangles, requires the full diagonalization of a matrix whose size grows with the number of spins, at each minimal point. Variational approaches allow to show that special (usually planar) configurations are critical points for µ (the first derivative of µ vanishes at these points), but to show that these configurations are global minima requires additional techniques.

As illustrated in Section III, the local geometry of the minimal classical set plays a very important role. Points near which the classical minimal set is a smooth manifold are now quite well understood from a mathematical point of view. On a point where exactly two manifolds cross, there is a chance that quantum order by disorder selects the crossing, especially in symmetrical situations for which the function µ reaches a local minimum at the crossing. Conversely, if three or more manifolds cross at a point, with model the boundary of a hypercube, then the crossing point has no reason to be selected by the quantum system.

We believe that, near planar configurations on the Kagome lattice, the local structure of the classical minimal set is a direct product of structures with two manifolds crossing [START_REF]R 4 with potential x 2 y 2 + z 2 t 2[END_REF] , with quartic non-degenerate part (that is, they follow the model case above). Indeed, the quadratic and quartic terms in the energy, near a planar configuration, do not depend on the particular planar configuration, so that as soon as for one configuration one has a product of structures as above, it is the case for all configurations.

On systems where the classical minimal set is non smooth, such as the Kagome lattice, the parametrisation of this set is already a challenge. Numerical techniques which do not involve knowledge of the minimal set should be of help in tackling this problem.

C. Tunnelling

To conclude with, we address the issue of exponential precision in estimates related to Toeplitz operators. This problem is relevant in the context of tunnelling: it is generally hoped that, in the presence of symmetries, the ground state will tunnel between various configurations, and the spectral gap (or the inverse time needed for a quantum state to go from one configuration to another) will be of order exp(-cS) in the large spin limit, where c is a "tunnelling rate", related to some classical action.

Various attempts 1,3,4,8,11,15,16 have been made to study this phenomenon in the setting of spin systems, mainly by removing two antipodal points on the phase space (the sphere), thus formally transforming the phase space into R × S 1 in which usual (Weyl) quantization takes place with quantum state space L 2 (S 1 ). However, it is doubtful that these attempts yield the correct tunnelling rate. First, this manipulation changes the quantization procedure, and it is unclear whether there is a way to perform the computations which is consistent with the initial problem up to an error of order S -1 , let alone an exponentially small error. Second, rates of decay of order exp(-c -1 ) are notoriously delicate even in the simplest geometrical setting of Weyl quantization on R 2n , as detailed by Martinez [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF] . The basic difficulty is that one needs to extend data in complex space, which can be done only if the classical energy is real analytic, and only to a small distance from the real space. This puts a limit on the actual tunnelling rate. Lower bounds (Agmon estimates) on the tunnelling rate for Toeplitz operators were recently obtained by the author 10 .

Appendix: Computation of Toeplitz operators on the sphere

For the particular case of the sphere, one can build Toeplitz operators as on C d via the stereographic projection, which maps the sphere minus the north pole onto C in a holomorphic way.

Via this transformation, quantum states are holomorphic functions on C which have finite norm under the following Hermitian structure:

f |g H k = k + 1 π C f (w)g(w) (1 + |w| 2 ) k+2 dw.
The space H k consists of polynomials of degree less than k. As for the flat case, the monomials are orthogonal but not normalized. A Hilbert basis of H k is given by e j,k : z → k + 1 π k j z j .

To prove this (and perform further computations), we use the fact that, for 0 ≤ j ≤ k: Since this function is radial, the matrix elements e j,k |T k (z)|e j ,k H k are zero for j = j . Moreover,

e j,k |T k (z)|e j,k H k = 2j -k k + 2 .
Hence, in this basis, the operator T k (z) is k k + 2 times a diagonal operator with equidistributed diagonal values from -1 to 1; that is, the spin operator S z with 2S = k.

The states e j,k corresponds to spin states |S, m with m = j -S.

The abscissa x is mapped, via the stereographic projection, into the map

w → 2Re(w) 1 + |w| 2 .
This is the sum of a function of winding number -1 and a function of winding number 1. Hence the matrix of T k (x) in the natural basis is zero except on the over-and underdiagonal. The matrix elements are e j,k |T k (x)|e j+1,k H k = (k -j)(j + 1) k + 2 .

In this basis the matrix of the operator T k (x) is k k+2 S x .

By this method, the Toeplitz quantization of any polynomial in the coordinates can be computed; this yields Table I.

FIG. 1 .

 1 FIG. 1. Pieces of the Husimi tree (left) and the Kagome lattice (right)

FIG. 2 .

 2 FIG. 2. Numerical plot of µ for a loop of 4 triangles (insets), along C1 (left) and C2 (right).

1 .

 1 Let us compute the Toeplitz quantization of simple functions defined on the sphere.The height z is mapped, via the stereographic projection, in the map w → |w| 2 -1 |w| 2 + 1 .


Charles Bertucci

Jean-Michel Lasry

Pierre-Louis Louis Lions

Some remarks on Mean Field Games

come

Introduction

Introduction

This paper presents some results on three topics in the theory of Mean Field Games (MFG). It is divided in three parts, independent of each other for the most part. We present first some results on the uniqueness of solutions for MFG in the case of coupled dynamics. Then, we introduce MFG models with noise in the discrete state space case. We give some conditions for the problem to be well-posed. Finally we discuss the limit of MFG models as the intertemporal preference rate of the players goes to infinity. Those three subjects are described in details later on in each section. We briefly recall now a few well known facts on MFG.

MFG have been introduced by the last two authors in [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] Lasry | Mean field games[END_REF] and the field has known a tremendous development in the past ten years. MFG models address differential games involving an infinite number of indistinguishable players. In the absence of a common noise, to find a Nash equilibrium of the game reduces to solve a system of forward backward partial differential equations (PDE). This system consists in a backward Hamilton-Jacobi-Bellman equation, which is solved by the value function of the differential game a generic player is facing, and a forward Fokker-Planck equation which models the evolution of the population. Namely in the case of a game set on the d dimensional torus T d with a time horizon T , the MFG reduces to systems of the following kind (1)

