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Schauder Estimates for a Class of Potential Mean Field Games of Controls

  • J. Frédéric Bonnans
  • Saeed Hadikhanloo
  • Laurent PfeifferEmail author
Open Access
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Abstract

An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray–Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.

Keywords

Mean field games of controls Extended mean field games Strongly coupled mean field games Potential formulation Hölder estimates 

Mathematics Subject Classification

91A13 49N70 

1 Introduction

The goal of this work is to prove the existence and uniqueness of a classical solution to the following system of partial differential equations:
$$\begin{aligned} \left\{ \begin{array}{l l l} (i) \quad &{}-\partial _t u -\sigma \varDelta u + H(x,t, \nabla u(x,t) + \phi (x,t)^\intercal P(t) ) \quad &{} \\ &{} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = f(x,t,m(t))\quad &{}(x,t) \in Q, \\ (ii) \quad &{}\partial _t m - \sigma \varDelta m + \mathrm {div} (v m)= 0 \quad &{}(x,t) \in Q, \\ (iii) \quad &{}P(t) = \varPsi \left( t, \int _{ \mathbb {T}^d} \phi (x,t) v(x,t) m(x,t) \; \mathrm {d} x \right) \quad &{}t \in [0,T],\\ (iv) \quad &{} v(x,t) = - H_p(x,t, \nabla u (x,t) + \phi (x,t)^\intercal P(t) ) &{}(x,t) \in Q, \\ (v) \quad &{}m(x,0)= m_0(x), \quad u(x,T) = g(x) \quad &{} x \in \mathbb {T}^d,\\ \end{array} \right. \end{aligned}$$
(MFGC)
where \(u=u(x,t) \in \mathbb {R}\), \(m=m(x,t) \in \mathbb {R}\), \(v=v(x,t) \in \mathbb {R}^d\), \(P= P(t) \in \mathbb {R}^k\), with \((x,t) \in Q:=\mathbb {T}^d\times [0,T]\). The parameters \(T>0\), \(\sigma >0\) are given and
$$\begin{aligned} \begin{array}{ll} H :(x,t,p) \in Q \times \mathbb {R}^d \rightarrow \mathbb {R}, \qquad &{} \varPsi :(t,z) \in [0,T] \times \mathbb {R}^k \rightarrow \mathbb {R}^k, \\ \phi :(x,t) \in Q \rightarrow \mathbb {R}^{k \times d}, &{} f :(x,t,m) \in Q \times \mathcal {D}_1(\mathbb {T}^d)\rightarrow \mathbb {R}, \\ m_0 \in \mathcal {D}_1(\mathbb {T}^d), &{} g :x \in \mathbb {T}^d\rightarrow \mathbb {R}\end{array} \end{aligned}$$
are given data. The set \(\mathcal {D}_1(\mathbb {T}^d)\) is defined as
$$\begin{aligned} \mathcal {D}_1(\mathbb {T}^d)= \Big \{ m \in L^\infty (\mathbb {T}^d) \,|\, m \ge 0,\, \int _{\mathbb {T}^d} m(x) \; \mathrm {d} x =1 \Big \}. \end{aligned}$$
(1)
We work with \(\mathbb {Z}^d\)-periodic data and we set the state set as the d-dimensional torus \(\mathbb {T}^d\), that is a quotient set \(\mathbb {R}^d / \mathbb {Z}^d\). The Hamiltonian H is assumed to be such that \(H(x,t,p)= L^*(x,t,-p)\), for some mapping L, where \(L^*(x,t,p)\) denotes the Fenchel transform with respect to p:
$$\begin{aligned} { H(x,t,p) := \sup _{v \in \mathbb {R}^d } -\langle p,v\rangle - L(x,t,v). } \end{aligned}$$
The mapping L is assumed to be convex in its third variable.
The function u, as a solution to the Hamilton–Jacobi–Bellman (HJB) in equation (i) (MFGC) is the value function corresponding to the stochastic optimal control problem:
$$\begin{aligned} u(x,t)=&\inf _\alpha \, \mathbb {E}\Big [ \int _{t}^{T} {L(X_s ,s, \alpha _s)} + \langle \phi (X_s , s)^\intercal P (s),\alpha _s \rangle \; \mathrm {d} s \nonumber \\&+ \int _t^T f(X_s,s,m(s)) \; \mathrm {d} s + g(X_T) \Big ], \end{aligned}$$
(2)
subject to the stochastic dynamics \(\mathrm {d} X_s = \alpha _s \, \mathrm {d} s + \sqrt{2\sigma } \, \mathrm {d} B_s , \; X_t = x \in \mathbb {T}^d\). The feedback law v given by (iv) (MFGC) is then optimal for this stochastic optimal control problem. Equation (ii) (MFGC) is the Fokker–Planck equation which describes the evolution of the distribution m(t) of the agents, when the optimal feedback law is employed. At last, (iii) (MFGC) makes the quantity \(P(t)\) endogenous.

