1 Introduction

In this paper we consider a class of systems of degenerate elliptic PDEs of Hörmander type arising from certain ergodic differential games, more specifically, from the mean field game (MFG) theory of Lasry and Lions [43,44,45]. These systems have been introduced to model differential games with a large number of players or agents with dynamics described by controlled diffusion processes, under simplifying features such as homogeneity of the agents and a coupling of mean field type. This allows to carry out a kind of limit procedure as the number of agents tends to infinity which leads to simpler effective models. Lasry and Lions have shown that for a large class of differential games (either deterministic or stochastic) the limiting model reduces to a Hamilton–Jacobi–Bellman equation for the optimal value function of the typical agent coupled with a continuity (or Fokker–Planck) equation for the density of the typical optimal dynamic, the so-called mean field game equations. Solutions to these equations can be used to construct approximated Nash equilibria for games with a very large but still finite number of agents. The rigorous proof of the limit behaviour in this sense has been established by Lasry and Lions [43, 45] for ergodic differential games and extended by one of the authors to several homogeneous populations of agents [31]. The time-dependent case with nonlocal coupling has been addressed in a general context by [19]. For a general overview on mean field games, we refer the reader to the lecture notes of Guéant et al. [37], Cardaliaguet [17], the lecture videos of Lions at his webpage at Collège de France, the first papers of Lasry and Lions [43,44,45] and of Huang et al. [39, 40], the survey paper [36], the book by Gomes et al. [34] and by Bensoussan et al. [13], the two special issues [8, 9] and the recent paper [19] on the master equation and its application to the convergence of games with a large population to a MFG. For applications to economics see e.g. [3, 21, 34, 37, 42, 46]. From the mathematical side, there are several important questions related to both the convergence and then the study of the limit MFG system itself, e.g. long time behaviour [18, 20], ergodic MFG systems [11, 23, 35], for homogenisation [22]. For further contributions see also [2, 4, 33]. The literature on Mean Field Games is very vast so the previous list is only partial and we refer to the references therein for a more extended bibliography.

The novelty of this paper consists in assuming that the dynamic of the average player is a diffusion of Hörmander type and hence the differential operators arising in the system are degenerate: the second order operator is not elliptic but only subelliptic. Roughly speaking this means that the operators are elliptic only along certain directions of derivatives. Nevertheless the Hörmander condition ensures that the Laplacian induced by these selected derivatives is hypoelliptic. From the perspective of a single agent this means that the state cannot change in all directions, but the agent can move only along admissible directions: a subspace of the tangent space. This subspace depends on the state (position) of the agent. Similarly the growth conditions on the Hamiltonian are restricted to some selected directions of derivatives. This extension is not trivial and relies on recent deep achievements in the theory of Hörmander operators and subellipitc quasilinear equations. When the known regularity results will not be sufficient to proceed, we will use heat kernel estimates to overcome the problem. Moreover the techniques used here are different from the standard elliptic case and can also be used in other contexts to gain a-posteriori regularity.

Hamilton–Jacobi equations in the context of Hörmander regularity have been extensively studied, see e.g. [6, 24, 26, 28, 29], in particular because of the intriguing connection between the PDE theory and the underlying geometry induced by the admissible directions. This paper is to our knowledge the first one that connects these two recent and active areas.

We next state our main results:

  1. 1.

    Under suitable assumptions (see Sect. 3) and assuming in particular that the Hamiltonian grows at most quadratically in the subgradient, we prove that there exists a solution \((u, m) \in C_{\mathcal {X}}^2({\mathbb {T}}^d) \times C({\mathbb {T}}^d)\) of the system

    $$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}u + \rho u + H(x, D_{\mathcal {X}}u) = V[m] \\&\mathcal {L}^*m - \text {div}_{{\mathcal {X}}^* }( m g ( x, D_{\mathcal {X}}u ) ) = 0 \\&\int _{{\mathbb {T}}^d} m \,dx =1 , \quad m > 0, \end{aligned} \right. \end{aligned}$$

    where \(D_{\mathcal {X}}u\) is a subgradient associated to a family of Hörmander vector fields (e.g. \(D_{\mathcal {X}}u=\left( u_x-\frac{y}{2}u_z,u_y+\frac{x}{2}u_z\right) ^T\) on \({\mathbb {R}}^3\) in the Heisenberg case) and \(\mathcal {L}\) is a hypoelliptic operator, \(\mathcal {L}^* \) is the dual operator of \(\mathcal {L}\) and \(\text {div}_{\mathcal {X}^*} \) is the corresponding divergence operator. Moreover by \(C_{\mathcal {X}}^2({\mathbb {T}}^d)\) we indicate the sets of functions whose first and second derivatives in the selected directions exist and are continuous (see Sect. 2 for more formal definitions).

  2. 2.

    Under suitable assumptions (see Sect. 4) and assuming in particular that the Hamiltonian grows at most linearly in the subgradient, we prove that there exists a solution \((\lambda , u, m) \in {\mathbb {R}}\times C_{\mathcal {X}}^2({\mathbb {T}}^d) \times C({\mathbb {T}}^d)\) of the system

    $$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}u + \lambda + H(x, D_{\mathcal {X}}u) = V[m] \\&\mathcal {L}^*m - \text {div}_{\mathcal {X}^*} \big ( m g ( x, D_{\mathcal {X}}u ) \big ) = 0 \\&\int _{{\mathbb {T}}^d} u \, dx =0, \quad \int _{{\mathbb {T}}^d} m\, dx =1 , \quad m > 0 . \end{aligned} \right. \end{aligned}$$

    We also show uniqueness for both the systems under standard monotonicity assumptions.

Those results are applied to the feedback synthesis of MFG solutions and of Nash equilibria of a large class of N-player differential games.

The paper is organised as follows: in Sect. 2 we introduce the Hörmander condition and the corresponding first and second order operators and we state several regularity results and estimates which will be key in the proofs of our main results. In Sect. 3 we show existence for a stationary MFG system for at most quadratic Hamiltonians by a fixed-point argument in the presence of a regularisation. In Sect. 4 we remove this regularisation for Hamiltonians of at most linear growth and prove our main existence result. In the Appendix we show the convergence of Nash-equilibria as motivation for the MFG system studied. Since these results are very well-known in the non degenerate case and they do not lead to any substantial technical difference in the Hörmander case, we will omit the proofs, only reporting briefly the results.

2 Preliminaries and notations

Let us consider \(x\in {\mathbb {T}}^d\) the d-dimensional torus and \(\mathcal {X}=\{X_1, \ldots , X_m \}\) a family of smooth vector fields defined on \({\mathbb {T}}^d\) satisfying the Hörmander condition, i.e.

$$\begin{aligned} Span\bigg (\mathcal {L}\big (X_1(x),\dots ,X_m(x)\big )\bigg )=T_x {\mathbb {T}}^d \equiv {\mathbb {R}}^d,\quad \forall \, x\in {\mathbb {T}}^d, \end{aligned}$$
(2.1)

where \(\mathcal {L}\big (X_1(x),\dots ,X_m(x)\big )\) denotes the Lie algebra induced by the given vector fields and by \(T_x{\mathbb {T}}^d\) we denote the tangent space at the point \(x\in {\mathbb {T}}^d\). For more details on Hörmander vector fields we refer to [48]. Given a family of vector fields \(\mathcal {X}=\{X_1, \ldots , X_m \}\) and \(u:{\mathbb {T}}^d\rightarrow {\mathbb {R}}\), we define:

$$\begin{aligned} D_{{\mathcal {X}}} u&=(X_1 u,\dots ,X_m u)^T\in {\mathbb {R}}^m, \end{aligned}$$
(2.2)
$$\begin{aligned} \mathcal {L}u&= -\, \frac{1}{2} \sum _{j=1}^m X_j^2u\in {\mathbb {R}}. \end{aligned}$$
(2.3)

For any vector-valued function \(g:{\mathbb {T}}^d\rightarrow {\mathbb {R}}^m\), we will consider the divergence induced by the vector fields \(\mathcal {X}=\{X_1, \ldots , X_m \}\), that is

$$\begin{aligned} \text {div}_{{\mathcal {X}}}g=X_1\,{g_1}+\dots +X_m \,{g}_m, \end{aligned}$$
(2.4)

where \(g_i\) indicates the i-component of g, for \(i=1,\dots ,m\). In particular, later on, we will consider the divergence \( \text {div}_{{\mathcal {X}}^*}g \) induced by the dual vector fields \(X^*_i=-X_i-\text {div}X_i\) where \(\text {div}X_i\) indicate the standard (Euclidean) divergence of the vector fields \(X_i:{\mathbb {T}}^d\rightarrow {\mathbb {R}}^d\), for \(i=1,\dots ,m\). Given the family of vector fields \(\mathcal {X}=\{X_1, \ldots , X_m \}\) we recall that any absolutely continuous curve \(\gamma :[0,T]\rightarrow {\mathbb {T}}^d\) is called horizontal (or admissible) if there exists a measurable function \(\alpha :[0,T]\rightarrow {\mathbb {R}}^m\) such that

$$\begin{aligned} {\dot{\gamma }}(t)=\sum _{i=1}^m\alpha _i(t)X_i(\gamma (t)),\quad \text {a.e.}\; t\in (0,T), \end{aligned}$$
(2.5)

where \(\alpha _i(t)\) is the i-component of \(\alpha (t)\) for \(i=1,\dots ,m\).