     -∂ t u -ν∆u + H(x, ∇ x u, m) = 0 in (0, T ) × T d ; ∂ t m -ν∆m -div(D p H(x, ∇ x u, m)m) = 0 in (0, T ) × T d ; u(T ) = G(x, m); m(0) = m 0 in T d ;
where H is the Hamiltonian of the optimal control problem for the players, G the terminal cost of this problem and m 0 the initial distribution of players. The function u represents here the value function for a generic player and m is the density of players. Uniqueness (under suitable monotonicity conditions) and existence have been proved for this system in [START_REF] Lasry | Mean field games[END_REF][START_REF] Lions | Cours au college de france[END_REF]. Numerical methods are being developed, let us cite [START_REF] Achdou | Mean field games: Numerical methods[END_REF][START_REF] Lm Briceno-Arias | Proximal methods for stationary mean field games with local couplings[END_REF] for examples of this growing literature. Let us also mention the questions of long time average [START_REF] Cardaliaguet | Long time average of mean field games[END_REF][START_REF] Cardaliaguet | Long time behavior of the master equation in mean-field game theory[END_REF] or learning [START_REF] Cardaliaguet | Learning in mean field games: the fictitious play[END_REF]. In some very particular cases (the so-called "potential case") solutions of the MFG system can be obtain from a PDE optimal control problem [START_REF] Benamou | Variational mean field games[END_REF]. This approach is particularly successful to find weak solutions of (1) [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF]. This particular class of MFG is also somewhat similar to the optimal transport theory. Moreover a major axis of development of MFG is the probabilistic interpretation of such differential games. We refer to [START_REF] Carmona | Probabilistic analysis of mean-field games[END_REF][START_REF] Lacker | Mean field games via controlled martingale problems: existence of markovian equilibria[END_REF] for examples of such a point of view and to [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF][START_REF] Carmona | Probabilistic theory of mean field games: vol. ii, mean field games with common noise and master equations[END_REF] for a complete presentation of this probabilistic approach. Let us also recall that MFG are not limited to continuous control problem and that other setting have been studied like optimal stopping [START_REF] Bertucci | Optimal stopping in mean field games, an obstacle problem approach[END_REF][START_REF] Nutz | A mean field game of optimal stopping[END_REF][START_REF] Carmona | Mean field games of timing and models for bank runs[END_REF] or impulse control [START_REF] Bertucci | Fokker-planck equation of jumping particles and mean field games of impulse control[END_REF] for instance. Moreover we insist that in the presence of a common noise, the reduction of the MFG to the system (1) collapses and the study of the master equation is crucial for the understanding of the MFG. We refer the reader to [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF][START_REF] Lions | Cours au college de france[END_REF] for a detailed study of the master equation and to [START_REF] Cardaliaguet | Long time behavior of the master equation in mean-field game theory[END_REF][START_REF] Delaure | From the master equations to mean field games limits, fluctuations and large deviations[END_REF] for examples of applications.

1. Uniqueness in strongly coupled Mean Field Games 1.1. Formulation of the problem. In this section, we are interested in MFG in the case when the coupling between the players depend on their strategies. We refer to [START_REF] Lions | Cours au college de france[END_REF] for details on this question. In the case without common noise, the study of this strong coupling is the study of MFG system in which the Hamiltonian of the problem now depends on the measure associated to the distribution of the strategies of the players. We work here on the d dimensional torus T d and the system we are interested in is :

(2)

     -∂ t u -ν∆u + H(x, ∇u, m, µ) = 0 in (0, T ) × T d ; ∂ t m -ν∆m -div(D p H(x, ∇u, m, µ)m) = 0 in (0, T ) × T d ; m(0) = m 0 ; u(T ) = φ(m) in T d ;
where

H : T d × R d × P(T d ) × P(R d ) → R
denotes the Hamiltonian associated with the optimal control problem the players are facing. It is assumed to be smooth and D p H stands for the derivative with respect to its second variable. We denote by u the value function of the players, m the measure which describes their distribution in the state space T d and µ the measure associated to the controls of the players. Let us remark that because µ is the measure of the optimal controls used by the players, it satisfies the following relation :

(3) µ = (-D p H(x, ∇u, m, µ)) # m;
where G # m stands for the image measure of m by the measurable application G.

Existence of solutions of such a system was proven for particular Hamiltonians in [START_REF] Diogo A Gomes | On the existence of classical solutions for stationary extended mean field games[END_REF] ; we refer to [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] for a more general result of existence for (2) and to [START_REF] Carmona | A probabilistic weak formulation of mean field games and applications[END_REF][START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF] for the study of such a coupling in a probabilistic setup. In general, uniqueness of such solutions is not known. Before presenting the main result of this part, we make some preliminaries observations. More precisely, we state that the quantity (4)

T d
∇um is conserved in time under the assumption that the Hamiltonian H is of the form :

H(∇u, m)(x) = H(∇u(x)) -f (m(x)
). This result is quite simple but we believe it is worth mentioning as it does not seem to appear in the literature. And it shows that there may be properties enjoyed by the solutions which may play a role in uniqueness properties. Moreover, it will be useful for the study of the example we give at the end of this part.

Lemma 1. Assume that H does not depend on the space variable x and that it is of the form H(∇u, m)(x) = H(∇u(x))f (m(x)), then for any solution (u, m) of (1) the quantity Proof. We only prove the statement in the case of a smooth solution (u, m) as it easily extends to weaker notions of solutions. Multiplying by ∂ x i m the equation satisfied by u and integrating the PDE satisfied by u on (0, t) × T d , we obtain

0 = t 0 T d (-∂ t u -ν∆u + H(∇u) -f (m))∂ x i m; = t 0 T d (-∂ t m + ν∆m)∂ x i u + H(∇u)∂ x i m - T d f (m)∂ x i m + T d (u(0)∂ x i m 0 -u(t)∂ x i m(t)); = t 0 T d -div(D p H(∇u)m)∂ x i u + H(∇u)∂ x i m + T d (u(0)∂ x i m 0 -u(t)∂ x i m(t)).
We then deduce by integrating by parts all the terms of the last line :

0 = t 0 T d mD p H(∇u) • ∂ x i ∇u -D p H(∇u) • m∂ x i ∇u - T d (m 0 ∂ x i u(0) -m(t)∂ x i u(t)); = - T d (m 0 ∂ x i u(0) -m(t)∂ x i u(t)).
Remark 1. We thus see that the invariance by translation of the Hamiltonian implies that (5) is independent of time. Such a quantity is closely related to the average control of the players. Hence such a lemma leads us to think that one can obtain information on the average control using assumptions on the space dependence of the Hamiltonian.

As we are led to distinguish local and global dependence of the Hamiltonian in ∇u, we introduce the following notation :

• We note G(f ) when the dependence of the function G in the function

f is local, i.e. G(f )(x) = G(f (x)). • We note G(f (•))
for a general dependence of the function G in the function f . We shall only use this notation in the case when there might be an ambiguity.

1.2.

A uniqueness result. We present here a uniqueness result for the system :

(6)          -∂ t u -ν∆u + H(∇u) -G(∇u(•), m(•)) • ∇u = f (u, m) in (0, T ) × T d ; u(T ) = φ(m(T )) in T d ; ∂ t m -ν∆m -div(m(D p H(∇u) -G(∇u(•), m(•)))) = g(u, m) in (0, T ) × T d ; m(0) = m 0 in T d .
Such a system models a MFG in which the players are "pushed" by the mean field control of the other players. We recall the usual conditions under which we expect uniqueness to hold for solutions of ( 6) when G = 0:

• The hamitlonian H is convex.

• The terminal cost φ is monotone.

• There exist α and β such that (g, f ) + (αm, βu) is monotone in (u, m).

• Either H is strictly convex, (g, f )+(αm, βu) is strictly monotone in (u, m), φ is strictly monotone or (g, f ) is local in (u, m).

