SEMI-LINEAR PARABOLIC EQUATIONS ON HOMOGENOUS LIE GROUPS ARISING FROM MEAN FIELD GAMES

A BSTRACT . The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean ﬁeld games system deﬁned on an homogenous Lie group.

In the last decades, the existence and the regularity of solutions to linear, semi-linear and fully nonlinear equations defined on sub-Riemannian structures have attracted a lot of attention.Besides their own interests, these equations arise from several models such as diffusion process, control theory and human vision (we refer, for instance, to [19], [5]).In all these settings such equations have in common the fact that their ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space and all the remaining directions are recovered from commutators.It follows that the underlying geometric structure of the state space is of anisotropic type and this plays a crucial role in the analysis of solutions to elliptic and parabolic equations.
In the elliptic case, several results are present in literature on divergence form operators both for linear and semi-linear equations: Harnack's inequality, regularity results, existence and size estimates of the Green's function can be found, for instance, in [26], [34].To the authors' knowledge, only few results are available for non-divergence form operators, for which we refer to [10], [12], [41].The parabolic counterpart, despite its great relevance from the applicative viewpoint, has been less investigated, and we mention [3] and references therein.A general and self-contained introduction to the subject can be found in [13], [9], [3].
In this work we focus our attention on two semi-linear parabolic equations posed on an homogenous Lie group: Hamilton-Jacobi equation and Fokker-Plank equation, both arising from mean field games (MFG) theory.In particular, we are interested in showing the existence of classical solutions to such equations in order to obtain, as a consequence, the existence of classical solutions to the MFG system.In the framework of evolutive noncoercive MFG, we quote here the results in [35,36,1,2,15,38] but in these cases a key assumption is that the coefficients in the dynamics grow at most linearly.Up to the authors' knowledge, no results are available for parabolic MFG problem in unbounded domain for general Lie groups.
Next, we give an overview of the main results proved in this paper and the strategy of proofs.
1.1.Main results and strategy of proof.
1.1.1.Fokker-Plank equation.The first problem we address is the existence of classical solutions to the Fokker-Planck equation where T > 0, σ > 0, ∆ G and div G are respectively the horizontal Laplacian and the horizontal divergence with respect of a family of left invariant vector fields {X i } i=1,...m associated to a homogeneous Lie group (see Section 2.1) and the drift b is regular and bounded w.r.t.these vector fields.Note that, with such regularity assumption on b, equation (1.1) can be written as We show the existence of a global classical solution to (1.1) which, besides its own interests, it will be a fundamental tool to study the Hamilton-Jacobi equation looking at (1.1) as the dual of such equation for a suitable choice of the drift b.It will also play a crucial role in the study of MFG.In this case the model we have in mind is obtained taking b(t, x) = γ|∇ G u| γ−2 ∇ G u where γ ≥ 2, u is the solution to a Hamilton-Jacobi equation and ∇ G is the horizontal gradient.We also find uniqueness of the solution of the Fokker-Planck equation by showing uniqueness of the solution to a general linear equation with bounded coefficients.In our opinion this result has its own interest.Let us remark that in [17] a similar result is obtained with a different approach under stronger assumptions on the coefficients and for a different linear subelliptic equation.Moreover we also find an Hölder regularity result w.r.t. the flat Wasserstein metric d 0 which will be defined in subsection 3.2.1.
Throughout this paper the space C 1,ν G (R d ), defined in Section 2.1, is the Hölder space w.r.t. the horizontal derivatives and the metric associated to the group.
Then, equation (1.1) has a unique classical bounded solution ρ ∈ C((0, T ); (iii) There exists a constant C ρ ≥ 0 such that where ρ t is the measure associated to the density ρ(t, •).
The existence part of the statement is obtained approximating problem (1.1) by truncation and exploiting some regularity results which strongly rely on the Hörmander condition.The uniqueness part of the statement is obtained finding a suitable subsolution used as a test function related to the fundamental solution of the horizontal heat equation.