For all horizontal curves it is possible to define the length as:

$$\begin{aligned} l(\gamma )=\int _0^T\sqrt{\sum _{i=1}^m\alpha _i^2(t)}\;dt. \end{aligned}$$

The Carnot–Carathéodory distance induced by the family \(\mathcal {X}=\{X_1, \ldots , X_m \}\) is denoted by \(d_{CC}(\cdot ,\cdot )\), and defined as

$$\begin{aligned} d_{CC}(x,y)=\inf \left\{ l(\gamma )\,|\, \gamma \; \text {satisfying } (2.5) \text { with}\; \gamma (0)=x, \gamma (T)=y \right\} . \end{aligned}$$

The Hörmander condition implies that the distance \(d_{CC}(x,y)\) is finite and continuous w.r.t. the original Euclidean topology induced on \({\mathbb {T}}^d\) (see e.g. [48]). It is also known that there exists \(C> 0\) such that

$$\begin{aligned} C^{-1}|x-y|\le d_{CC}(x,y) \le C |x-y|^{1/k} \end{aligned}$$
(2.6)

for all \(x,y \in {\mathbb {T}}^d\), where \(k\in {\mathbb {N}}\) is the step, i.e. the maximum of the degrees of the iterated brackets occurring in the fulfillment of the Hörmander condition, see [49]. It was proved in [51, Lemma 5] and independently in [49] that there exists some \(Q>0\), called the homogenous dimension, such that, for all \(\delta >0\) sufficiently small and for some \(C>0\),

$$\begin{aligned} C^{-1} \delta ^Q \le |B_{d_{CC}} (x, \delta ) | \le C \delta ^Q, \end{aligned}$$

for all \(x\in {\mathbb {T}}^d\), where \(B_{d_{CC}} (x, \delta )\) is the ball of centre x and radius \(\delta \) w.r.t. the distance \(d_{CC}\) and, for any \(B\subset {\mathbb {T}}^d\), |B| denotes the standard Lebesgue measure of B.

2.1 Hölder spaces and Hölder regularity estimates

Next we recall the definition of Hölder and Sobolev spaces associated to the family of vector fields \(\mathcal {X}\) (we refer to [55, 56] for more details on these spaces). For every multi-index \(J=(j_1 , \ldots , j_m) \in \mathbb {Z}_+^m\) let \(\mathcal {X}^J = X_{j_1} \cdots X_{j_m}\). The length of a multi-index J is \(|J|= j_1+ \cdots +j_m\), thus \(\mathcal {X}^J\) is a linear differential operator of order |J|. For \(r\in {\mathbb {N}}\) and \(\alpha \in (0, 1)\) we define the function spaces

$$\begin{aligned} C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d)= & {} \left\{ u \in L^{\infty }({\mathbb {T}}^d) \; : \; \sup _{ \begin{array}{c} x, y \in {\mathbb {T}}^d \\ x\ne y \end{array} } \frac{|u(x) - u(y)| }{d_{CC}(x,y)^\alpha } <\infty \right\} ,\\ C_{\mathcal {X}}^{r, \alpha }({\mathbb {T}}^d)= & {} \left\{ u \in L^{\infty }({\mathbb {T}}^d) \; : \; \mathcal {X}^J u \in C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d) \;\; \forall |J| \le r\right\} . \end{aligned}$$

For any function \(u\in C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d) \) one can define a seminorm as

$$\begin{aligned}{}[ u ]_{C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d)}= \sup _{ \begin{array}{c} x, y \in {\mathbb {T}}^d\\ x\ne y \end{array} } \frac{|u(x) - u(y)| }{d_{CC}(x,y)^\alpha }, \end{aligned}$$

and, for every \(u\in C_{\mathcal {X}}^{r, \alpha }({\mathbb {T}}^d)\), the norm is defined as

$$\begin{aligned} \left\| u\right\| _{C_{\mathcal {X}}^{r, \alpha }({\mathbb {T}}^d)}= \left\| u\right\| _{L^\infty ({\mathbb {T}}^d)} +\sum _{1\le |J|\le r} [\mathcal {X}^J u ]_{C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d)}. \end{aligned}$$

Endowed with the above norm, \(C_{\mathcal {X}}^{r, \alpha }({\mathbb {T}}^d) \) are Banach spaces for any \(r\in {\mathbb {N}}\) and \(\alpha \in (0,1)\).

From estimates (2.6), it follows immediately

$$\begin{aligned}&C^{-1}\left\| u\right\| _{C^{0, \frac{\alpha }{k}}({\mathbb {T}}^d)} \nonumber \\&\quad \le \left\| u\right\| _{C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d)}\le C\left\| u\right\| _{C^{0, \alpha }({\mathbb {T}}^d)}\quad \implies \quad C_{}^{0, \alpha }({\mathbb {T}}^d)\subset C_{\mathcal {X}}^{0, \alpha }({\mathbb {T}}^d) \subset C_{}^{0, \frac{\alpha }{k}}({\mathbb {T}}^d),\qquad \end{aligned}$$
(2.7)

where \(\left\| u\right\| _{C^{0, \alpha }({\mathbb {T}}^d)}\) is the standard Hölder norm, k is the step in the Hörmander condition and \(C>0\) is a global constant depending only on the dimension d and the family of vector fields \(\mathcal {X}=\{X_1,\dots ,X_m\}\). More in general, for all \(r\in {\mathbb {N}}\), \( C^{r, \alpha }({\mathbb {T}}^d)\subset C_{\mathcal {X}}^{r, \alpha }({\mathbb {T}}^d).\)

Let r be a non-negative integer and \(1\le p \le \infty \). We define the space

$$\begin{aligned} W_{\mathcal {X}}^{r, p}({\mathbb {T}}^d) = \left\{ u \in L^p({\mathbb {T}}^d) \; : \; \mathcal {X}^J u \in L^p({\mathbb {T}}^d) , \; \forall J \in \mathbb {Z}_+^m, \; |J| \le r \right\} \,. \end{aligned}$$

Endowed with the norm \( \left\| u\right\| _{ W_{\mathcal {X}}^{r,p} ({\mathbb {T}}^d) } = \left( \sum _{|J|\le r } \int _{{\mathbb {T}}^d } |\mathcal {X}^J u |^p \, dx \right) ^{1/p} , \) \(W_{\mathcal {X}}^{r, p}({\mathbb {T}}^d)\) is a Banach space. For \(p=2\) we write \(H_{\mathcal {X}}^{r}({\mathbb {T}}^d)\) instead of \(W_{\mathcal {X}}^{r, p}({\mathbb {T}}^d) \) and in this case the space is Hilbert when endowed with the corresponding inner product. Moreover, for any \(1\le p< \infty \), the embeddings

$$\begin{aligned}&C_{\mathcal {X}}^{kr, \alpha }({\mathbb {T}}^d) \hookrightarrow C^{r, \frac{\alpha }{k}}({\mathbb {T}}^d) ,\\&W_{\mathcal {X}}^{r, p}({\mathbb {T}}^d)\hookrightarrow W^{r/k, p}({\mathbb {T}}^d) \,, \end{aligned}$$

hold true. The first is proved in [54] and the second in [53].

In proving one of our main results we will also need the following compact embedding.

Lemma 2.1

\(W_{\mathcal {X}}^{1, p}({\mathbb {T}}^d)\) is compactly embedded into \(L^{p}({\mathbb {T}}^d)\).

This follows from the previous embedding and the fact that the fractional Sobolev space \(W^{k/m, p}({\mathbb {T}}^d)\) is compactly embedded into \(L^p({\mathbb {T}}^d)\) (see e.g. [27]).