We say that the hypothesis (H1) is satisfied if those four requirements are satisfied. We also recall the notion of monotonicity for the sake of completeness. To define this notion properly one usually needs to make precise the functions spaces on which f and g are defined and take their values. To fix ideas, we recall the notion of monotonicity in the case of maps of L 2 : a map F from L 2 (Ω) into itself is said to be (resp. strictly) monotone if for any u 1 , u 2 ∈ L 2 (Ω) :

Ω (F (u 1 ) -F (u 2 ))(u 1 -u 2 ) ≥ 0 (resp > 0 if u 1 = u 2 )
We define by T the group of transformation :

T := {(v, m) → (v • τ x , m • τ x )|τ x is the translation by x ∈ T d }
We are now able to state the following result :

Theorem 1. Let us assume that G and φ are invariant by translations, i.e. that they do not depend on the space variable x and that G is invariant under the group of transformation T . Then under the usual condition that (H1) is satisfied, there exists at most one solution (u, m) of (6).

Proof. We denote by (u 1 , m 1 ) and (u 2 , m 2 ) two solutions of the system [START_REF] Cardaliaguet | Learning in mean field games: the fictitious play[END_REF]. For i ∈ {1, 2} we define (ũ i , mi ) by

(ũ i , mi )(t, x) = (u, m)(t, x + t 0 G(∇u i (s), m i (s))ds).
Because G and φ are invariant by translations, we deduce that for i ∈ {1, 2}, (ũ i , mi ) is a solution of the MFG system :

     -∂ t ũi -ν∆ũ i + H(∇ũ i ) = f (ũ i , mi ) in (0, T ) × T d ; ∂ t mi -ν∆ mi -div(D p H(∇ũ i ) mi ) = g(ũ i , mi ) in (0, T ) × T d ; mi (0) = m 0 ; ũi (T ) = φ( mi (T )) in T d .
Let us note that the assumption that H does not depend explicitly on the space variable is crucial. Because the assumption (H1) holds, there is a unique solution of this system and (ũ 1 , m1 ) = (ũ 2 , m2 ). Now let us remark that because G is invariant by the transformation which sends (u, m) on (ũ, m) we obtain that

G(∇u 1 , m 1 ) = G(∇u 2 , m 2 ) = G(∇ũ, m).
Thus, (u 1 , m 1 ) = (u 2 , m 2 ) and there is a unique solution of (6).

G(∇u(•), m(•)) = T d
g(∇u(y))m(y)dy seem to be quite general for applications. See the example studied in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] for example. Moreover such a term satisfies the assumption of theorem 1. However it is true that the assumption that this term is linear in the "p" variable for the Hamiltonian is rather strong (i.e. that G appears in the Hamiltonian only through G(∇u, m) • ∇u).

Concerning the question of the invariance by translation, the following example shows that this assumption can be extended in some cases. But it also shows that there is a complex interaction between the uniqueness and the type of dependence of the Hamiltonian in the control of the other players. Let us introduce the following system which satisfies all the assumptions of the theorem 1 except for the fact that the terminal cost φ now depends on the space variable.

(7)

     -∂ t u -ν∆u + 1 2 |∇u| 2 -λ( ∇um) • ∇u = 0 in (0, T ) × T d ; ∂ t m -ν∆m + div((∇u -λ( ∇um))m) = 0 in (0, T ) × T d ; u(T ) = φ(x); m(0) = m 0 (x) in T d .
Even though this section is not particularly concerned with the question of existence, let us mention that a solution of such a system exists as soon as φ is a Lipschitz function. The following lemma gives a general property satisfied by the solutions of [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF]. Proposition 1 details a precise example of a function φ which yields uniqueness even though it is not constant : Lemma 2. Let us denote by (u, m) a solution of (7), we define A = T d ∇um and we denote by u 0 the solution of the HJB equation

-∂ t u 0 -ν∆u 0 + 1 2 |∇u 0 | 2 = 0; u 0 (T ) = φ.
Then the following holds

(8) A = T d ∇u 0 (x + λAT, 0)m 0 (x)dx.
Proof. Let us remark that the proof of lemma 1 applies also for [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], hence (u, m) satisfies :

d dt T d ∇um = 0.
Thus A is well defined. Now let us remark that a simple change of variable yields:

u(t, x) = u 0 (t, x + λA(T -t)).
Hence, evaluating T d ∇um at t = 0 we obtain that

A = T d ∇u 0 (x + λAT, 0)m 0 (x)dx.
Each solution A of this equation produces a solution for the system [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF]. Hence if φ and m 0 are such that the previous equation admits a unique solution then we have proved uniqueness. Such a situation is possible if we obtain a precise bound on ∇u 0 for instance. We give now an example of such a situation : Proposition 1. If D 2 φ(x) ≤ c Id holds for some c > 0 in the sense of distributions, then uniqueness in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF]

holds if λ < 1 + cT cT .
Proof. Under those assumptions, in view of classical semi-concavity estimates for HJB equations, the Hessian matrix D 2 u 0 of u 0 also satisfies in the sense of distributions for any 0 ≤ t ≤ T :

D 2 u 0 (t) ≤ c 1 + c(T -t)
Id.

Hence the equation ( 8) satisfied by A admits a unique fixed point if :

λ < 1 + cT cT ; because then A -T d ∇u 0 (x + λAT, 0)m 0 (x)dx is a strictly increasing function of A.
Remark 2. It is clear from [START_REF] Cardaliaguet | Long time behavior of the master equation in mean-field game theory[END_REF] that without an assumption on φ, uniqueness may not hold. Moreover such a proof can be adapted to a case in which we add a dependence f (m) in the HJB equation.

Remark 3. Let us note that the condition in this proposition can be interpreted as the bigger c is (i.e. the more the system depends on the space variable x), the more the effect of the average strategy (the parameter λ) needs to be small. Also, if λ < 1, then the condition is satisfied independently of c, thus the condition λ < 1 yields uniqueness even in the case c = +∞, that is for general φ.

Common noise in discrete state space MFG

The addition of a common noise in the MFG setting remains one of the most important questions in the MFG theory. Even though [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF][START_REF] Lions | Cours au college de france[END_REF] provide a detailed study of the so-called master equation, several problems remain open. We can cite for example wether or not there exist weaker regularity assumptions than the one needed in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF], or the possibility to give an extended sense of solutions which yields existence results in the non-monotone case. We present here a discrete state space master equation which models the presence of a common noise in the MFG. We first recall some facts on the master equation without common noise before explaining what we mean by common noise in the discrete state space. We then prove the well-posedness of such an equation as well as some limit equations which can be obtained from our model.

2.1.

The master equation for a discrete state space. In a discrete state space, without a common noise, the infinite dimensional master equation reduces to a first order non conservative hyperbolic system of the following form (see [START_REF] Lions | Cours au college de france[END_REF] for instance) :

(9) ∂ t U(t, x) + (F (x, U) • ∇)U = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d .
Let us recall that in such a system, U is a map from (0, t f ) × R d to R d , F and G are applications from R 2d to R d . The time t f is the final time (fixed in (0, ∞)) and the initial condition is U 0 . We also recall that the system has to be understood as follows : for all 1 ≤ i ≤ d, the ith component U i of U satisfies :

∂ t U i (t, x) + F (x, U) • ∇U i = G i (x, U) in (0, t f ) × R d U i (0, x) = (U 0 ) i (x) in R d .
The MFG interpretation of ( 9) is that x represents the number of players in the d possible states, meaning that x 1 is the number of players in the first state, x 2 the number of players in the second and so on. The interpretation of the map U is that the ith component U i (t, x) of U(t, x) is the value of the optimal control problem for a generic player in the state i, with the repartition of other players being x at the time t. Let us note that contrary to the classical MFG system [START_REF] Achdou | Mean field games: Numerical methods[END_REF] or to the master equation presented in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF], the equation is written here forward in time, instead of backward (i.e. we reversed time with t → Tt).