An interpretation of the system (MFGC) is as follows. Consider a stock trading market. A typical trader, with an initial level of stock \(X_0=x\), controls its level of stock \((X_t)_{ t \in [0,T]}\) through the purchasing rate \(\alpha _t\) with stochastic dynamic \(\mathrm {d} X_t = \alpha _t \mathrm {d} t+ \sqrt{2 \sigma } \mathrm {d} B_t\). The agent aims at minimizing the expected cost (2) where \(P(t)\) is the price of the stock at time t. The agent is considered to be infinitesimal and has no impact on \(P(t)\), so it assumes the price as given in its optimization problem. On the other hand, in the equilibrium configuration, the price \(P(t) \; (t \in [0,T])\) becomes endogenous and indeed, is a function of the optimal behaviour of the whole population of agents as formulated in (iii) (MFGC). The expression \(D(t) := \int _{ \mathbb {T}^d} \phi (x,t) v(x,t) m(x,t) \; \mathrm {d} x\) can be considered as a weighted net demand formulation and the relation \(P = \varPsi (D)\) is the result of supply-demand relation which determines the price of the good at the market. Concerning the role of the mapping \(\phi \), one can think for example to the case of two exchangeable goods, i.e. \(x \in \mathbb {R}^2\), with a price given by \(P(t)= \varPsi ( \int _{\mathbb {T}^d} (\phi _1(x,t)v_1(x,t)+\phi _2(x,t) v_2(x,t)) m(x,t) \, \text {d} x )\), where \(\varPsi :\mathbb {R}\rightarrow \mathbb {R}\). The use of a mapping \(\phi \), which is valued in \(\mathbb {R}^{1 \times 2}\) and whose values depend on the scale choosed for the goods, is in such a situation necessary. Thus, the system (MFGC) captures an equilibrium configuration. Similar models have been proposed in the electrical engineering literature, see for example [2, 10, 11] and the references therein.

In most mean field game models, the individual players interact through their position only, that is, via the variable m. The problem that we consider belongs to the more general class of problems, called extended mean field games, for which the players interact through the joint probability distribution \(\mu \) of states and controls. Several existence results have been obtained for such models: in [13] for stationary mean field games, in [15] for deterministic mean field games. In [6, Section 5], a class of problems where \(\mu \) enters in the drift and the integral cost of the agents is considered. We adopt the terminology mean field games of controls employed by the authors of the latter reference. Let us mention that our existence proof is different from the one of [6], which includes control bounds. In [3, Section 1], a model where the drift of the players depends on \(\mu \) is analyzed. In [14], a mean field game model is considered where at all time t, the average control (with respect to all players) is prescribed. We finally mention that extended mean field games have been studied with a probabilistic approach in [1, 8] and in [7, Section 4.6], and that a class of linear-quadratic extended mean field games has been analyzed in [20].

A difficulty in the study of mean field games of controls, directly related to the supply-demand relation mentioned above, is the fact that the control variable, at a given time t, cannot be expressed in an explicit fashion as a function of \(m(\cdot ,t)\) and \(u(\cdot ,t)\). Instead, one has to analyze the well-posedness and the stability of a fixed point equation (see for example [6, Lemma 5.2]). In our model, if we combine (iii) and (iv) (MFGC), we obtain the fixed point equation
$$\begin{aligned} v= -H_p(\nabla u + \varPsi (\smallint \phi v m )) \end{aligned}$$
(3)
for the control variable v. A central idea of the present article is the following: equation (3) is equivalent to the optimality conditions of a convex optimization problem, when L is convex and \(\varPsi \) is the gradient of a convex function \(\varPhi \). This observation allows to show the existence and uniqueness of a solution v (to equation (3)) and to investigate its dependence with respect to \(\nabla u\) and m in a natural way. More precisely, we prove that this dependence is locally Hölder continuous.