Next we want to recall some Hölder regularity results for linear and quasilinear subelleptic PDEs, key for the later existence results. Hölder and Schauder estimates for subelliptic linear and quasilinear equations have been proved by Xu [53, 55], Xu-Zuily [47, 56]; see also the references therein. In particular we will consider the results proved in [56], but we will rewrite them in a stronger form, by combining them with some \(L^p\)-estimates proved by Sun-Liu-Li-Zheng [52]. The results in [56] are proved for subelliptic systems but we will apply them to the case of a single equation. We first consider linear equations of the form:

$$\begin{aligned} \text {div}_{\mathcal {X}^*}\big (A(x)D_{\mathcal {X}} u\big )+ g(x) \cdot D_{\mathcal {X}} u+c(x)\,u =f(x). \end{aligned}$$
(2.8)

and assume that

$$\begin{aligned} A(x)\; \text {is a }m\times m-\text { uniformly elliptic matrix}. \end{aligned}$$
(2.9)

Note that in the case of the sub-Laplacian the previous assumption is trivially satisfied since A(x) is equal to the identity \(m\times m\)-matrix.

Theorem 2.2

(\(C^{2,\alpha }_{\mathcal {X}}\)-regularity for linear subelleptic PDEs, [52, 56].) Assuming (2.9) and that all coefficients of A(x), g(x), c(x) and f(x) are Hölder continuous, then any weak solution \(u\in H_{\mathcal {X}}^1({\mathbb {T}}^d)\) of (2.8) belongs to \(C_{\mathcal {X}}^{2,\alpha }({\mathbb {T}}^d)\) for some \(\alpha \in (0,1)\).

Moreover there exists a constant \(C>0\) (depending only on the Hölder norms of the coefficients of the equation, on d and on the vector fields \(\mathcal {X}\)) such that

$$\begin{aligned} \left\| u\right\| _{C_{\mathcal {X}}^{2, \alpha }({\mathbb {T}}^d)}\le C. \end{aligned}$$

Proof

First we recall that, if the coefficients are \(C^{0,\alpha }\) then they are also \(C^{0,\alpha }_{\mathcal {X}}\) (see (2.7)). Then Theorems 3.4 and 3.5 in [56] ensure that, given any u weak \(H_{\mathcal {X}}^1\)- solution, u belongs to \( C_{\mathcal {X}}^{2,\alpha }({\mathbb {T}}^d)\), and the \(C_{\mathcal {X}}^{2,\alpha }\)-Hölder norm of u is bounded by a constant depending on the Hölder norms of the coefficients, on the geometry of the problem (i.e. the step r, the dimension d and the number of vector fields m), but also on a constant M such that \( \left\| u\right\| _{H_{\mathcal {X}}^{1}({\mathbb {T}}^d)} \le M. \)

We can now use the uniform \(L^p\) estimates proved in Theorem 1.4 in [52] to show that the constant C is actually independent of M, i.e. independent of the \(H^{1}_{\mathcal {X}}\)-norm of u. Note that Hölder regularity on a compact domain implies all the necessary \(L^p\)-bounds to apply the result in [52]. \(\square \)

Let us now consider a subelliptic quasilinear equation of the form:

$$\begin{aligned} \text {div}_{\mathcal {X}^*}\big (A(x)D_{\mathcal {X}} u\big )=f(x,u,D_{\mathcal {X}}u). \end{aligned}$$
(2.10)

and assume that f(x, z, q) is a Hölder function with at most quadratic grow, i.e.

$$\begin{aligned} |f(x,z,q)|\le a|q|^2+b, \end{aligned}$$
(2.11)

for some non-negative constants a and b.

Theorem 2.3

(\(C^{1,\alpha }_{\mathcal {X}}\)-regularity for quasilinear subelleptic PDEs, [52, 56].) Assuming (2.9), (2.11) and that all the coefficients of the equation are Hölder continuous, then any weak solution \(u\in H_{\mathcal {X}}^1({\mathbb {T}}^d)\cap C({\mathbb {T}}^d)\) belongs to \(C_{\mathcal {X}}^{1,\alpha }({\mathbb {T}}^d)\) for some \(\alpha \in (0,1)\) and there exists a constant \(C>0\) (depending only on the Hölder norms of the coefficients of A(x) and of f, on a and b in (2.11), on the step r, on d and m) such that

$$\begin{aligned} \left\| u\right\| _{C_{\mathcal {X}}^{1, \alpha }({\mathbb {T}}^d)}\le C. \end{aligned}$$

Proof

Combining once again the \(L^p\)-estimates in [52] with Theorem 4.1 in [56] one can immediately deduce the result. \(\square \)

Theorem 2.4

(\(C^{\infty }\)-regularity, Theorem 4.2, [56]). Under the assumptions of Theorem 2.3, if in addition all coefficients in Eq. (2.10) are \(C^{\infty }({\mathbb {T}}^d)\) then \(u\in C^{\infty }({\mathbb {T}}^d)\).

3 Discounted systems with at most quadratic Hamiltonians

In this section we consider a subelliptic MFG system with a first order nonlinear term that grows at most quadratic w.r.t. the horizontal gradient. We assume:

(II-Q) :

For \(q=\sigma (x)p\in {\mathbb {R}}^m\) there exists a constant \(C \ge 0\) such that

$$\begin{aligned} |H(x, q )| \le C( |q |^2 + 1 ) \quad \forall x\in {\mathbb {T}}^d\,,\; q \in {\mathbb {R}}^m . \end{aligned}$$
(3.1)
(III) :

The vector-valued function \( g:{\mathbb {T}}^d \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^m \) is Hölder-continuous.

(Note that since \({\mathbb {T}}^d\) is compact and we will later prove global bounds for \(D_{\mathcal {X}} u\), the continuity of g implies also that g is globally bounded).

(IV) :

Set \({\mathcal {A}}:=\left\{ m \in C({\mathbb {T}}^d) \; : \; m>0 \,, \; \int _{{\mathbb {T}}^d} {m(x) \, dx =1 } \right\} \), then the map \(V:{\mathcal {A}}\rightarrow L^\infty ({\mathbb {T}}^d)\) is assumed continuous and bounded. Moreover, we assume that V is regularising, that is, \(V[m] \in C^\alpha _\mathcal {X}({\mathbb {T}}^d)\) for all \(m \in {\mathcal {A}}\), and \( \sup _{m \in {\mathcal {A}}} \Vert V[m]\Vert _{C^\alpha _{\mathcal {X}}({\mathbb {T}}^d)} < \infty . \)

Theorem 3.1

Assume (2.1), (II-Q), (III), (IV) and that H(x, q) is locally Hölder, then given \(\mathcal {L}\) defined in (2.3) with dual operator \(\mathcal {L}^*\) and \(\mathrm {div}_{\mathcal {X}^*}\) defined as in (2.4) w.r.t. the dual vector fields \(X^*_i=-X_i-\text {div}X_i\), for every \(\rho >0\) the system

$$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}u + \rho u + H(x, D_{\mathcal {X}}u) = V[m] \\&\mathcal {L}^*m - \text {div}_{{\mathcal {X}}^* }( m g ( x, D_{\mathcal {X}}u ) ) = 0 \\&\int _{{\mathbb {T}}^d} m \,dx =1, \quad m > 0 \end{aligned} \right. \end{aligned}$$
(3.2)

has a solution \((u, m) \in C_{\mathcal {X}}^2({\mathbb {T}}^d) \times C({\mathbb {T}}^d)\). (Note that u solves the system in the classical sense while m is a weak solution in the distributional sense.)

To prove the existence for the system (3.2) we need to look at both the equations involved, starting first from the associated linear PDE for u.