We also recall that there exists a system of characteristics for [START_REF] Cardaliaguet | Long time average of mean field games[END_REF]. It is defined for any x 0 ∈ R d by ( 10)

     d ds V (s) = G(Y (s), V (s)); d ds Y (s) = F (Y (s), V (s)); Y (0) = x 0 ; V (0) = U 0 (x 0 ).
Using the previous characteristics, we can define a solution U of ( 9) by :

U(t, Y (t)) = V (t);
One can show ( [START_REF] Lions | Cours au college de france[END_REF]) that it is possible to define a solution of ( 9) using the characteristics under a monotonicity assumptions. The system [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] is the analogous of the usual MFG system (1) in a discrete state space. Thus, many results on ( 9) can be proven using this system of characteristics [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]. However, addressing the regularity and the well-posedness of (9) without using (10) leads to techniques and proofs which are adaptable to the addition of a common noise, whereas the use of the system [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] collapses in the presence of a common noise.

As the notion of monotonicity is crucial in all the results we present in this section, we define here what we mean by monotone. A map

V from R d into itself is said to be monotone if ∀x, y ∈ R d , < V (x) -V (y), x -y >≥ 0.
The map V is said to be α monotone if

∀x, y ∈ R d , < V (x) -V (y), x -y >≥ α|x -y| 2 .
Here, < x, y >= x • y denotes the usual scalar product between x and y.

2.2.

The master equation in the presence of noise. We propose here a master equation in finite dimension with additional terms which model jumps which can occur in the population. Those jumps can be random and thus can be assimilated to noise in the MFG. The jumps are described by an application T . We give two examples of such maps T to fix ideas, and explain how the jumps have to interpreted, depending on the nature of the noises.

• If the map T is defined by :

T (x) = (x 2 , x 1 , x 3 , ..., x d );
then the map T models jumps which only concerns players in the first or second state. The players in the first state are sent to the second one and vice versa. If all the players face this jump at the same time, then all the players in the first state and the ones in the second one are exchanged. If only a proportion 0 ≤ θ ≤ 1 of the players in each states jump simultaneously, then the state of the population after the jump is given by θT (x) + (1θ)x, if x is the state of the population before. • If the map T is defined by :

T (x) = ( x 1 2 + δ, x 2 + x 1 2 , x 3 , .., x d );
where δ > 0 is fixed, then if a player is in the first state, it has a probability one half to jump to the second state. This jump is naturally associated with a structure of noise, independently of the randomness that may exist concerning the players this jump affects or the time at which it occurs. If a proportion 0 ≤ θ ≤ 1 of the players in all states are simultaneously affected by this jump, then a fraction θ 2 of the players in the first state jump in the second one and an amount of players corresponding to θδ is added to the game in the first state.

The noise structures modeled by this kind of terms are numerous. Indeed with this second example, we show that there can be randomness in the fact that a player will jump or not, just as there can be randomness on the state in which it is going to jump. There can also be randomness on the times at which those jumps occur. The correlations between those times can be quite general as we shall now see through some examples of master equations.

We now make precise how jumps are taken into account in the master equation, depending on the correlations of the jumps. We give three examples of master equation which models jumps occurring at deterministic times, or when the times of the jumps are random but common to every players and finally when they are random and independent identically distributed (iid) for the players. In those examples, T is a differentiable map from R d to itself.

If there is only one jump characterized by T which occurs at the deterministic time t 1 ∈ (0, t f ), then the master equation is :

(11)          ∂ t U(t, x) + (F (x, U) • ∇)U = G(x, U) in (t 1 , t f ) × R d ; U(t + 1 , x) = (DT (T x)) * U(t - 1 , T (x)) in R d ; ∂ t U(t, x) + (F (x, U) • ∇)U = G(x, U) in (0, t 1 ) × R d ; U(0, x) = U 0 (x) in R d ;
where t f > 0 is the time horizon and (DT ) * stands for the adjoint of the differential of the map T .

If at random times given by an exponential law of parameter λ, all the players jump according to T , then the master equation is :

(12) ∂ t U(t, x) + (F (x, U) • ∇)U + λ(U -(DT (T (x))) * U(t, T (x))) = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d ;
where λ is the parameter of the Poisson process which describes the jumps. In [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF], the term

λ(U -(DT (T x)) * U(t, T x))
is understood as the players anticipating the noise. The proper derivation of such a term is technical and we do not enter in such calculations here. We refer to [START_REF] Conze | A system of non-linear functional differential equations arising in an equilibrium model of an economy with borrowing constraints[END_REF][START_REF] Conze | Borrowing constraints and international comovements[END_REF] for a proper derivation of this term in a different setting, when T is a linear map.

If all the players jump according to T at times given by iid Poisson processes of paramter λ, then the master equation is given by : (

         G(x, U) =∂ t U(t, x) + (F (x, U) • ∇)U + λ(Id -(DT (x)) * )U(t, x) + λ((Id -T (x))x • ∇)U in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d . 13)
In this equation, the drift term λ((Id -T (x))x • ∇)U is interpreted as the effect of the flow of players which are jumping. Indeed, because the jumps are independent and there is an infinity of players, there is always a constant proportion of players which are jumping. This phenomenon induces a dynamic in the population which is taken into account in this drift term. Remark 4. Let us remark that a master equation can be written in a similar way if some of the above three phenomena happen together in the MFG.

Propagation of monotonicity and regularity.

In this section, we shall show that under some assumptions on the form of T , the solutions of ( 11), [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF] and ( 13) have some monotonicity properties. We then show how we can establish regularity properties from the propagation of the monotonicity. Proposition 2. Assume that T is affine. Let U be a classical solution of ( 11), [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF] or [START_REF] Carmona | Probabilistic theory of mean field games: vol. ii, mean field games with common noise and master equations[END_REF]. If (G, F ) (seen as a map from R 2d into itself ) is monotone and U 0 is monotone, then U is monotone for all time.

Proof. The proof of this result follows the arguments in [START_REF] Lions | Cours au college de france[END_REF]. We write the affine map as T = S + e where e ∈ R d and S is a linear map.

The case of (11) :

Let U be a classical solution of (11), we denote by V another classical solution of (11) and we define W by W (t, x, y) =< U(t, x) -V (t, y), xy > .

Let us note that W satisfies the following equation in ((0, t 1 ) ∪ (t 1 , t f )) × R d :

∂ t W + F (x, U) • ∇ x W + F (y, V ) • ∇ y W = < G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) >;
Let us now take V = U. Because U 0 is monotone, we deduce that W (t, x, y) ≥ 0 for all x, y ∈ R d . We also deduce from the monotonicity of (G, F ) that the right hand side of the previous equation is positive everywhere. Thus we deduce form the maximum principle (lemma 3 in the appendix) that U is monotone (W is positive) in (0, t 1 ). Now let us remark that the relation satisfied by U at t 1 yields for W :

W (t + 1 , x, y) = W (t - 1 , T x, T y). Thus W is positive at t +
1 and thus W is positive for all time t ∈ (0, t f ), by repeating the same argument.