Note that, as proved in [6] and [40], the measure {ρ t } t∈[0,T ] represents the law of the stochastic process where • denotes the Stratonovich stochastic integral, {B j } j=1,...,m are d dimensional independent Brownian motions and X 0 is the drift term explicitly provided in [40].However, since the data can grow more than linearly, we are not able to provide a bound on the moments of such a process which is a key result to get the solution in the non degenerate case.
1.1.2.Hamilton-Jacobi equation.We are interested in studying the well-posedness of the following Hamilton-Jacobi equation with γ ≥ 2 under the assumptions To do this we will study a general semi-linear parabolic equation of the following type From the structure of (1.2) we assume the following on f : there hold The first result is the existence and uniqueness of a classical solution of (1.5) for small times.
Theorem 1.2.Let f satisfy (HP) and (HP') and let Then, there exists T 0 > 0 such that for any T ≤ T 0 there exists a unique classical solution for some κ(T 0 ) > 0.
In order to prove the above small-time existence of classical solutions to the semi-linear Hamilton-Jacobi equation (1.5) we use the Duhamel formula and the decay estimate of the subelliptic heat semigroup (Lemma 4.1).
Next, we state the second main result on the global in time existence of classical solutions to the Hamilton-Jacobi equation (1.2).To this end, we establish an estimate of the horizontal gradient of u using the Bernstein method suitably adapted to sub-Riemannian framework.In order to apply this method, we need an extra technical assumption for a set of right-invariant vector fields.
3) and let u 0 satisfy (1.4).Let {Y 1 , . . ., Y m } be a set of right-invariant smooth vector fields on R d and assume that Let T > 0 be arbitrary.Then, there exists a classical solution u ∈ C To obtain the global in time existence, we first show that the solution is bounded in space and time-uniformly w.r.t. the second order coefficient σ-(Proposition 4.4) and that also its horizontal gradient is bounded globally in time and locally in space (Proposition 4.6)-uniformly w.r.t.σ.The common main technical point of the above results is the use of the duality between Hamilton-Jacobi equation and Fokker-Plank equation: given a suitable choice of the drift function and the corresponding solution to the Fokker-Plank equation we deduce properties for the solution u.This duality property have been investigated in several other settings, see for instance [18], [24], [29] .Moreover, in order to get the estimate on the horizontal gradient we use the Bernstein method adapted to the sub-Riemannian framework.More precisely, one cannot apply a vector field generating the Lie group to the equation because, otherwise, one will get some extra terms involving commutators which are difficult to deal with.In order to overcome this issue, we first differentiate the equation by the family of smooth right-invariant vector fields introduced in the statement.Hence, we obtain a local bound for the gradient of the solution on the distribution generated by such right-invariant vector fields.Finally, since the right-invariant distance is locally Lipschitz equivalent to the left-invariant one, we get the desired estimate.
We conclude this part, observing that the assumption γ ≥ 2 is needed in order to gain regularity on the solution to the Hamilton-Jacobi equation when the initial data is regular enough.In a forthcoming paper, we will address a similar problem investigating the sub-quadratic case with merely local integrability of the solution.In this case, the integrability assumptions of the drift of the transport equation and the duality approach still allow us to obtain classical solutions to the Fokker-Plank equation, to the Hamilton-Jacobi equation and, consequently, to the MFG system.
1.1.3.Mean Field Games.We conclude this work investigating the existence for small times of solutions to the MFG system x ∈ R d .
Such a system couples the equations we have previously studied, i.e., the Hamilton-Jacobi equation in (1. 2) in which the potential F MFG is a strongly regularizing nonlocal term which depends on the distribution ρ and the Fokker-Planck equation in (1.7) whose drift is defined by the optimal feedback associated with (1.2).In the following, we consider a coupling function and we assume the following: MFG theory, introduced in [31,32,33], is devoted to the study of differential games with a very large number of interacting agents.A typical model is described by a system of PDEs: a backward-in-time Hamilton-Jacobi equation whose solution is the value function for the generic player (and also provides their optimal choices) and a forward-in-time Fokker-Planck equation which describes how the distribution of individuals changes.The two equations are coupled in a way that takes into account both the state of a single agent and how he/she is influenced by the others.The system of PDEs describes a model with a continuum of players which is, clearly, not realistic.However, the solution of such a system is expected to capture the behavior of Nash equilibria for differential N-players game as the number of agents goes to infinity.For an extensive and detailed introduction to the subject we refer to [7], [16], [28] and references therein.