Lemma 3.2

Assume (2.1) and that \(\mathcal {L}\) is the corresponding sub-Laplacian defined in (2.3), then for every \(\rho >0\) and \(f\in C_{}^{0,\alpha }({\mathbb {T}}^d) \)

$$\begin{aligned} \mathcal {L}u + \rho u = f \text { in } {\mathbb {T}}^d \end{aligned}$$
(3.3)

has a unique solution \(u \in C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d) \). Moreover \(\exists \;C \ge 0\) (independent of u and f) such that

$$\begin{aligned} \Vert u\Vert _{ C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d) } \le C\, \Vert f\Vert _{C_{\mathcal {X}}^{0,\alpha } ({\mathbb {T}}^d)}. \end{aligned}$$
(3.4)

Proof

The solution is unique by the strong maximum principle of Bony [15] (see also Bardi and Da Lio [10]). We show the existence by vanishing viscosity methods, i.e. for all \(\varepsilon >0\) we consider the operator \(\mathcal {L}_\varepsilon = - \,\varepsilon \Delta + \mathcal {L}\), with \(\varepsilon >0\) and the corresponding problem (3.3), replacing \(\mathcal {L}\) by \(\mathcal {L}_\varepsilon \). Note that \(\mathcal {L}_\varepsilon u+\rho u=f\) is a linear uniformly elliptic equation. It is well-known that such a problem has a unique classical solution \(u_\varepsilon \), which is of class \(C^{2,\alpha }\) since \(f\in C^{0,\alpha }\) (see e.g. [11, Lemma 2.7] and [32]). Moreover \( \Vert u_\varepsilon \Vert _\infty \le \frac{1}{\rho }\Vert f\Vert _\infty \,. \) This implies that (up to a subsequence) \(u_{\varepsilon } \rightarrow u\) in the weak\(^*\)-topology of \(L^\infty ({\mathbb {T}}^d)\). Therefore u is a distributional solution of \(\mathcal {L}u+\rho u =f\). Furthermore, if f is smooth then, by Hörmader’s hypoellipticity Theorem [38], u is smooth. So let us assume for the moment that \(f\in C^{\infty }({\mathbb {T}}^d)\); then u is in particular a classical solution satisfying the assumption of Zuily and Xu [56], thus Theorem 2.2 gives directly estimate (3.4).

If \(f\in C_{\mathcal {X}}^{0,\alpha }({\mathbb {T}}^d) \), one can bypass this obstacle by mollifications and noticing that estimate (3.4) is stable w.r.t. the mollification parameter. More precisely, when f is not smooth but only Hölder, we introduce \(f_\zeta := f * \varphi _\zeta \), where \(\varphi _\zeta (x) := \zeta ^{-d} \varphi ( x/\zeta )\) for \(\zeta >0\) and \(x\in {\mathbb {R}}^d\), and \(\varphi \) is a mollification kernel, that is, a nonnegative function of class \(C^\infty \), with support in the unit ball of \({\mathbb {R}}^d\) and \(\int _{{\mathbb {R}}^d} \varphi (x)\,dx =1\). One can easily check that \(f_\zeta \rightarrow f\) as \(\zeta \rightarrow 0\) in \(C_{\mathcal {X}}^{0,\alpha }({\mathbb {T}}^d)\). Let \(\{\zeta _n\}_{n\in {\mathbb {N}}}\) be a sequence of positive numbers converging to zero. For every \(n\in {\mathbb {N}}\) there exists a unique solution \(u_n \in C^{2,\alpha }({\mathbb {T}}^d)\) to (3.3) for , and by estimate (3.4), we have \(\Vert u_n- u_m \Vert _{ C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d) } \le C\, \Vert f_n-f_m\Vert _{C_{\mathcal {X}}^{0,\alpha } ({\mathbb {T}}^d)}\) for some constant \(C>0\) that does not depend on \(n, m \in {\mathbb {N}}\). Thus \(\{u_n\}_{n\in {\mathbb {N}}}\) is Cauchy in \(C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d)\) (using \(f_n \rightarrow f\) in \(C_{\mathcal {X}}^\alpha ({\mathbb {T}}^d)\)), hence it converges to some u in \(C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d)\). Passing to the limit as \(n\rightarrow \infty \) in the equation \(\mathcal {L}u_{n} + \rho u_{n} =f_n\) and in the estimates \(\Vert u_n \Vert _{C_{\mathcal {X}}^{2, \alpha }({\mathbb {T}}^d ) } \le C \Vert f_n\Vert _{C^\alpha _{\mathcal {X}}({\mathbb {T}}^d) }\) we find that u is a solution to (3.3) and that estimate (3.4) is satisfied. \(\square \)

The existence and uniqueness for the subelliptic linear equation for m is more technical. We first recall some heat kernel estimates and an ergodic result which will be key for the later results. Consider the Cauchy problem

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial z}{\partial t} - \mathcal {L}z - g \cdot D_{{\mathcal {X}}} z = 0 \\&z(0,x) = \phi (x) \end{aligned}\right. \end{aligned}$$
(3.5)

where \(\phi \) is Borel and bounded and g is Hölder-continuous. Then we have the following representation for the unique solution of (3.5):

$$\begin{aligned} z(t,x) = \int _{{\mathbb {T}}^d}K(t, x,y) \phi (y) \, dy\,, \end{aligned}$$

where the function \((t,x,y)\mapsto K(t,x,y)\), defined for \(t>0\), \(x,y\in {\mathbb {T}}^d\), \(x\ne y\), is the heat kernel associated to the ultraparabolic operator \(\partial _t - \mathcal {L}- g \cdot D_{{\mathcal {X}}}\). We next recall some known Gaussian estimates satisfied by the heat kernel K(t, x, y): there exist constants \(C=C(T)\,{>}\,0\) and \(M>0\) (depending only on the Hölder norm of g) such that

$$\begin{aligned} \frac{C^{-1}}{|B_{d_{CC}}(x, t^{1/2}) |}e^{-M \,d_{CC}(x,y)^2/t } \le K(t,x,y) \le \frac{C}{|B_{d_{CC}}(x, t^{1/2}) |}e^{-M\, d_{CC}(x,y)^2/t } , \end{aligned}$$
(3.6)

for all \(T>t>0\) and \(x\in {\mathbb {T}}^d\), where by \(|B_{d_{CC}}(x, t^{1/2}) |\) we indicate the Lebesgue measure of the Carnot-Carathéodory ball centred at x and of radius \(R=t^{1/2}\). This estimate has been firstly proved in the subelliptic case by [41] for “sums of squares” operators on compact manifolds and later generalised by many authors: in particular we refer to [16].

We now need to recall the following ergodic result.

Theorem 3.3

([12], Theorem II.4.1). Let \((S, \Sigma )\) be a compact metric space equipped with its Borel \(\sigma \)-algebra \(\Sigma \). Let P be a linear operator defined on the Banach algebra of Borel bounded functions on S. We assume that \( \Vert P \Vert \le 1 \) and \( P(1) =1, \) and there exists \(\delta >0\) such that

$$\begin{aligned} P\mathbf {1} _E(x) - P\mathbf {1} _E(y) \le 1- \delta , \qquad \forall x,y \in S , \; E\in \Sigma \,, \end{aligned}$$
(3.7)

where by \(\mathbf {1} _E(\cdot )\) we indicate the characteristic function of the Borel set E.

Under these assumptions there exists a unique probability measure \(\pi \) on S such that

$$\begin{aligned} \left| P^n \phi (x) - \int _S \phi \, d\pi \right| \le C e^{- k n } \Vert \phi \Vert _\infty \quad \forall x \in S\,, \end{aligned}$$
(3.8)

where \(C= 2/(1-\delta )\), \(k= - \ln {(1-\delta )}\). Then the measure \(\pi \) is the unique invariant measure of the operator P, that is the unique probability measure satisfying

$$\begin{aligned} \int _S P\phi \, d\pi = \int _S \phi \, d\pi , \end{aligned}$$

for every bounded Borel function \(\phi \) on S.

The measure \(\pi \) is called the ergodic measure of the operator P (for more details on ergodic measure see e.g. [25]). Property (3.8) is a “strong” ergodic property: it implies the convergence

$$\begin{aligned} \lim _{n\rightarrow \infty } P^n\phi = \int _S \phi \, d\pi \quad \text {uniformly} \end{aligned}$$

but also provides an exponential decay estimate on the convergence rate.

Remark 3.4

As noted also in [12], when applying the ergodic theorem above usually one checks if the so-called Doeblin condition is satisfied. More precisely, we assume that \((S, \Sigma )\) is equipped with a probability measure \(\mu \) and that P has the form

$$\begin{aligned} P\phi (x) = \int _S k(x,y) \phi (y) \, d\mu (y ) , \end{aligned}$$

for some Borel and bounded kernel \(k:S\times S \rightarrow {\mathbb {R}}\), and that there exist a set U with \(\mu (U)>0\) and \(\delta _0>0\) such that (Doeblin condition)

$$\begin{aligned} k(x,y) \ge \delta _0 >0 \qquad \forall x \in S, \; y \in U \,. \end{aligned}$$
(3.9)

It is easy to check that (3.9) implies (3.7) with \(\delta = \mu (U) \delta _0\). In fact, using \(S=\big (S\cap E\big ) \cup \big (S\cap E^c\big ) \):

$$\begin{aligned} P\mathbf {1} _E(x) - P\mathbf {1} _E(y)= & {} 1-\int _S k(y,z)\mathbf {1} _{E}(z)\, dz- \int _S k(x,z)\mathbf {1} _{E^c}(z)\, dz \le 1\\&-\,\delta _0\bigg [\big |E^c\cap U\big |+\big | E\cap U\big |\bigg ]\!\!\!=1\!\!-\delta _0|U|. \end{aligned}$$

Next we show existence and uniqueness for the weak solution of the subelliptic linear equation associated to m.