The case of (12):

Let U be a classical solution of ( 12), we denote by V another classical solution of ( 12) and we define again W by W (t, x, y) =< U(t, x) -V (t, y), xy > .

Let us note that W satisfies the following equation :

∂ t W +F (x, U) • ∇ x W + F (y, V ) • ∇ y W + λW -λ < S * U(t, T x) -S * V (t, T y), x -y > =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) >;
which can be rewritten in :

∂ t W +F (x, U) • ∇ x W + F (y, V ) • ∇ y W + λW -λW (t, T x, T y) =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) > .
By the maximum principle (lemma 3 in the appendix), we deduce that U is monotone for all time. This is a straightforward consequence of the monotonicity of (G, F ) and U 0 , taking V = U in W .

The case of ( 13) :

Let U be a classical solution of (13), we denote by V another classical solution of (13) and we define once again W by

W (t, x, y) =< U(t, x) -V (t, y), x -y > .
The appliccation W satisfies the following equation :

∂ t W + F (x, U)•∇ x W + F (y, V ) • ∇ y W + λ(S -Id)x • ∇ x W +λ(S -Id)y • ∇ y W + λ < U -V, (Id -S)(x -y) > =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) > +λ < U-V, (Id -S)(x -y) >;
Thus W satisfies :

∂ t W + F (x, U)•∇ x W + F (y, V ) • ∇ y W + λ(S -Id)x • ∇ x W +λ(S -Id)y • ∇ y W =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) >;
Once again, by the maximum principle (lemma 3 in the appendix), we deduce that U is monotone for all time by taking V = U and using the monotonicity of U 0 and (G, F ).

Remark 5. Let us note that in particular, under the assumptions of proposition 2, there exists at most one classical solution of ( 11), ( 12) and ( 13). Indeed uniqueness follows readily from the positivity of W (see [START_REF] Lions | Cours au college de france[END_REF]).

As we can see in the proof of the previous statement, the propagation of the monotonicity of U is really a "strong" property of ( 11), ( 12) and ( 13), in the sense that it does not depend on λ, t 1 or on T . The different terms in each system only reinforce the propagation of monotonicity.

We now prove that under stronger assumptions on U 0 and (G, F ), we can deduce from the propagation of monotonicity, some regularity for the system [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF]. Similar results can be obtain for [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF] and ( 13) by similar arguments.

Theorem 2. Assume that the map T is affine and that U 0 , F and G are Lipschitz continuous. Let U be a solution of [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF]. If both U 0 and (G, F ) are monotone and either U 0 and G are α monotone or F is α monotone (for some α > 0), then U is Lipschitz in space, uniformly on (0, t f ), for all t f > 0.

Proof. We here only prove the result in the case in which both U 0 and (G, F ) are α monotone. The proof of the complete statement is given in the appendix. This proof also follows the argument in [START_REF] Lions | Cours au college de france[END_REF]. Once again we note the affine map T = S + e with S linear and e ∈ R d constant. We define W and Z β by

W (t, x, ξ) =< U(t, x), ξ >; Z β (t, x, ξ) =< ξ, ∇ x W (t, x, ξ) > -β|∇ x W (t, x, ξ)| 2 .
We now write the PDE satisfied by Z β :

(14) ∂ t Z β + < F (x, ∇ ξ W ), ∇ x Z β > + < D p F (x, ∇ ξ W )∇ ξ Z β , ∇ x W > -< D p G(x, ∇ ξ W )∇ ξ Z β , ξ > + λ(Z β -Z β (t, T x, T ξ -e)) =< ∇ x G(x, ∇ x W )ξ, ξ > -< D p G(x, ∇ ξ W )∇ x W, ξ > -< ∇ x F (x, ∇ ξ W )∇ x W, ξ > + < D p F (x, ∇ ξ W )∇ x W, ∇ x W > -2β < ∇ x G(x, ∇ x W )ξ, ∇ x W > +2β < ∇ x F (∇ ξ W )∇ x W, ∇ x W > + βλ |∇ x W | 2 -2 < ∇ x W (t, T x, T ξ -e), S∇ x W > +|∇ x W (t, T x, T ξ -e)| 2 .
For the sake of clarity, we do not detail the calculation which leads to the previous equation, which easily follows from the chain rule. Let us remark that the following holds :

|∇ x W | 2 -2 < ∇ x W (t, T x, T ξ-e), S∇ x W > +|∇ x W (t, T x, T ξ-e)| 2 ≥ |∇ x W | 2 -|S∇ x W | 2 .
Because (G, F ) and U 0 are α monotone, we deduce that :

∂ t Z β + < F (x, ∇ ξ W ), ∇ x Z β > + < D p F (x, ∇ ξ W )∇ ξ Z β , ∇ x W > -< D p G(x, ∇ ξ W )∇ ξ Z β , ξ > + λ(Z β -Z β (t, T x, T ξ -e)) ≥ α(|ξ| 2 + |∇ x W | 2 ) + βλ(|∇ x W | 2 -|S∇ x W | 2 ) -2β < ∇ x G(x, ∇ x W )ξ, ∇ x W > +2β < ∇ x F (∇ ξ W )∇ x W, ∇ x W >; ≥ α(|ξ| 2 + |∇ x W | 2 ) + βλ(1 -||S|| 2 )|∇ x W | 2 -2β(||∇ x G|| • |ξ| • |∇ x W | + ||∇ x F || • |∇ x W | 2 ).
Hence, if β satisfies the inequality [START_REF] Carmona | A probabilistic weak formulation of mean field games and applications[END_REF], then Z β is positive for all time (by lemma 3 in appendix). [START_REF] Carmona | A probabilistic weak formulation of mean field games and applications[END_REF] β ≤ min( α λ(||S|| 2 -1)

+ + ||∇ x G|| + 2||∇ x F || , ||∇ x U 0 ||).
From the fact that Z β is positive for all time, we deduce that :

∀t, x, ξ : |∇ x W (t, x, ξ)| ≤ β -1 |ξ|.
Hence U is β -1 Lipschitz, uniformly in time.

Remark 6. Let us notice that the constant β involved in the proof only depends on α, t f , λ(||DT || 2 -1) + and on the Lipschitz constants of U 0 , F and G. Thus if the application S is non-expansive, then the constant β can be chosen independently of λ and T , thus we can expect that the sequence of solutions (U λ ) λ>0 has a limit as λ goes to infinity, extracting a subsequence if necessary.