The existence of a classical solution of (MFGC) is established with the Leray–Schauder theorem and classical estimates for parabolic equations. A similar approach has been employed in [16, 17], and [18] for the analysis of a mean field game problem proposed by Chan and Sircar [9]. In this model, each agent exploits an exhaustible resource and fixes its price. The evolution of the capacity of a given producer depends on the price set by the producer, but also on the average price (with respect to all producers).

The application of the Leray–Schauder theorem relies on a priori bounds for fixed points. These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are also solutions to an optimal control problem of the Fokker–Planck equation. We are not aware of any other publication making use of such a potential formulation for a mean field game of controls, with the exception of [17] for the Chan and Sircar model. Let us mention that besides the derivation of a priori bounds, the potential formulation of the problem can be very helpful for the numerical resolution of the problem and the analysis of learning procedures (which are out of the scope of the present work).

The article is structured as follows. We list in Sect. 2 the assumptions employed all along. The main result (Theorem 1) is stated in Sect. 3. We provide in Sect. 4 a first incomplete potential formulation of the problem, incomplete in so far as the term f(m) is not integrated. We also introduce some auxiliary mappings, which allow to express P and v as functions of m and u. We give some regularity properties for these mappings in Sect. 5. In Sect. 6 we establish some a priori bounds for solutions to the coupled system. We prove our main result in Sect. 7. In Sec. 8, we give a full potential formulation of the problem, prove the uniqueness of the solution to (MFGC) and prove that (uPf(m)) is the solution to an optimal control problem of the HJB equation, under an additional monotonicity condition on f. Some parabolic estimates, used all along the article, are provided and proved in the appendix.

2 Assumptions on Data

Let us introduce the main notation used in the article. Recall that \(\mathcal {D}_1(\mathbb {T}^d)\) was defined in (1). For all \(m \in \mathcal {D}_1(\mathbb {T}^d)\), for all measurable functions \(v :\mathbb {T}^d\rightarrow \mathbb {R}^d\) such that \(|v(\cdot )|^2 m(\cdot )\) is integrable, the following inequality holds true,
$$\begin{aligned} \Big | \int _{\mathbb {T}^d} v(x) m(x) \; \mathrm {d} x \Big |^2 \le \int _{\mathbb {T}^d} |v(x)|^2 m(x) \; \mathrm {d} x, \end{aligned}$$
(4)
by the Cauchy–Schwarz inequality.

The gradient of the data functions with respect to some variable is denoted with an index, for example, \(H_p\) denotes the gradient of H with respect to p. The same notation is used for the Hessian matrix. The gradient of u with respect to x is denoted by \(\nabla u\). Let us mention that very often, the variables x and t are omitted, to alleviate the calculations. We also denote by \(\int \phi v m\) the integral \(\int _{\mathbb {T}^d} \phi v m \; \mathrm {d} x\) when used as a second argument of \(\varPsi \). For a given normed space X, the ball of center 0 and radius R is denoted B(XR).

Along the article, we use the following Hölder spaces: \(\mathcal {C}^\alpha (Q)\), \(\mathcal {C}^{2+ \alpha }(\mathbb {T}^d)\), and \(\mathcal {C}^{2+ \alpha ,1+\alpha /2}(Q)\), defined as usual with \(\alpha \in (0,1)\). Sobolev spaces are denoted by \(W^{k,p}\), the order of derivation k being possibly non-integral (see their definition in [19, section II.2]). We fix now a real number p such that
$$\begin{aligned} p > d+2. \end{aligned}$$
We will also make use of the following Banach space:
$$\begin{aligned} W^{2,1,p}(Q)= L^p(0,T;W^{2,p}(\mathbb {T}^d)) \cap W^{1,p}(Q). \end{aligned}$$