Lemma 3.5

Assume (2.1) and that \(g:{\mathbb {T}}^d \rightarrow {\mathbb {R}}^m\) is Hölder continuous. Then the problem

$$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}^* m - \text {div}_{\mathcal {X}^*}(m g) =0 \;\; \text {in } {\mathbb {T}}^d \,,\\&\int _{{\mathbb {T}}^d} m \, dx =1\,, \end{aligned} \right. \end{aligned}$$
(3.10)

has a unique weak solution m in \(H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\). Moreover \(0< \delta _0 \le m \le \delta _1\), for some \(\delta _1, \delta _0\) depending only on the Hölder norm of g and the coefficients of \(\mathcal {L}\) (i.e. the coefficient of the vector fields \(X_1,\dots , X_m\)).

A solution m of the PDE in (3.10) is to be understood in the weak (or \(H^1_\mathcal {X}\)) sense, i.e. we define the bilinear form

(3.11)

and its dual for all \(u, v\in H_\mathcal {X}^1({\mathbb {T}}^d)\). Then m is a solution of the PDE in (3.10) if \(\langle m, v\rangle ^* =0\) for all \(v\in H^1_{\mathcal {X}}({\mathbb {T}}^d)\).

Proof

The proof follows the approach introduced in [14, Theorem 3.4] for uniformly elliptic operators and in [12, Theorem II.4.2 ]. We want first to show that, for \(\eta >0\) large enough and for every \(\varphi \in L^2({\mathbb {T}}^d)\), the problem

$$\begin{aligned} \mathcal {L}u - g \cdot D_{\mathcal {X}} u + \eta \,u = \varphi \, \end{aligned}$$
(3.12)

is well-posed in \(H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\) in the standard weak sense, that is

$$\begin{aligned} \int _{{\mathbb {T}}^d} \left( -\,\frac{1}{2} \sum _{i=1}^mX_iu\; X^*_iv - (g \cdot D_{\mathcal {X}} u)\, v + \eta \,u\, v \right) d x =\int _{{\mathbb {T}}^d}\varphi \, v\,dx, \quad \forall \; v\in C^{\infty }_0\big ({\mathbb {T}}^d\big ). \end{aligned}$$

The previous well-posedness is proved by standard Hilbert space arguments. In fact, on the space \(H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\), we consider the bilinear form

$$\begin{aligned}&\langle \cdot , \cdot \rangle _{\eta } :H_{{\mathcal {X}}}^1({\mathbb {T}}^d) \times H_{{\mathcal {X}}}^1({\mathbb {T}}^d) \rightarrow {\mathbb {R}}\\&\quad (u, v) \mapsto \langle u, v\rangle _{\eta } := \langle u, v\rangle + \int _{{\mathbb {T}}^d} \eta \, u\, v \,dx \end{aligned}$$

for all \(u, v \in H^1_{\mathcal {X}}({\mathbb {T}}^d)\), where \(\langle u, v \rangle \) is defined in (3.11). For \(\eta >0\) large enough and for some \(c_1 > 0\), \(c_2 \ge 0\) we claim that for all \(u,v \in H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\)

$$\begin{aligned} \langle u, u \rangle _{\eta }&\ge c_1 \Vert u\Vert ^2_{ H_{{\mathcal {X}}}^1({\mathbb {T}}^d) }, \end{aligned}$$
(3.13)
$$\begin{aligned} |\langle u, v \rangle _{\eta }|&\le c_2 \Vert u\Vert _{ H_{{\mathcal {X}}}^1({\mathbb {T}}^d) } \Vert v\Vert _{ H_{{\mathcal {X}}}^1({\mathbb {T}}^d) }. \end{aligned}$$
(3.14)

We first check estimate (3.13). Since g and \(\text {div} X_i\) are by assumption continuous, hence bounded on \({\mathbb {T}}^d\), there exists \(M\ge 0\) such that \(\Vert g\Vert _\infty \le M\) and \(\Vert \text {div} X_i\Vert _\infty \le M\). Moreover

$$\begin{aligned} \langle u, u \rangle _{\eta }&= \int _{{\mathbb {T}}^d} \bigg ( \frac{1}{2} \sum _{i=1}^m |X_i u|^2 + \frac{1}{2} \sum _{i=1}^m X_i u \, (\text {div}X_i) u - \big (g \cdot D_{\mathcal {X}} u\big )u + \eta \,u^2 \bigg ) \, dx\\&\ge \int _{{\mathbb {T}}^d} \bigg ( \frac{1}{2} \sum _{i=1}^m |X_i u|^2 - \frac{1}{2} \sum _{i=1}^m |X_i u| |\text {div}X_i| |u| - |g| | D_{\mathcal {X}} u | |u| + \eta \,u^2 \bigg ) \, dx\,. \end{aligned}$$

Using the inequality \(ab \le (1/4)a^2 + b^2\) and recalling \( | D_{\mathcal {X}} u |^2=\sum _{i=1}^m |X_i u|^2\), we find

$$\begin{aligned} \langle u, u \rangle _{\eta }&\ge \int _{{\mathbb {T}}^d} \bigg ( \frac{1}{2}| D_{\mathcal {X}} u |^2 - \frac{1}{8} | D_{\mathcal {X}} u |^2 - |\text {div}X_i|^2 |u|^2 - \frac{1}{4} | D_{\mathcal {X}} u |^2 - |g|^2 |u|^2 + \eta \, u^2 \bigg ) \, dx \\&\ge \frac{1}{8} \int _{{\mathbb {T}}^d} | D_{\mathcal {X}} u |^2 \, dx + (\eta - 2M^2) \int _{{\mathbb {T}}^d} u^2 \,dx \,, \end{aligned}$$

from which, taking \(\eta > 2M^2\), we obtain the first estimate (3.13) for a suitable \(c_1 >0\) (in particular \(c_1=\min \big \{1/8,\eta -2M^2\big \}>0\)). For estimate (3.14) similarly

$$\begin{aligned} |\langle u, v \rangle _{\eta }| \le \frac{1}{2} \int _{{\mathbb {T}}^d} |D_{\mathcal {X}} u| |D_{\mathcal {X}}v| \, dx +M \int _{{\mathbb {T}}^d} |D_\mathcal {X}u| |v|\, dx + \eta \int _{{\mathbb {T}}^d} |u|\,|v| \, dx, \end{aligned}$$

and by the Cauchy–Schwarz inequality for integrals

$$\begin{aligned} |\langle u, v \rangle _{\eta }|&\le \frac{1}{2} \Vert D_\mathcal {X}u \Vert _{L^2({\mathbb {T}}^d) } \Vert D_\mathcal {X}v \Vert _{L^2({\mathbb {T}}^d) } +(M + \eta ) \Vert u \Vert _{L^2({\mathbb {T}}^d) } \Vert v \Vert _{L^2({\mathbb {T}}^d) }\\&\le \left( M+\eta +\frac{1}{2}\right) \left\| u\right\| _{H^1_{\mathcal {X}}({\mathbb {T}}^d)}\, \left\| v\right\| _{H^1_{\mathcal {X}}({\mathbb {T}}^d)}, \end{aligned}$$

where we have used simply \(a\,b+c\,d\le (a+c)\,(b+d)\) for every non-negative scalars a, b, c and d. This gives (3.14) with \(c_2 =\left( M+\eta +\frac{1}{2}\right) > 0\). Then the claim is proved.

Thus the bilinear form \(\langle \cdot ,\cdot \rangle _{\eta }\) is coercive and continuous. Clearly,

$$\begin{aligned} H_{{\mathcal {X}}}^1({\mathbb {T}}^d) \ni u \mapsto \int _{{\mathbb {T}}^d} u \varphi \, dx\, \in {\mathbb {R}}\end{aligned}$$

is a continuous linear functional on \(H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\). Therefore by the Lax-Milgram Theorem there exists a unique \(u\in H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\) such that, for all \(v \in H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\),

$$\begin{aligned} \langle u, v \rangle _{\eta } = \int _{{\mathbb {T}}^d}\varphi v \, dx. \end{aligned}$$

For every \(\eta >0\) large enough (i.e. \(\eta >2M^2\)), we define the following linear operator \( T_\eta :L^2({\mathbb {T}}^d) \rightarrow L^2({\mathbb {T}}^d)\, \) by \(T_\eta \varphi :=u,\) where u is the unique solution to (3.12).