This result emphasis the fact that the monotonicity of the system "implies" the regularity of the solution. Indeed the Lipschitz regularity is enough to prove higher order regularity see [START_REF] Lions | Cours au college de france[END_REF] for instance. Finally we end this discussion on the well-posedness of this system of conservation laws with this straightforward application : Theorem 3. Let X be a compact set of affine functions and µ a measure on X . Then, under the assumptions of theorem 2, there exists a unique smooth monotone solution U of :

∂ t U(t, x) + (F (x, U) • ∇)U + λ X (U -(DT ) * U(t, T x))dµ(T ) = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d .
2.4. Asymptotic differential operators. In this section, we present how we can derive, from a common noise term, higher order terms in [START_REF] Carmona | Probabilistic theory of mean field games: vol. i, mean field fbsdes, control, and games[END_REF](first order and second order terms) which conserve the monotonicity property, and thus the wellposedness of this equation. In the previous section we showed that adding the term (T is assumed to be linear in all the rest of this section) :

λ(U -T * U(T •))
does not alter the propagation of monotonicity. Formally, if we set λ = ǫ -1 and T = Id + ǫS, then

ǫ -1 (U -T * U(T •)) -→ ǫ→0 -(Sx • ∇)U(x) -S * U(x).
Also if we define λ = ǫ -2 ; T + = Id+ǫS and T -= Id-ǫS, then once again formally :

ǫ -2 (U -T * + U(T + •) + U -T * -U(T -•)) -→ ǫ→0 -2S * DU(x).Sx -D 2 U.Sx.Sx
Thus the propagation of monotonicity shall hold for these two equations : (16)

∂ t U(t, x) + (F (x, U) • ∇)U -(Sx • ∇)U(x) -S * U(x) = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d ; (17) ∂ t U(t, x) + (F (x, U) • ∇)U -S * (Sx • ∇)U(x) -2(Sx • ∇ 2 • Sx)U = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d .
To avoid going into technical difficulties we do not address here the question of the regularity of ( 16) and [START_REF] Conze | A system of non-linear functional differential equations arising in an equilibrium model of an economy with borrowing constraints[END_REF]. We restrict ourselves to the proof of the fact that the monotonicity indeed propagates, as it is expected. The following result presents this fact. Theorem 4. Assume that U 0 and (G, F ) are monotone and let U be a solution of either [START_REF] Carmona | Mean field games of timing and models for bank runs[END_REF] or (17), then U is monotone for all time.

Proof. We begin with the case of ( 16). We consider U and V two solutions of ( 16). As we did in the proof of 2, we define W by

W (t, x, y) =< U(t, x) -V (t, y), x -y > .
Let us remark that W satisfies :

∂ t W +F (x, U) • ∇ x W + F (y, V ) • ∇ y W -< (∇ x W -∇ y W ), Sx -Sy > =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) > .
Hence, concluding as we did in proposition 2, if W is positive at the initial time, it stays positive for all time and thus the propagation of monotonicity is proved.

If U and V are solutions of ( 17), then W satisfies :

∂ t W +F (x, U) • ∇ x W + F (y, V ) • ∇ y W -Sx∇ xx W Sx -Sy∇ yy W Sy =< G(x, U) -G(y, V ), x -y > + < F (x, U) -F (y, V ), U(t, x) -V (t, y) > .
Once again, we can conclude with lemma 3 (in appendix) that if W is positive at the initial time it remains positive for all time.

From MFG to agent based models

In this last part, we investigate to which extent MFG models can approach agent based models (ABM). The terminology ABM is often used in Economics and other sciences to refer to a model in which macroscopic phenomena are captured by aggregating individual actions. The main difference with MFG models is that in the ABM, the agents do not anticipate on the behavior of the other agents, hence they are not entirely rational but obey a given "mechanical" rule. For this reason, ABM are purely forward models. The main idea of this section is that one can measure the weight of anticipation for a given agent in a MFG model through the inter temporal preference parameter, thus taking this parameter to infinity should allow to recover an agent-based model from a MFG model. Indeed the players will no longer anticipate as they have no interest in the future. In the following MFG system the inter temporal preference parameter is always denoted by λ (which has nothing to do with the parameter λ in the part on the common noise) :

(18)      -∂ t u -ν∆u + H(t, x, ∇u, m) + λu = 0 in (0, T ) × R d ; ∂ t m -ν∆m -div(D p H(t, x, ∇u, m)m) = 0 in (0, T ) × R d ; u(T ) = φ(m(T )); m(0) = m 0 in R d .
In this system, H(x, p, m) still denotes the Hamiltonian of the optimal control problem faced by the players. As λ increases, the players are more and more interested in the present over the future. The main idea of this section is that letting λ go to infinity, one expects that u goes to 0 and that (18) reduces to [START_REF] Delaure | From the master equations to mean field games limits, fluctuations and large deviations[END_REF] ∂ t m -ν∆mdiv(D p H(t, x, 0, m)m) = 0 in (0, T ) × R d ; m(0) = m 0 .

In the first section we work in a particular case and show how we can establish such a convergence. In the second section, we indicate some extensions of this model without giving proofs. Because this remark on MFG is quite general, we prefer to explain the approach on a simple example rather than trying to obtain the most general results.

3.1. An example of convergence results. We here give some assumptions we make on the Hamiltonian for this section :

• H is Lipschitz in x, t and p, uniformly in m.

• H : m → ((t, x, p) → H(t, x, p, m)) is continuous from C((0, T ), P) to the set of Lipschitz functions. The following result captures the idea that MFG models approach ABM. Theorem 5. For any λ > 0, we denote by (u λ , m λ ) ∈ C 1,2,α × C((0, T ), P) a solution of (18) (with 0 < α < 1). This sequence is bounded and any limit point m * of the sequence (m λ ) λ>0 is a weak solution of :

∂ t m -ν∆m -div(D p H(t, x, 0, m)m) = 0 in (0, T ) × R d ; m(0) = m 0 in R d .
Proof. From classical estimates on Hamilton-Jacobi-Bellman equations and on Fokker-Planck equations, we know that solutions of [START_REF] Conze | Borrowing constraints and international comovements[END_REF] exist for all λ > 0, as a consequence of Schauder's fixed point theorem. For any sequence (u λ , m λ ) λ of solutions of ( 18), (u λ ) λ>0 is bounded in L 2 (H 2 ). Indeed, if we multiply the equation satisfied by u λ by -∆u λ and we integrate, we obtain (20)

- T t R d 1 2 d dt |∇u λ | 2 +ν T t R d (∆u λ ) 2 +λ T t R d |∇u λ | 2 = T t R d
∆u λ H(s, x, ∇u λ , m λ )dxds;

we then deduce

ν T t R d (∆u λ ) 2 + 1 2 R d |∇u λ | 2 (t) ≤ 1 2 R d |∇u λ | 2 (T )+ T t R d ∆u λ H(s, x, ∇u λ , m λ )dxds.
We obtain from the growth assumption on H that (∆u λ ) λ is a bounded sequence of L 2 ((0, T ) × R d ). Thus, (u λ ) λ converges to 0 in L 2 ((0, T ),

H 1 (R d )). Because (D p H(•, •, ∇u λ , m λ )) λ is a bounded sequence in L ∞ , (m λ )
λ is a bounded sequence of C((0, T ), P) where we have equipped P with the Monge-Kantorovich distance. Hence, by passing to the limit in the weak formulation of the equation satisfied by m λ , we deduce that any limit point of this sequence is a weak solution of the Fokker-Planck equation [START_REF] Delaure | From the master equations to mean field games limits, fluctuations and large deviations[END_REF].