The motivation for studying our model is the following: heuristically, given X 1 , . . ., X m smooth vector fields on R d , each single player can move only along the directions generated by the set of vector fields.In [22,25] stationary subelliptic MFG systems have been studied in the torus.We also cite the papers [35,36,2,15,38], where, in the whole space, the generic player has some forbidden directions because it follows either a dynamic generated by the vector fields on Heisenberg group, or a Grushin dynamic or it controls its acceleration.In particular, the results in [35,36] are obtained under the key assumption that the coefficients of the vector fields X i grow at most linearly.Up to the authors' knowledge, no results are available for parabolic MFG problem in unbounded Lie groups.Clearly, the unboundedness of the state space, eventually with unbounded vector fields, gives rise to several difficulties to overcome.
From the optimization point of view, a generic player wants to minimize the cost where γ * is the conjugate index of γ and ξ(•) denotes the stochastic process that solves where α is the control chosen by the player while • and W j are as before.
We prove a small time existence result of a classical solution.Note that we cannot prove the existence for any time T because of the lack of compactness in the results of Theorem 1.3.

SUB-RIEMANNIAN SETTING
Let (G, * ) be a homogeneous Lie group and let {δ λ } λ>0 be a family of dilations which are automorphisms of the group, i.e., for all x, y ∈ G and λ > 0. Let g be the Lie algebra associated with the group G.The dilations of the group induce a direct sum decomposition on g, i.e., In particular, V 1 is called the horizontal layer and its elements are left-invariant vector fields.We can identify (G, * ) with R d via the so-called exponential map exp : g → G which turns out to be a diffeomorphism.Given a basis X 1 , . . ., X d adapted to the stratification, any x ∈ G can be written in a unique way as and one can identify x with (x 1 , . . ., x d ) and G with (R d , •) where the group law is given by the Baker-Campbell-Hausdorff formula.
Hence, hereafter we work on R d and we consider an orthonormal basis X 1 , . . ., X m of the horizontal layer V 1 .We assume that such a family of vector fields satisfies the Hörmander condition, i.e., Lie(X 1 , . . .X m )(x) = R d for all x ∈ R d .and to be homogeneous of degree one w.r.t. the family of dilations.
Moreover, given a, b ∈ R we say that an absolutely continuous curve γ : and the length of γ is defined as Under the Hörmander condition, a well-known result by Chow states that any two points on R d can be connected by an horizontal curve.Hence, the definition of Carnot-Carathéodory distance is well-posed d SR (x, y) = inf{ℓ(γ) : γ is an horizontal curve joining x to y}.

One can prove a variational interpretation of the above distance as
Note that, the Carnot-Carathéodory distance is not equivalent to the Euclidean one.Indeed, it is well-known that, for any K compact set, there exists a constant C > 0 such that, for any y and x in K, we have where σ(x) ∈ N is the nonholonomic degree at x ∈ R d , that is, the maximum of the degrees of the iterated brackets occurring to fulfill the Hörmander condition.Using the family of dilations and the sub-Riemannian distance one can define a norm on R d tailored from the Lie group, However, from the homogeneity of the vector fields X 1 , . . ., X m and the stratification of R d , one can define an homogeneous norm • G and the homogeneous dimension Q of the group as Example 2.1.Examples of homogeneous Lie groups are Heisenberg-type groups, Engel group and Martinet group (see, for instance, [39]).
For completeness, we write the following result on homogenous functions of Lie groups which will be useful later.