2.1 Convexity Assumptions

We collect below the required assumptions on the data. As announced in the introduction, H is related to the convex conjugate of a mapping \(L :Q \times \mathbb {R}^d \rightarrow \mathbb {R}\) as follows:
$$\begin{aligned} H(x,t,p)= L^*(x,t,-p)= \sup _{v \in \mathbb {R}^d} - \langle p,v \rangle - L(x,t,v). \end{aligned}$$
(5)
The mapping L is assumed to be strongly convex in its third variable, uniformly in x and t, that is, we assume that L is differentiable with respect to v and that there exists \(C> 0\) such that
$$\begin{aligned} \langle L_v(x,t,v_2) -L_v(x,t,v_1), v_2 -v_1 \rangle \ge \frac{1}{C} |v_2- v_1|^2, \end{aligned}$$
(A1)
for all \((x,t) \in Q\) and for all \(v_1\) and \(v_2 \in \mathbb {R}^d\). This ensures that H takes finite values and that H is continuously differentiable with respect to p, as can be easily checked. Moreover, the supremum in (5) is reached for a unique v, which is then given by \(v= -H_p(x,t,p)\), i.e.
$$\begin{aligned} H(x,t,p)+ L(x,t,v) + \langle p,v \rangle = 0 \Longleftrightarrow v= -H_p(x,t,p), \end{aligned}$$
(6)
for all \((x,t,p,v) \in Q \times \mathbb {R}^d \times \mathbb {R}^d\).
We also assume that \(\varPsi \) has a potential, that is, there exists a mapping \(\varPhi :[0,T] \times \mathbb {R}^k \rightarrow \mathbb {R}\), differentiable in its second argument, such that
$$\begin{aligned} \varPsi (t,z)= \varPhi _z(t,z), \quad \forall (t,z) \in [0,T] \times \mathbb {R}^k. \end{aligned}$$
(7)

2.2 Regularity Assumptions

We assume that \(L_v\) is differentiable with respect to x and v and that \(\phi \) is differentiable with respect to x. All along the article, we make use of the following assumptions.

2.3 Growth Assumptions

There exists \(C>0\) such that for all \((x,t) \in Q\), \(y \in \mathbb {T}^d\), \(v \in \mathbb {R}^d\), \(z \in \mathbb {R}^k\), and \(m \in \mathcal {D}_1(\mathbb {T}^d)\),
$$\begin{aligned} \bullet \quad L(x,t,v) \le C |v|^2 + C\hbox {~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{aligned}$$
(A2)
$$\begin{aligned}&\bullet \quad |L(y,t,v)-L(x,t,v)| \le C |y-x| (1 + |v|^2)\hbox {~~~~~~~~~~~~~~~~~~~~~~} \end{aligned}$$
(A3)
$$\begin{aligned}&\bullet \quad |\varPsi (t,z)| \le C |z| + C\hbox {~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{aligned}$$
(A4)
$$\begin{aligned}&\bullet \quad |f(x,t,m)| \le C.\hbox {~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{aligned}$$
(A5)

2.4 Hölder Continuity Assumptions

  • For all \(R>0\), there exists \(\alpha \in (0,1)\) such that
    $$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{l} L \in \mathcal {C}^{\alpha }(B_R), \\ L_v \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^d), \\ L_{vx} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d}), \\ L_{vv} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d}), \end{array} \end{array}\right. } \quad {\left\{ \begin{array}{ll} \begin{array}{l} \varPsi \in \mathcal {C}^{\alpha }(B_R',\mathbb {R}^d), \\ \phi \in \mathcal {C}^{\alpha }(Q,\mathbb {R}^{k \times d}), \\ D_x \phi \in \mathcal {C}^{\alpha }(Q,\mathbb {R}^{k \times d \times d}), \end{array} \end{array}\right. } \qquad \end{aligned}$$
    (A6)
    where \(B_R= Q \times B(\mathbb {R}^d,R)\) and \(B_R'= [0,T] \times B(\mathbb {R}^k,R)\).
  • There exists \(\alpha \in (0,1)\) and \(C>0\) such that
    $$\begin{aligned} \begin{aligned}&\qquad \qquad | f(x_2,t_2,m_2) - f(x_1,t_1,m_1) | \\&\qquad \qquad \qquad \le C \big ( |x_2 - x_1| + |t_2 - t_1|^\alpha + \Vert m_2 - m_1 \Vert _{L^{\infty }(\mathbb {T}^d)}^\alpha \big ), \end{aligned} \end{aligned}$$
    (A7)
    for all \((x_1,t_1)\) and \((x_2,t_2) \in Q\) and for all \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\).
  • $$\begin{aligned} \text {There exists }\alpha \in (0,1)\text { such that }m_0 \in \mathcal {C}^{2+ \alpha }(\mathbb {T}^d)\text {, }g \in \mathcal {C}^{2 +\alpha }(\mathbb {T}^d). \end{aligned}$$
    (A8)
Let us mention here that the variables \(C>0\) and \(\alpha \in (0,1)\) used all along the article are generic constants. The value of C may increase from an inequality to the next one and the value of the exponent \(\alpha \) may decrease.
Some lower bounds for L and for \(\varPhi \) can be easily deduced from the convexity assumptions. By assumption (A6), L(xt, 0) and \(L_v(x,t,0)\) are bounded. It follows then from the strong convexity assumption (A1) that there exists a constant \(C> 0\) such that
$$\begin{aligned} \frac{1}{C} |v|^2 - C \le L(x,t,v), \quad \text { for all }(x,t,v) \in Q \times \mathbb {R}^d. \end{aligned}$$
(8)
Without loss of generality, we can assume that \(\varPhi (t,0)=0\), for all \(t \in [0,T]\). Since \(\varPhi \) is convex, we have that \(\varPhi (t,z) \ge \langle \varPsi (t,0), z \rangle \), for all \(z \in \mathbb {R}^k\). We deduce then from assumption (A4) that
$$\begin{aligned} \varPhi (t,z) \ge -C | z |, \quad \text {for all }z \in \mathbb {R}^k, \end{aligned}$$
(9)
where C is independent of t and z.