Note that \(T_\eta \varphi =u\in H_{\mathcal {X}}^1({\mathbb {T}}^d) \subset L^2({\mathbb {T}}^d)\). Since the embedding of \(H_{{\mathcal {X}}}^1({\mathbb {T}}^d)\) into \(L^2({\mathbb {T}}^d)\) is compact (see Lemma 2.1), \(T_\eta \) is a linear compact operator. Thus the equation

$$\begin{aligned} \mathcal {L}^* m - \text {div}_{\mathcal {X}^*}(m g) =0 \quad \text {in } {\mathbb {T}}^d \end{aligned}$$

is equivalent to

$$\begin{aligned} (I - \eta T_\eta )^* m = 0\,, \end{aligned}$$
(3.15)

where I is the identity operator of \(L^2({\mathbb {T}}^d)\).

Since \(T_\eta \) is compact, the Fredholm alternative applies. Indeed \(I - \eta T_\eta \) is a Fredholm operator of index zero (see e.g. [1, Lemma 4.45]). This means that the kernels of \(I - \eta T_\eta \) and \((I - \eta T_\eta )^*\) have the same dimension, in other words, the number of linearly independent solutions of the equation \((I - \eta T_\eta )^* =0 \) is equal to the number of linearly independent solutions of the equation \(I - \eta T_\eta =0\). Then we must find the number of linearly independent solutions of \( (I - \eta T_\eta ) u= 0\,, \) that is

$$\begin{aligned} \mathcal {L}u + g (x) \cdot D_{\mathcal {X}}u =0\,. \end{aligned}$$

By [56, Theorem 3.3] the solution u belongs to \(C^{2, \alpha }_{\mathcal {X}}({\mathbb {T}}^d)\) for some \(\alpha \in (0,1)\). Moreover the operator \(\mathcal {L}+ g \cdot D_{{\mathcal {X}}}\) satisfies the strong maximum principle, see [10, 15]. Thus by the considerations above (Fredholm alternative) the Eq. (3.15), and hence (3.12), admits a unique solution \(m\in H^{1}_{\mathcal {X}}({\mathbb {T}}^d)\) up to a multiplicative constant.

The upper and lower bounds for m (that imply in particular the positivity of m) are shown by its interpretation as the ergodic measure of the diffusion having generator \(\mathcal {L}+ g \cdot D_{\mathcal {X}}\). They rely on an ergodic theorem and on the Gaussian estimates (3.6). In fact, using that \(d_{CC}(x,y)\) is continuous on \({\mathbb {T}}^d\) (compact), we can easily see from (3.6) (by simply taking the maximum and the minimum of \(d^2_{CC}(x,y)\) on \({\mathbb {T}}^d\)) that there exits \(\delta _0, \delta _1>0\) such that

$$\begin{aligned} \delta _0\le K(1,x, y) \le \delta _1\qquad \forall x,y \in {\mathbb {T}}^d. \end{aligned}$$
(3.16)

Therefore we can apply Theorem 3.3 and Remark 3.4 with \(S={\mathbb {T}}^d\), \(\Sigma \) the Borel \(\sigma \)-algebra on \({\mathbb {T}}^d\), \(\mu \) the Lebesgue measure on \({\mathbb {T}}^d\) and operator P defined by

$$\begin{aligned} P\phi (x) = z(1, x) = \int _{{\mathbb {T}}^d} K(1,x,y) \phi (y) \, dy \,. \end{aligned}$$

Note that \(P^n\phi (x) = z(n, x)\). Then Theorem 3.3 implies the existence of a unique invariant probability measure \(\pi \) such that

$$\begin{aligned} \left| z(n, x) - \int _{{\mathbb {T}}^d} \phi (y) \, d \pi (y) \right| \le C e^{-kn } \Vert \phi \Vert \,. \end{aligned}$$
(3.17)

Using m defined as the unique solution of (3.10) and z(t, x) defined as unique solution of (3.5), we want to show the following claim:

$$\begin{aligned} \int _{{\mathbb {T}}^d} z(t, x) m (x) \, dx = \int _{{\mathbb {T}}^d} \phi (x) m(x) \, dx, \; \forall \, t\ge 0. \end{aligned}$$
(3.18)

To prove the previous claim, first note that for \(t=0\) (3.18) is trivially satisfied by the initial condition. We want to show that the right-hand side in (3.18) is constant in time, so we look at

$$\begin{aligned} \frac{d}{dt} \int _{{\mathbb {T}}^d} z(t, \cdot ) m \, dx= & {} \int _{{\mathbb {T}}^d} \partial _t z m \, dx = \int _{{\mathbb {T}}^d} \big (\mathcal {L}z +g\cdot D_{\mathcal {X}} z\big ) \, m \, dx\\= & {} \int _{{\mathbb {T}}^d} \big (\mathcal {L}^*m- \text {div}_{\mathcal {X}^*} (m\,g)\big ) z\,dx =0. \end{aligned}$$

Then \(\int _{{\mathbb {T}}^d} z(t, x) m (x) \, dx =\int _{{\mathbb {T}}^d} z(0, x) m (x) \, dx\), for all \(t\ge 0\), that proves the claim (3.18).

By using (3.17) and taking \(t=n\) in (3.18) and passing to the limit as \(n\rightarrow +\infty \), we can deduce:

$$\begin{aligned} \int _{{\mathbb {T}}^d} \phi (x) \,m (x)\, dx =\int _{{\mathbb {T}}^d} \bigg (\int _{{\mathbb {T}}^d} \phi (x)\, d\pi (x) \bigg ) m(x)d x=\int _{{\mathbb {T}}^d} \phi (x)\, d\pi (x), \end{aligned}$$

for any Borel bounded function \(\phi \) on \({\mathbb {T}}^d\) (where we have used \( \int _{{\mathbb {T}}^d} m =1 \)). Thus m is the density measure of the probability measure \(\pi \) and therefore \(m \ge 0\) a.e. on \({\mathbb {T}}^d\).

Using (3.16) together with (3.18) for \(t=1\), it follows that

$$\begin{aligned} \delta _1 \int _{{\mathbb {T}}^d } \phi (y) \, dy \ge \int _{{\mathbb {T}}^d} \phi (y) m(y) \, dy \ge \delta _0 \int _{{\mathbb {T}}^d } \phi (y) \, dy, \end{aligned}$$

for any bounded and Borel function \(\phi \ge 0\) on \({\mathbb {T}}^d\). Since \(\phi \ge 0\) is arbitrary, one can deduce \( \delta _0 \le m \le \delta _1 , \) thus Lemma 3.5 is proved. \(\square \)

We can now prove our first existence result for a subelliptic MFG system.

Proof of Theorem 3.1

The proof is based on a corollary of Schauder’s fixed point theorem. More precisely, we apply [32, Theorem 11.3] which states that, if \(T:\mathcal {B}\rightarrow \mathcal {B}\) is a continuous and compact operator in the Banach space \(\mathcal {B}\) such that the set \(\{ u \in \mathcal {B}\; : \; s T u = u ,\; 0\le s \le 1 \}\) is bounded, then T has a fixed point, that is, there exists \(u \in \mathcal {B}\) such that \(Tu =u\). We define the Banach space \( \mathcal {B} = C_{\mathcal {X}}^{1, \alpha }({\mathbb {T}}^d)\,, \) where \(0<\alpha < 1\) is to be fixed later, and the operator \( T :\mathcal {B} \rightarrow \mathcal {B} \,, \) according to the scheme \( v \mapsto m \mapsto u \,. \) This means that, given \(v \in \mathcal {B}\), we solve the second equation together with the corresponding conditions

$$\begin{aligned}\left\{ \begin{aligned}&\mathcal {L}^*m - \text {div} ( m g ( x, D_{\mathcal {X}}v ) ) = 0 \, \text { in }{\mathbb {T}}^d\,, \\&\int _{{\mathbb {T}}^d} m \, dx =1 \,, \quad m>0 \, \text { in } {\mathbb {T}}^d \end{aligned}\right. \end{aligned}$$

and by Lemma 3.5 we find a unique solution \(m \in H^1_\mathcal {X}({\mathbb {T}}^d)\cap L^\infty ({\mathbb {T}}^d)\). Moreover m is bounded. By assumption \(\mathbf{(IV)}\), \(V[\cdot ]\) is regularizing, hence the function \(f(x) = V[m](x) - H(x, D_{\mathcal {X}}v (x) )\) belongs to \(C^\alpha _{\mathcal {X}}({\mathbb {T}}^d)\). Thus we apply Lemma 3.2 and deduce that

$$\begin{aligned} \mathcal {L}u + \rho u + H(x, D_{\mathcal {X}}v) = V[m] \end{aligned}$$
(3.19)

admits a unique solution \(u \in C^{2}_{\mathcal {X}}({\mathbb {T}}^d)\). Set \(T v = u,\) where u is the unique solution of (3.19), it is easy to check that T is continuous and compact, using that \(C_{\mathcal {X}}^2({\mathbb {T}}^d)\) is compactly embedded into \(C^{1,\alpha }_{\mathcal {X}}({\mathbb {T}}^d)\) for all \(\alpha \in (0,1)\). Therefore, in order to apply [32, Theorem 11.3] we need to show that

$$\begin{aligned} \mathcal {A}=\{ u\in \mathcal {B} \;: \; \exists \,0\le s\le 1 \text { such that } u = s T u\} \end{aligned}$$

is bounded in \(C_{\mathcal {X}}^{1, \alpha }({\mathbb {T}}^d)\). So note that: if u is a fixed point of sT (i.e. \(sTu=u\)), then it is also a solution of

$$\begin{aligned} \mathcal {L}u + \rho u + s H(x, D_{\mathcal {X}}u) = s V[m] . \end{aligned}$$
(3.20)