So, we conclude, observing that 2.1.Subelliptic equations.Let us consider a family of smooth left-invariant vector fields X 1 , . . ., X m as in the beginning of this section, satisfying the Hörmander condition and a function u : R d → R. We define the horizontal gradient and the horizontal Laplacian of u as and respectively For any vector-valued function u : R d → R m , we will consider the divergence operator induced by the vector fields, that is, div u m where u i denotes the i-th component of u for i = 1, . . ., m. Next, we recall the definition of Hölder space associated with the family of vector fields.For every multi-index We introduce C 0 (R d ) as the set of continuous (possibly unbounded) functions on R d and we associate the norm where d SR is the Carnot-Carathéodory distance defined before.We introduce and the corresponding norm are Banach spaces for any r ∈ N and any α ∈ (0, 1].We conclude this preliminary section recalling, also, the definition of horizontal Sobolev spaces.Let r ∈ N and 1 ≤ p ≤ ∞.We define the space We denote by C r,α G,loc (R d ) and W r,p G,loc (R d ) the horizontal Hölder space and horizontal Sobolev space, respectively, as in the above manner with R d replaced by any compact subset Ω.
In order to have a self-contained presentation of the work, next we recall the main results used below on linear parabolic subelliptic equations and a Sobolev embedding result.
where C p depends only on p and on the holomorphic constant of the semi-group e t∆ G .
Then, there exists a positive constant C, depending only on U, Ω, T and p such that for α = 1 p (p − (Q + 2)) and for every u (Recall that Q is the homogeneous dimension of the group defined before).

FOKKER-PLANCK EQUATION
3.1.Existence of solutions.We consider the Fokker-Planck equation ), for some δ ∈ (0, 1).Proposition 3.1.Under the assumptions of Theorem 1.1 equation (3.1) has a classical bounded solution ρ ∈ C((0, T ); C 2,δ G,loc (R d )).Moreover, , and Proof.Consider the initial-boundary value problem where B R is the ball of radius R w.r.t. the Carnot-Carathèodory distance.We shall first solve problem (3.3) establishing several properties of ρ R and after, letting R → ∞, we obtain a solution to problem (3.1) with the desired properties.Invoking Lions' Theorem (see [14, theorem X.9]), we infer that there is a unique function where where the right hand side belongs to L 2 ((0, T ) × B R ).The results in [42,Theorem 18] ensures that: In particular, we deduce that the differential equation in (3.3) is satisfied for a.e.(t, x) ∈ (0, T ) × B R .We claim that ρ R ∈ C([0, T ] × B R ) and that, for every domain Ω ⊂ (0, T ) × B R , there exist a constant K(Ω, R) (depending on the assumptions, on Ω and on R) and a constant K ′ (Ω) (depending on the assumptions and on Ω), such that . Moreover, the results in [11, Theorem 1.1] (with k = 0) ensure that ρ R,n fulfills (3.5) with constants K and K ′ , both independent of n.Letting n → ∞, we accomplish the proof of our claim (3.5).