We now show that an ABM can be approximated by a MFG system. The result we present here is that, if it is regular enough, there is always at least one solution of the agent based model which is the limit of a MFG. Indeed let us take an ABM defined by a smooth vector field B such that :

∃C > 0, ∀t ∈ (0, T ), ∀x ∈ R d , ∀m ∈ C((0, T ), P), ||B(t, x, m)|| L ∞ ≤ C.
The corresponding so called agent-based model is then :

(21) ∂ t m -ν∆m -div(B(t, x, m)m) = 0 in (0, T ) × R d ; m(0) = m 0 .
Let us take any δ > 0 and define by H the following hamiltonian :

H(t, x, p, m) = B(t, x, m) • p + δ|p| 2 .
Proposition 3. For any sequence of solutions (u λ , m λ ) λ of :

(22)      -∂ t u -ν∆u + H(t, x, ∇u, m) + λu = 0 in (0, T ) × R d ; ∂ t m -ν∆m -div(D p H(t, x, ∇u, m)m) = 0 in (0, T ) × R d ; u(T ) = 0; m(0) = m 0 in T d ;
any accumulating point of the bounded sequence (m λ ) λ is a solution of the ABM [START_REF] Lacker | Mean field games via controlled martingale problems: existence of markovian equilibria[END_REF].

Proof. In view of theorem 5, the sequence of solutions of the MFG system ( 22) is compact and its accumulating points are solutions of the agent-based model(21).

3.2. More general models. In this section we give examples of how one can apply the previous remark to more general models. The first one is a MFG, where a part of the running cost has a proportion of the total cost, which varies with the parameter λ. The second one is a higher order approximation of the limit model. We conclude with cases in which a common noise can occur.

3.2.1. Relative running cost. We here present the limit as λ goes to infinity of the following MFG system :

     -∂ t u -ν∆u + H(x, ∇u, m) + λ(u -ψ(m)) = 0 in (0, T ) × R d ; ∂ t m -ν∆m -div(mD p H(x, ∇u, m)) = 0 in (0, T ) × R d ; u(T ) = φ(m(T )); m(0) = m 0 in R d ; (23)
where ψ is a smooth function of m. The interpretation of such a system, in particular of such an HJB equation, is that ψ(m) stands for a cost which has an increasing importance for the players as λ goes to infinity. This cost should be interpreted as a cost the players anticipate with the same weight, whatever the length of the game is. The ABM limit of ( 23) is the following :

(24) ∂ t m -ν∆m -div(mD p H(x, ∇ x (ψ(m)), m)) = 0 in (0, T ) × R d ; m(0) = m 0 in R d .
Let us remark that from the results on uniqueness in MFG ( [START_REF] Lasry | Mean field games[END_REF][START_REF] Lions | Cours au college de france[END_REF]), the usual conditions under which (23) has a unique solution, implies in particular that ψ is monotone in m and H is convex in p. On the other hand, the higher order term in the first line of ( 24) is (in the case of a local ψ to simplify notation):

-ν∆m -mψ ′ (m)tr(D pp H(x, ∇ x (ψ(m)), m) • ∇ 2 xx m);
where tr stands for the trace operator. Hence, if the Hamiltonian H is convex in p and ψ is monotone in m, we deduce that ( 24) is a non-linear parabolic equation. This remark makes an obvious link between the well-posedness of the two models and the monotonicity of ψ.

3.2.2. Higher order approximation. We present here a higher order approximation of the Fokker-Planck equation in [START_REF] Conze | Borrowing constraints and international comovements[END_REF] in the limit λ goes to infinity. Let us observe that, at least formally, we have the following in (0, T ) × T d :

u(t, x) ≈ λ→+∞ -1 λ H(x, 0, m) + O(λ -2 ).
Thus if we neglect terms of order λ -2 , the resulting Fokker-Planck equation is :

(25) ∂ t m -ν∆m -div(mD p H(x, ∇ x ( -1 λ H(x, 0, m)), m)) = 0 in (0, T ) × R d ; m(0) = m 0 in R d .
As we did for the relative running cost, we show that under the usual condition of uniqueness for [START_REF] Conze | Borrowing constraints and international comovements[END_REF], the equation ( 25) is parabolic. The higher order term in m in (25) is, when H is local in m :

-ν∆m + 1 λ mD z H(x, 0, m)tr(D pp H(x, ∇ x ( -1 λ H(x, 0, m)), m) • ∇ 2 xx m);
where tr still stands for the trace operator and D z H for the derivative of H with respect to its third variable. We recall here the usual conditions under which (18) has a unique solution ( [START_REF] Lions | Cours au college de france[END_REF]) : for any (x, p, z)

∈ T d × R d × R + : zD pp H(x, p, z) z 2 D pz H(x, p, z) z 2 D pz H(x, p, z) -D z H(x, p, z) ≥ 0.
Thus, as in the previous case, the uniqueness of solutions of (18) implies the wellposedness of [START_REF] Lions | Cours au college de france[END_REF]. This strongly suggests that the previous assumption on the derivative of H is close from being necessary for the uniqueness in (18).

3.2.3.

The case of common noise : discrete state space. We now explain why, formally, the addition of a common noise should not alter the property that, in the limit λ goes to ∞, a MFG converges toward an ABM. We present first a discrete state space case (cf section 2), and next a continuous state space case with the notations of [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF].

For the discrete state space case, we work with a common noise described by an intensity of jump β > 0 and a linear map T . The map T is assumed to be linear for the sake of clarity of the derivation of the equation ( 28), but this result apply to more general maps T . We introduce the solution U of (26)

∂ t U(t, x) + (F (x, U) • ∇)U + λU + β(U -T * U(t, T x)) = G(x, U) in (0, t f ) × R d ; U(t f , x) = U 0 (x) in R d .
We fix a probability space (Ω, A, P). We denote by (τ i ) i∈N a sequence of independent random variables of exponential law of parameter β. We define the sequence (t i ) i∈N by

t i = i k=0 τ k .
We define for any t > 0, x 0 ∈ R d the couple (V s , Y s ) 0≤s≤t by the following :

(27)         
d(e λs V s ) = G(Y s , U(s, Y s ))ds; ∀s ∈ (t i , t i+1 ); dY s = F (Y s , U(s, Y s ))ds; ∀s ∈ (t i , t i+1 );

Y t + i = T Y t - i ; V t + i = T * V t - i Y t = x 0 ; V 0 = U 0 (Y (0)).
This couple (V s , Y s ) represents the characteristic associated to [START_REF] Nutz | A mean field game of optimal stopping[END_REF] in the sense that if we defined Ũ by Ũ (t, x 0 ) = E[V t ] with (V s , Y s ) s≥0 being defined by (27), then a formal calculation yields that Ũ solves : (28)

∂ t Ũ (t, x) + (F (x, U) • ∇) Ũ + λ Ũ + β( Ũ -T * Ũ (t, T x)) = G(x, U) in (0, t f ) × R d ; U(0, x) = U 0 (x) in R d .
Thus formally Ũ = U. Hence, the process (Y s ) s≥0 should represent a realization of the evolution of a population of player facing the MFG modeled by [START_REF] Nutz | A mean field game of optimal stopping[END_REF] and starting from the configuration x 0 . Formally, in the limit λ goes to infinity, we obtain that that U goes to 0 uniformly in x, and thus the evolution of the process (37)