Then looking at the minimum and maximum of u, we find

$$\begin{aligned} \Vert u\Vert _{\infty , {\mathbb {T}}^d} \le \frac{s}{\rho }\sup _{m\in H^1_{{\mathcal {X}}}({\mathbb {T}}^d)} \Vert V[m] -H (\cdot , 0 )\Vert _{\infty , {\mathbb {T}}^d}, \end{aligned}$$

which is finite since \(V[\cdot ]\) is by assumption bounded.

The key step is now to apply \(C_{\mathcal {X}}^{1, \alpha }\)-regularity for semilinear equation (Theorem 2.3) that gives

$$\begin{aligned} \Vert u \Vert _{C_{\mathcal {X}}^{1, \alpha } ({\mathbb {T}}^d) } < C \end{aligned}$$
(3.21)

for some constant \(C>0\) and \(\alpha \in (0,1)\) independ of u and \(s\in [0,1]\).

Note that, in order to apply the given theorem, we should write our equation in divergence form, which we can easily do by using the relation \(X_i^* = -X_i - \text {div} X_i\) (by adding the term \(-\sum _{j=1}^n (\text {div}X_j) X_j\) to the Hamiltonian). Observe that the new Hamiltonian has the same properties of the original Hamiltonian; in particular, it grows at most quadratically in \(D_{\mathcal {X}}u\) (in fact the functions \(\text {div} X_j\) are bounded due to the \(C^{\infty }\)-regularity of the vector fields \(X_j\)). Using estimate (3.21) we can look at the semilinear PDE (3.20) as a linear PDE with an Hölder right-hand side \( f(x)= s V[m] -s H(x, D_{\mathcal {X}}u)-\sum _{j=1}^m (\text {div} X_j) X_ju \); hence we can apply the Schauder type result for linear equations proved in [56] (see Theorem 2.2), that implies \(u \in C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d) \) and \( \Vert u \Vert _{C_{\mathcal {X}}^{2, \alpha } ({\mathbb {T}}^d) } < C. \) To conclude we need only to remark that \(\text {div} X_j \in C^{0, \alpha }_{\mathcal {X}}({\mathbb {T}}^d)\) (since the vector fields are smooth on a compact domain) in order to apply the previous \(C_{\mathcal {X}}^{2, \alpha }\)-estimates. \(\square \)

4 Ergodic system with linear growth

We now want to study the ergodic problem that can be obtained by letting \(\rho \rightarrow 0^+\) in (3.2). However, to study this, we need a more restrictive assumption on the Hamiltonian, i.e. we assume that H grows at most linearly in \(|D_{\mathcal {X}}u|\). More precisely:

(II-L) :

\(H(x,q)=H(x,\sigma (x) p)\) grows at most linearly w.r.t. q, i.e. \(\exists \, C \ge 0\) such that

$$\begin{aligned} |H(x, q )| \le C( |q | + 1 ) \quad \forall x\in {\mathbb {T}}^d\,,\; q \in {\mathbb {R}}^m . \end{aligned}$$
(4.1)

We prove existence of solutions for the system of ergodic PDEs under condition (II-L).

Theorem 4.1

(Existence). Assume (2.1), (II-L), (III), (IV) and that H(x, q) is locally Hölder, then the system

$$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}u + \lambda + H(x, D_{\mathcal {X}}u) = V[m] \\&\mathcal {L}^*m - \text {div}_{\mathcal {X}^*} \big ( m g ( x, D_{\mathcal {X}}u ) \big ) = 0 \\&\int _{{\mathbb {T}}^d} u \, dx =0, \quad \int _{{\mathbb {T}}^d} m\, dx =1 , \quad m > 0 \end{aligned} \right. . \end{aligned}$$
(4.2)

has a solution \((\lambda , u, m) \in {\mathbb {R}}\times C_{\mathcal {X}}^2({\mathbb {T}}^d) \times C({\mathbb {T}}^d)\).

Proof

For \(\rho >0\) let \((u_\rho , m_\rho ) \in C^2_\mathcal {X}({\mathbb {T}}^d)\times \big ( H_\mathcal {X}^1({\mathbb {T}}^d)\cap L^\infty ({\mathbb {T}}^d) \big )\) be a solution of (3.2) by the existence result given in Theorem 3.1. Looking at the minima and maxima of \(u_\rho \), we have

$$\begin{aligned} \Vert \rho u_{\rho } \Vert _{\infty } \le \sup _{m\in H_\mathcal {X}^1({\mathbb {T}}^d)}\Vert H(\cdot , 0)- V[m]\Vert _{\infty } \,. \end{aligned}$$
(4.3)

Let \(<u_\rho >:= \int _{{\mathbb {T}}^d} u_\rho \,dx \) be the average of \(u_\rho \), the key estimate is given in the following claim: there exist \(\rho _0>0\) and \(C>0\) (independent of \(\rho \)) such that

$$\begin{aligned} \Vert u_{\rho } -<u_{\rho }>\Vert _{\infty } \le C, \quad \forall \; 0<\rho <\rho _0. \end{aligned}$$
(4.4)

To prove (4.4) we adapt some ideas from [5]. Assume by contradiction that there is a sequence \(\rho _n \rightarrow 0\) such that \(\Vert u_{\rho _n} - <u_{\rho _n}>\Vert _{\infty }\rightarrow +\infty \) or equivalently, such that the sequence

$$\begin{aligned} \varepsilon _n:=\Vert u_{\rho _n} - <u_{\rho _n}>\Vert _{\infty }^{-1}\rightarrow 0. \end{aligned}$$

Then the renormalised functions \(\psi _n:=\varepsilon _n(u_{\rho _n} - <u_{\rho _n}>)\) satisfy

$$\begin{aligned} \mathcal {L}{\psi _n}+\varepsilon _n H\left( x, \frac{D_{\mathcal {X}}\psi _n}{\varepsilon _n}\right) +\rho _n\psi _n= \varepsilon _n(V[m_{\rho _n}] - \rho _n<u_{\rho _n}>) . \end{aligned}$$
(4.5)

We now apply [53, Theorem 17] to deduce that the sequence \(\{\psi _n\}\) is equi-Hölder continuous. In fact \(\psi _n\) solve quasilinear equations of the same form as in [53] with \(A_i(x,u,\xi )=\xi _i\) and \( B(x,u,\xi )=\varepsilon _nH\left( x,\frac{\xi }{\varepsilon _n}\right) -\rho _n u-\varepsilon _n\big (V[m_{\rho _n}]\big )-\rho _n <u>; \) then it is easy to check that all conditions on the equation are satisfied just taking \(g=0\), \(f=1\) and a \(\Lambda \) depending only on the bound for \(V[\cdot ]\), the constant in (II-L) and the Lebesgue measure of \({\mathbb {T}}^d\). Thus [53, Theorem 17] tells us that, taking \(\rho _n\le 1\) and \(\varepsilon _n\le 1\), the Hölder norms of the solutions \(\psi _{\rho _n}\) are equi-bounded independently on n, which implies that \(\psi _{\rho _n}\) are equi-Hölder. Therefore (up to a subsequence) we get that \(\psi _n\) converges uniformly to a function \(\psi \). Note that the functions \(\psi _n\) are all renormalised, then \(\Vert \psi \Vert _{\infty }=1\). Moreover, since \(\int _{{\mathbb {T}}^d} \psi _n \,dx=0\) by definition, then there exists a point \(x_n\in {\mathbb {T}}^d\) such that \(\psi _n(x_n)=0\). Thus (up to a further subsequence) we get \(\psi (\overline{x})=0\) for some \(\overline{x}\in {\mathbb {T}}^d\). By using assumption (II-L) into equation (4.5), one finds out that \(\psi _{\rho _n}\) are classical (and hence viscosity) subsolutions of

$$\begin{aligned} \mathcal {L}{\psi _n}-C \,|D_{\mathcal {X}}\psi _n|+\rho _n\psi _n- \varepsilon _n(V[m_{\rho _n}] - \rho _n<u_{\rho _n}>+C) =0. \end{aligned}$$
(4.6)

and classical (and hence viscosity) supersolutions of

$$\begin{aligned} \mathcal {L}{\psi _n}+C \,|D_{\mathcal {X}}\psi _n|+\rho _n\psi _n- \varepsilon _n(V[m_{\rho _n}] - \rho _n<u_{\rho _n}>-C) =0. \end{aligned}$$
(4.7)