Since ρ R is continuous and div G b is bounded, standard comparison principle entails and that, for each (t, x), the value ρ R (t, x) is nondecreasing with respect to R, i.e., From the properties proved so far we obtain that for each (t, x) there exists the limit lim which we denote by ρ(t, x) and, clearly, . By a standard diagonalization process, using (3.5), ρ ∈ C 2,δ G,loc (Ω) for every domain Ω ⊂ (0, T ) × R d .We now proceed with the proof of the second part of (3.2).First, we consider again the approximating problem (3.3).We integrate the equation on [0, T ] × B R , we use the divergence theorem and we note that ∂ρ R /∂ν ≤ 0 where ν is the outward pointing normal to ∂B R .Hence we get that ρ R (t ) such that ξ(x) = 1 for any x ∈ B 1 and ξ(x) = 0 for any x ∈ R d \B 2 , and define ξ R (x) = ξ( x R ) for each R > 0. Hence, multiplying (3.1) by ξ R and integrating by parts we have Observing that by dominated convergence theorem as R ↑ ∞, we conclude Proposition 3.2.Under the assumptions of Theorem 1.1, let ρ be the solution of (3.1) constructed in Proposition 3.1.Then: (i) there exists a constant K depending on b, σ and (3.6) Proof.For R > 0, consider the solution ρ R to (3.3) found before.For simplicity of notation, we shall denote by K a constant which may change from line to line but which always depends only on the assumptions (in particular it is independent of R).Assume for the moment that, for every t ∈ [0, T ] and R > 0, there holds (3.8) From (3.8), we deduce Hence, by Gromwall's lemma, we infer So, as R ↑ ∞ we obtain the former estimate in (3.6).On the other hand, integrating (3.8), we have where the last inequality is due to relation (3.9).Again by (3.9), we deduce that there exists a constant K (independent of R and ρ 0 ) such that As R ↑ ∞, we obtain the latter estimate in (3.6).It remains to prove estimate (3.8).To this end, using ρ R as test function for (3.3), we get Using Hölder inequality on the last term, we get (3.8).Finally, from the same reasoning we also get (3.7).

Uniqueness and regularity.
We recover the uniqueness of solutions to (3.1) by showing the uniqueness of the classical solution to the general linear equations with bounded coefficients of the form Let us remark that in [17] a similar result is obtained with a different approach under stronger assumptions on the coefficients and for a different linear subelliptic equation.
Let B and Q be bounded continuous functions on [0, T ] × R d and, moreover, assume that B has a continuous and bounded horizontal gradient.For j = 1, 2, let ρ j ∈ C((0, T ); C 2,ν G,loc (R d )) be two classical solutions to (3.10) such that for some positive constant β we have Then, The proof of the proposition is postponed after the following technical lemma.
Lemma 3.4.Let ρ j , for j = 1, 2, be two solutions to (3.10) such that for some β > 0 we have for τ ∈ (τ 0 , T ], with τ − τ 0 sufficiently small and β sufficiently large, and Proof.From the homogeneity of the norm • G and using Lemma 2.2 we have that Hence, setting for simplicity β1 = β 1 + β(t − τ 0 ) we have The proof of (3.12) is thus complete by choosing τ − τ 0 sufficiently small and β sufficiently large.Bounds (3.13) are an easy consequence of the choice of β 1 and assumption (3.11).
Proof of Proposition 3.3.Without any loss of generality, we assume Q ≥ 0 and, we proceed by contradiction assuming that ρ 1 = ρ 2 .Set The continuity of ρ j , for j = 1, 2, ensure that the function ρ For any ε > 0 define the function and observe that the following equalities hold Therefore, multiplying (3.14) by ρ w we get So, for any nonnegative test function v ∈ C ∞ ([τ 0 , T ] × R d ) with bounded support in space and for any t ∈ [τ 0 , T ] there holds Choose t ∈ [τ 0 , τ ] and v = ξ R Φ where τ and Φ are respectively the constant and the function introduced in Lemma 3.
Hence, we get Letting R ↑ ∞, by dominated convergence theorem and Lemma 3.4 we have that the right hand side converges to zero and we obtain By Proposition 3.3, the classical solution constructed in Proposition 3.1 is unique, hence we proved the following corollary.

Under the assumptions of Theorem 1.1, there exists a unique bounded classical solution
G,loc (R d )) of (3.1).

3.2.1.