∂ t Z β + < F (x, ∇ ξ W ), ∇ x Z β > + < D p F (x, ∇ ξ W )∇ ξ Z β , ∇ x W > -< D p G(x, ∇ ξ W )∇ ξ Z β , ξ > + λ(Z β -Z β (t, T x, T ξ -e)) =< ∇ x G(x, ∇ x W )ξ, ξ > -< D p G(x, ∇ ξ W )∇ x W, ξ > -< ∇ x F (x, ∇ ξ W )∇ x W, ξ > + < D p F (x, ∇ ξ W )∇ x W, ∇ x W > -2β < ∇ x G(x, ∇ x W )ξ, ∇ x W > +2β < ∇ x F (∇ ξ W )∇ x W, ∇ x W > + βλ |∇ x W | 2 -2 < ∇ x W (t, T x, T ξ -e), S∇ x W > +|∇ x W (t, T x, T ξ -e)| 2 - d dt β|∇ x W | 2 .
This equation reduces to (using this time only the α monotonicity of G) : (38)

∂ t Z β + < F (x, ∇ ξ W ), ∇ x Z β > + < D p F (x, ∇ ξ W )∇ ξ Z β , ∇ x W > -< D p G(x, ∇ ξ W )∇ ξ Z β , ξ > + λ(Z β -Z β (t, T x, T ξ -e)) ≥ α|ξ| 2 -2β < ∇ x G(x, ∇ x W )ξ, ∇ x W > +2β < ∇ x F (∇ ξ W )∇ x W, ∇ x W > + βλ(|∇ x W | 2 -|S∇ x W | 2 ) - d dt β|∇ x W | 2 .
We then deduce that : (39) ∂ t Z β + < F (x, ∇ ξ W ), ∇ x Z β > + < D p F (x, ∇ ξ W )∇ ξ Z β , ∇ x W > -< D p G(x, ∇ ξ W )∇ ξ Z β , ξ > + λ(Z β -Z β (t, T x, T ξe))

≥ α|ξ| 2 -β||∇ x G|| • |ξ| 2 -β||∇ x G|| • |∇ x W | 2 -2β||∇ x F (∇ ξ W )|| • |∇ x W | 2 + βλ(|∇ x W | 2 -|S∇ x W | 2 ) - d dt β|∇ x W | 2 .
Thus if we define β by : (40) β(t) = αe -(2||∇xF ||+||∇xG||+(||S|| 2 -1) + λ)t ; then Z β is positive for all time because of lemma 3. We then conclude the proof as we did in the case in which U 0 , G and F are α monotone.

B.2. F is α monotone. We still define W by (36) and we define Z β,γ by : Z β (t, x, ξ) =< ξ, ∇ x W (t, x, ξ) > -β(t)|∇ x W (t, x, ξ)| 2 + γ(t)|ξ| 2 .

Here, both β and γ are function of the time we shall define later. The PDE satisfied by Z β,γ is :

is positive for all time and the theorem follows immediately. Now let us remark that one can always choose β(0) small enough so that for all time t ∈ [0, t f ] : 0 ≤ β(t) ≤ αt + β(0); γ(t) ≤ β(0)||D x U 0 || 2 e (λ(1-||S|| 2 ) -+||DpF ||+2||DpG||)t + β(0)||D x U 0 || 2 α t 2 2 + β(0)||∇ x G||t. Thus the theorem is proved.

  1.3. A simple example. The previous proof relies strongly on the form of the Hamiltonian and on the invariance by translations of the problem. We now discuss to what extent those assumptions are needed. First, it is very natural to allow the measure of controls µ in (2) to appear in the Hamiltonian only through terms like G(∇u, m) because we are interested in a mean field interaction with the control of the other players. Thus terms of the form

This work is supported by a grant from the Fondation CFM pour la recherche.

B.1. U 0 and G are α monotone B.2. F is α monotone (Y s ) s≥0 is given by : (29) dY s = F (Y s , 0)ds; ∀s ∈ (t i , t i+1 ); Y 0 = x 0 ; Y t + i = T Y t - i ; where the sequence (t i ) i∈N has the same law as in (27). Hence, the "model" (29) is the limit ABM one should obtain in the limit λ goes to infinity for [START_REF] Nutz | A mean field game of optimal stopping[END_REF].

3.2.4. The case of common noise : the infinite dimensional case. Assume now that the MFG is in infinite dimension and that the master equation is given by : (30)

In this setting H is the hamiltonian of the optimal control problem faced by the players, F the running cost and G the terminal cost. We refer to [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] for a precise description of the MFG this master equation models and for definitions of the derivatives in the space of measures. We just recall here that β is positive parameter which describes the intensity of the common noise ; that F and G satisfies some strong regularity assumptions and that H is smooth, globally Lipschitz and coercive in its second argument. We denote by (Ω, A, P) a probability space. In the context modeled by (30), the forward-backward system (1) takes the form :

The process (W t ) 0≤t≤T is a standard d dimensional bronwnian motion under the probability space (Ω, A, P). The unknown are the three random functions (u t , m t , v t ) 0≤t≤T and (v t ) 0≤t≤T is a term which forces u t to be measurable with respect to the filtration generated by (W t ) 0≤t . Formally as λ goes to infinity, the solution U of (30) goes to 0 and thus (31) reduces to : (32)

This nonlinear stochastic Fokker-Planck equation is the limit ABM.

Appendix A. A maximum principle result

We present a maximum principle result which is quite general and that we use several times in this paper. Although this result is not new (see [START_REF] Lions | Cours au college de france[END_REF] for example), we recall it and give a simple proof for the sake of completeness. Lemma 3. Let u be a smooth function from (0, T ) × R d to R which satisfies :

where A is a uniform elliptic term which is linear in its third argument, B is linear with respect to its third argument,

Proof. Let us assume that there exists (t 0 , x 0 ) ∈ (0, T )×R d such that u(t 0 , x 0 ) < 0. Thus, for any λ ≥ 0, there exists δ > 0 such that (34) e -λt 0 u(t 0 , x 0 ) + δt 0 < 0.

We define v by v(t, x) = e -λt u(t, x) + δ(t + 1).

This function satisfies

Thus, there exists (t 1 , x 1 ) such that :

Because ∇ x v(t 1 , x 1 ) = 0 implies that ∇ x u(t 1 , x 1 ) = 0, evaluating the PDE satisfied by u, we deduce that -δ(1 + λ(t 1 + 1)) ≥ 0.

Thus we obtain the desired contradiction.

Appendix B. Proof of Theorem 2 under general assumptions

The proof of Theorem 2 follows the argument from [START_REF] Lions | Cours au college de france[END_REF]. We show first the case U 0 and G are α monotone, and then the case F is α monotone. B.1. U 0 and G are α monotone. As we did above, we define W and Z β by ( 36)

except that now β can be a function of the time. The PDE satisfied by Z β is :

Thus, we deduce the inequality : (42)

Hence, if β and γ satisfies

0; then the right hand side of (42) is positive for all time t ∈ (0, t f ). Moreover let us remark that, at the initial time, because U 0 is monotone :

Thus if we define β and γ by then Z β,γ is positive at t = 0 and the right hand side of (42) is positive for all time. Thus, if β(t) > 0 for all t ∈ [0, t F ], then by lemma 3 we conclude that Z β,γ