Finally by taking \(n\rightarrow \infty \) in (4.6) and (4.7) and by using the stability for viscosity subsolutions and viscosity supersolutions under uniform convergence (see e.g. [7]), \(\psi \) is a viscosity subsolution of \( \mathcal {L}{\psi } -C|{D_{{\mathcal {X}}}\psi }| = 0 \) and a viscosity supersolution of \( \mathcal {L}{\psi } +C|{D_{{\mathcal {X}}}\psi }| = 0 \). Since \(\mathcal {L}\) is the subelliptic Laplacian associated to smooth Hörmander vector fields and \(\psi \) is periodic, we deduce from the strong maximum principle (see [10, 15]) that \(\psi \) must be a constant, which contradicts \(\Vert \psi \Vert _{\infty }=1\) and \(\psi (\overline{x})=0\), proving thus (4.4).

We complete the proof of the theorem by showing that there exists a sequence \(\rho _n \rightarrow 0\) such that, for \(w_\rho :=u_{\rho } - <u_{\rho }>\),

$$\begin{aligned} \left( \rho _n <u_{\rho _n}>, \,w_{\rho _n}, \, m_{\rho _n} \right) \rightarrow (\lambda , \, u, \, m) \qquad \text{ in } {\mathbb {R}}\times C_{\mathcal {X}}^2({\mathbb {T}}^d) \times H_{\mathcal {X}}^1({\mathbb {T}}^d), \end{aligned}$$
(4.8)

where \((\lambda , \, u, \, m)\) is a solution of (4.2); the convergence \(m_{\rho _n} \rightarrow m\) is in the weak topology of \(H_{\mathcal {X}}^1({\mathbb {T}}^d)\). Indeed, we note that \((w_{\rho }, m_{\rho })\) solves

$$\begin{aligned} \left\{ \begin{aligned}&\mathcal {L}{w_{\rho }}+\rho w_\rho +H(x, D_{\mathcal {X}}{w_{\rho }})= V[m_{\rho } ] - \rho <u_{\rho }> \quad \text{ in }\; {\mathbb {T}}^d,\\&{\mathcal {L}}^*{m_{\rho }}-\text {div}_{{\mathcal {X}}^*}\left( g(x, D_{\mathcal {X}}{w_{\rho }} ){m_{\rho }}\right) =0,\\&\int _{{\mathbb {T}}^d} m_\rho (x) dx = 1,\quad m_\rho >0 \,. \end{aligned} \right. \end{aligned}$$
(4.9)

By the a-priori Hölder estimates for quasilinear subelliptic equations recalled in Theorem 2.3 we know that

$$\begin{aligned} \Vert {w_{\rho }}\Vert _{C^{1, \alpha }_{\mathcal {X}}({\mathbb {T}}^d)} \le C, \end{aligned}$$

for some \(\alpha \in (0,1)\) and \(C>0\) depending only on an upper bound of \(\Vert {w_{\rho }}\Vert _{\infty }\) and on the data of the problem, in particular on the supremum norm of \(V[m_{\rho }] - \rho <u_{\rho }>\,,\) which is bounded uniformly in \(\rho \) by (IV) and (4.3). In other words, \(\alpha \) and C can be chosen independent of \(\rho \). Next by Schauder local estimates for subelliptic linear equations [56, Theorem 3.5], we have

$$\begin{aligned} \Vert {w_{\rho }}\Vert _{C_{\mathcal {X}}^{2,\alpha }({\mathbb {T}}^d )} \le C, \end{aligned}$$
(4.10)

for some C and \(\alpha \) independent of \(\rho \). On the other hand, by Lemma 3.5 and assumption (III)

$$\begin{aligned} \Vert m_{\rho }\Vert _{H_{\mathcal {X}}^{1}({\mathbb {T}}^d)} \le C, \end{aligned}$$
(4.11)

for \(C\ge 0\) independent of small enough \(\rho \). Since \(C_{\mathcal {X}}^{2, \alpha }({\mathbb {T}}^d)\) is compactly embedded into \(C_{\mathcal {X}}^2({\mathbb {T}}^d)\), the previous estimates (4.10), (4.11) and the fact that the set \(\{\rho <u_\rho> \; : \; \rho >0 \} \) is bounded (in \({\mathbb {R}}\)) by (4.3), we can extract a sequence \(\rho _n \rightarrow 0\) such that  (4.8) holds. Furthermore, since g is locally Hölder by assumption (III) and \(D_{\mathcal {X}}w_{\rho _n } \rightarrow D_{\mathcal {X}}u\) in \(C_{\mathcal {X}}^1({\mathbb {T}}^d)\), then . Let \(\langle \cdot , \cdot \rangle _n\) denote the bilinear form associated with \(g_n\) in the same fashion as \(\langle \cdot , \cdot \rangle \) denotes the bilinear form associated with g after the statement of Lemma 3.5. Since \(m_{\rho _n}\) is the solution of the second equation in (4.9), \(\langle m_{\rho _n}, \varphi \rangle _n^*=0\) for all \(\varphi \in H_\mathcal {X}^1({\mathbb {T}}^d)\). From this and the fact that \(g_n \rightarrow {\bar{g}}(\cdot , D_\mathcal {X}u (\cdot ))\) in \(L^2({\mathbb {T}}^d)\), it is fairly easy to deduce that \(\langle m , \varphi \rangle ^* =0\) for all \(\varphi \in H^1_\mathcal {X}({\mathbb {T}}^d)\). Thus m is a solution of the second equation in (4.2). The normalising conditions in the third row of (4.2) are clearly preserved in the limit. Thus the triplet \((\lambda , u, m)\) is indeed a solution of (4.2). \(\square \)

Exactly as in the elliptic case, both the previous MFG systems have unique solutions under suitable monotonicity assumptions.

Recall that an operator V, defined on some subset of \(L^2({\mathbb {T}}^d)\) with values in \(L^2({\mathbb {T}}^d)\), is monotone if \( \int _{{\mathbb {T}}^d} \left( V[m_1]-V[m_2] \right) (m_1-m_2)\, dx \ge 0\), \( \forall m_1, m_2, \) and it is strictly monotone if the inequality is strict for all \(m_1 \ne m_2\). Given a function \(H :{\mathbb {T}}^d \rightarrow {\mathbb {R}}\) and a vector-valued map \(g:{\mathbb {T}}^d \rightarrow {\mathbb {R}}^m\), we say that H is g-convex if \( H( q_2) - H(q_1) - g(q_1) \cdot (q_2-q_1) \le 0, \) for all \(q_1, q_2 \in {\mathbb {T}}^d\). If the inequality is strict for \(q_1\ne q_2\), H is strictly g-convex.

Theorem 4.2

(Uniqueness) Assume that one of the two following assumptions holds:

(i) :

V is monotone in \(L^2\) and H is strictly \((-g)\)-convex, or

(ii) :

V is strictly monotone in \(L^2\) and H is \((-g)\)-convex.

Then the system (4.2) has a unique weak solution.

The proof is standard so we omit it.

Remark 4.3

  1. 1.

    Hamiltonians H coming from optimal control are “\((-g)\)-convex” and, under suitable assumptions, strictly \((-g)\)-convex.

  2. 2.

    The strict \((-g)\)-convexity can be relaxed requiring that \( H( q_2) - H(q_1) - g(q_1) \cdot (q_2-q_1) \le 0, \) implies \(g(x, q_1)= g(x, q_2)\), instead of \(q_1=q_2\). In this way one can cover also the case \(H(x, q)= |q|\) and \(g(x, q) = -q/|q|\) for \(q\ne 0\), \(g(x, 0)=0\).

  3. 3.

    Similarly one can state the uniqueness for the “discounted” system (3.2).