Hölder regularity and flat metric.Next, we prove Hölder regularity of the solution ρ w.r.t. the so-called flat metric d 0 distance here defined.There are many ways to metrize weak convergence of measures and the one we use here is the following (see, for instance, [43]): the bounded Lipschitz distance, also called Fortet-Mourier distance of flat Wasserstein metric Let ρ be the unique solution to (3.1) constructed in Proposition 3.1.Then, there exists C ρ ≥ 0 such that Proof.We argue adapting some ideas of [23,Proposition 6.6].We consider the smooth function For ε > 0, let be a smooth mollifier with support in B(0, ε) and where the constant C is independent of ε and such that ξ ε dx = 1 (we recall that δ ε −1 (x) denotes the dilation of radius ε −1 ).Note that, by homogeneity of the norm Let ϕ be a real valued function with ϕ C 0,1 G (R d ) ≤ 1 and let ϕ ε (x) = ξ ε ⋆ ϕ(x) where the symbol "⋆" denotes the convolution based on the operation of the group.Note that, by standard calculus and Lagrange theorem (see [9,Theorem 20.3.1],there holds: First, from (3.15) and standard calculus, we obtain In conclusion, minimizing over ε > 0, we get Proof of Theorem 1.1.From the above analysis we have that existence, uniqueness, (i) and (ii) follow from Proposition 3.1, Proposition 3.2 and Proposition 3.3.Finally, (iii) is proved in Proposition 3.6.

HAMILTON-JACOBI EQUATION
4.1.Small-time existence of solutions.Throughout this section we assume that assumptions (HP) and (HP' ) are in force and we study equation (1.5).We introduce, for simplicity of notation, the space (4.1) Before, for proving the existence of a small-time solution to (1.5) we need the following decay estimate for the heat semi-group e t∆ G generated by the horizontal Laplacian.Lemma 4.1.For any t ∈ [0, T ] and any ϕ ∈ L ∞ (R d ) we have that where c(T ) is a constant depending only on T .
Proof.Let Γ be the fundamental solution to the heat operator (∂ t − ∆ G ), found in [10].Following [10, Theorem 1.2] and by construction of the heat semigroup we get Similarly, still from [10, Theorem 1.2] we deduce the second estimate in (4.2).
Proof of Theorem 1.2.Consider T > 0, which will be chosen later on.For any k > 0 we denote by X k (T ) the closed ball of radius k in X (T ), defined in (4.1).Given k > 0, we consider the map Φ : defined by Next, we show that there exist T 0 > 0 and k > 0 such that the map Φ is well defined, i.e., Φu ∈ X k (T ) for u ∈ X k (T ) and Φ is a contraction for any T ≤ T 0 .
To do so, let us fix k > 0 and u ∈ X k (T ) for some T > 0.Then, from (4.2) and assumptions (HP), we have that Moreover, still from (4.2) and assumptions (HP) and (HP'), we have that where c(T ) is the constant introduced in (4.2) while c is a constant, that depends on C f , d and m (in particular, is independent of T and k) and may change from line to line.Moreover, by (4.2), standard arguments entail By relations (4.4), (4.5), (4.6) and (4.7), for k > 0 sufficiently large and T sufficently small, there holds Next, we proceed to show that Φ is a contraction.Let u, v ∈ X k (T ).Then, from (4.2) we have that

So, by (HP) we obtain
By using similar arguments, one gets Furthermore, for any i, j ∈ {1, . . ., m}, we have Moreover, we have Replacing the last inequality in the previous one, by assumption (HP'), we get (4.10) where c is a constant, that depends on C f , d and m (in particular, is independent of T and k) and may change from line to line.Hence, using (4.8), (4.9) and (4.10), we get and we conclude choosing k such that Thus, from the fixed point theorem we obtain the existence of a unique solution in X k (T ).Moreover, from the representation formula provided by the contraction argument, i.e., and the regularity of the fundamental solution (see [10,Theorem 1.2]) we deduce that u ∈ C 4.2.Global existence of solutions.In Theorem 1.2 we showed that for a sufficiently small time horizon T there exists a solution to the general semilinear parabolic equation (1.5) in X (T ).In this section, we go back considering the Hamilton-Jacobi (4.11) and we prove that there exists a solution for any T > 0.
In order to prove that such a solution exists for any arbitrary T > 0, the key point is the duality feature between the Hamilton-Jacobi equation and the Fokker-Plank equation studied so far.For this reason, we first show that the solution u constructed in Theorem 1.2 taking f (t, x, p) = F (t, x) − |p| γ solves problem (4.11) also in a suitable weak (energy) sense and, then, following a standard procedure (see for instance [18]), we provide the duality relation between the two equations in the sub-Riemannian setting.Lemma 4.2.Let F be as in (1.3), u 0 as in (1.4) and let u be the solution of (4.11) found in Theorem 1.2.For any T ≤ T 0 , we have Proof.Recall that the solution u constructed in Theorem 1. (i).Following the arguments of [4,Theorem B] we get that u(t) ∈ L 1 (R d ): let u solve ∂ t u − σ∆ G u = 0 with u(0) = u 0 ; we refer to [10,Theorem 1.2] for the representation formula of u and for the regularity of the fundamental solution for the heat equation.By comparison principle, using that u 0 ≥ 0, we get 0 ≤ u ≤ u and consequently u(t) ∈ L 1 (R d ).Hence, integrating (4.13) we deduce , by interpolation we conclude (i).
(ii).It is an immediate consequence of point (i) and assumption (1.3).
By a standard approximation argument, we infer (4.12).
In the following lemma we get a useful relation between the solutions u and µ respectively of the Hamilton-Jacobi and Fokker-Planck equations using the duality structure of these equations.Lemma 4.3.Let u ∈ X (T 0 ) be a solution to (4.11) as in Theorem 1.2.Let τ ∈ (0, T 0 ].For any found in Theorem 1.1.Then, for any s ∈ (0, τ ) we have that Proof.From the regularity of and µ, by using u as a test function in (4.14) and, respectively, µ as a test function in (4.12) and taking the sum of the two relations we obtain Hence we get (4.15).
In the following proposition we prove a key estimate using the duality argument of Lemma 4.3.
Proof.First, we prove a bound from above for u.To do so, fix τ ∈ [0, T 0 ] and consider the solution µ : By duality arguments, i.e., proceeding as in Lemma 4.3, we obtain (4.17) Since, from Theorem 1.1, µ(t, •) 1,R d = 1 for any t ∈ [0, τ ] and µ ≥ 0, we get Hence, and, thus, by passing to the supremum, over µ τ ≥ 0 with µ(t, To prove the lower bound for u, we first observe that combining (4.15) and (4.18) we get So, again by (4.17) we get which, by the arbitrariness of µ τ , yields to Therefore, (4.18) and (4.20) imply (4.16).Now we want to prove a local bound on ∇ G u. To do this we need the following remark.
Remark 4.5.Following [37], for any two Riemannian metrics g and g, we have that, by the very definition of the gradient, for every smooth function u it holds that g(D g u, •) = g(D g u, •).So, since any two Riemannian metrics are equivalent, if we denote by |D g u| 2 g = g(D g u, D g u), and the same for D g u, we have that for any compact set Ω ⊂ R d there exist constants K In particular, for any two sequences of metrics {g n } n∈N , { g n } n∈N such that g n → g and g n → g it can be proved that K 1 (Ω) and K 2 (Ω) in (4.21) can be chosen independently of n, i.e., there exist K gn .We now recall that, to obtain a bound on ∇ G u, we cannot directly apply the classical Bernstein method to X i u, where u solves the equation (4.11), but we have to adapt it because some extra terms, involving commutators, would appear.In order to overcome this issue, we consider the family of right-invariant vector fields {Y 1 , . . ., Y m } introduced in (1.6).Proposition 4.6.Under the assumptions of Theorem 1.3, let u ∈ X (T 0 ) be a solution to (4.11) and let τ ∈ [0, T 0 ].Then, for any compact subset Ω of R d there exists C(Ω), depending on Proof.Let {X ε 1 , . . ., X ε m , X ε m+1 , . . ., X ε d } be a completion of {X 1 , . . ., X m } to a Riemannian basis on R d with X ε j = X j for j = 1, . . ., m and, for some fields Xj , X ε j = ε Xj for j = m + 1, . .where ∆ ε and ∇ ε denotes, respectively, the Laplacian and the gradient w.r.t. the vector fields {X ε 1 , . . ., X ε d }.Fix j ∈ {1, . . ., d} and set v ε j = Y j u ε .From (4.24) we have that v ε j is a classical solution to the equation (4.25) ∂ t v ε j (t, x)−σ∆ ε v ε j (t, x)+γ|∇ ε u ε (t, x)| γ−2 ∇ ε u ε (t, x)∇ ε v ε j = Y j F (t, x), (t, x) ∈ (0, T )×R d .We proceed with a duality argument as before.To do so, for τ > 0, let µ solve (4.26) −∂ t µ(t, x) − σ∆ ε µ(t, x) − div ε (γ|∇ ε u ε (t, x)| γ−2 ∇ ε u ε (t, x)µ(t, x)) = 0, (t, x) ∈ (0, τ ) µ(τ, x) = µ τ (x) where div ε denotes the divergence operator w.r.t.{X ε 1 , . . ., X ε m , X ε m+1 , . . ., X ε d }.Thus, using µ as a test function in the weak formulation of (4.25) and v as a test function in the weak formulation of (4.26) we obtain which is bounded by (1.6) for any compact subset Ω of R d and some constant C(Ω) ≥ 0. So, as ε ↓ 0 we have that X ε j u ε → X j u if j ∈ {1, . . ., m} and X ε j u ε → 0 if j ∈ {m + 1, . . ., d}.Hence, this yields to sup Proof of Theorem 1.3.Reasoning as in [21, sect.2.1], from Proposition 4.4 and Proposition 4.6 we have that there exists a solution on [0, T 0 + ε] and thus, a solution u ∈ C([0, T ]; W 1,∞ G,loc (R d )) for T finite but arbitrary large.
Next, in order to complete the proof of gain regularity of the solution we proceed with a bootstrap argument.For any R ≥ 0 let u R be a solution to Then, as proved so far we have that u R ∈ C([0, T ]; W 1,∞ G (R d )).Then, the same equation can be seen as a subelliptic heat equation with bounded right hand-side.Hence, the right hand-side belongs to L p (B R ) for any p ≥ 1 and using Theorem 2.3, we obtain u R ∈ W 1,p ([0, T ]; W 2,p G (B R )).Taking p ≥ Q + 2 applying Theorem 2.4 we gain regularity on the solution, that is, u R ∈ C 1,α G ([0, T ] × B R ) with α = 1 p (p − (Q + 2)).So, following again the same reasoning we deduce that for any R ≥ 0 the solution u R belongs to C 2+α,1+ α 2 G ([0, T ] × B R ).Finally, as R ↑ ∞ by a diagonalization argument the proof is complete.

APPLICATION TO MFG
The goal of this section is the application of the above results to get the existence for small times of solutions to the following MFG system (5.1) x ∈ R d .
Note that we cannot prove the existence for any time T because of the lack of compactness in the results of Theorem 1.3.
As it is customary in MFG, the existence result is a consequence of the Schauder's fixed point theorem.However, due to the lack of control of the moments of the measure ρ t associated to the solution of the Fokker-Plank equation, the strategy is quite different w.r.t. the classical literature.
Proof of Theorem 1.4.We obtain the existence of a classical solution using the Schauder fixedpoint theorem.To do so, we endow the space C([0, T ]; W 2,∞ G (R d )) with the topology induced by the uniform convergence and we construct a map )) in the following way: given u ∈ C, with where T 0 and κ(T 0 ) are the constants constructed in Theorem 1.2 and let µ be the unique solution constructed in Theorem 1.1 to

Lemma 2 . 2 .
Let g : R d → R be a homogeneous function of degree 1 and assume |g sufficiently large (recall that δ is the Hölder exponent of ∇ G b). Indeed, consider a sequence {b n } n of drifts such that b n ∈ C ∞ and b n uniformly converges to b in [0, T ] × B R as n → ∞.Therefore, by the same arguments as before, problem (3.3) with b replaced by b n has a solution ρ R,n .Applying iteratively [42, Theorem 18], we infer that ρ R,n ∈ C ∞ and, by standard comparison principle, we get