Some regularity properties for the Hamiltonian can be deduced from the convexity assumption (A1) and the Hölder continuity of L and its derivatives (assumption (A6)). They are collected in the following lemma.

Lemma 1

The Hamiltonian H is differentiable with respect to p and \(H_p\) is differentiable with respect to x and p. Moreover, for all \(R>0\), there exists \(\alpha \in (0,1)\) such that \(H \in \mathcal {C}^{\alpha }(B_R)\), \(H_p \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^d)\), \(H_{px} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d})\), and \(H_{pp} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d})\)

Proof

For a given \((x,t,p) \in Q \times \mathbb {R}^d\), there exists a unique \(v:= v(x,t,p)\) maximizing the function \(v \in \mathbb {R}^d \mapsto -\langle p,v \rangle - {L(x,t,v)}\), which is strongly concave by (A1). It is then easy to deduce from (8) and the boundedness of L(xt, 0) that there exists a constant C, independent of (xtp), such that \(|v(x,t,p)| \le C( |p| +1)\). For all \((x,t,p) \in Q \times \mathbb {R}^d\), we have
$$\begin{aligned} p + L_v(x,t,v(x,t,p))= 0. \end{aligned}$$
(10)
Since \(L_v\) is continuously differentiable with respect to x and v, we obtain with the inverse mapping theorem that v(xtp) is continuously differentiable with respect to x and p. Let \(R > 0\) and let \((x_1,t_1,p_1)\) and \((x_2,t_2,p_2) \in Q \times B_R\). Let \(v_i= v(x_i,t_i,p_i)\) for \(i=1,2\). We have \(|v_i| \le C\), where C does not depend on \(x_i\), \(t_i\), and \(p_i\) (but depends on R). Moreover, we have
$$\begin{aligned}&\langle p_2 - p_1, v_2 -v_1 \rangle + \langle L_v(x_2,t_2,v_2)-L_v(x_1,t_1,v_2), v_2 - v_1 \rangle \\&\qquad + \langle L_v(x_1,t_1,v_2)-L_v(x_1,t_1,v_1), v_2- v_1 \rangle = 0. \end{aligned}$$
We deduce from (A1), Young’s inequality, and (A6) that there exists \(C>0\) and \(\alpha \in (0,1)\), both independent of \(x_i\), \(t_i\), and \(p_i\) such that
$$\begin{aligned} \frac{1}{C} |v_2 - v_1|^2&\le | \langle p_2 - p_1, v_2 -v_1 \rangle | + | \langle L_v(x_2,t_2,v_2)-L_v(x_1,t_1,v_2), v_2 - v_1 \rangle | \\&\le \frac{1}{2 \varepsilon } |p_2 - p_1|^2 + \varepsilon |v_2 - v_1|^2 + \frac{C}{\varepsilon } \big ( |x_2 - x_1|^\alpha + |t_2 - t_1|^\alpha \big ), \end{aligned}$$
for all \(\varepsilon > 0\). Taking \(\varepsilon = {1}/{2C}\), we deduce that the mapping \((x,t,p) \in B_R \mapsto v(x,t,p)\) is Hölder continuous. Since L is Hölder continuous on bounded sets, we obtain that the Hamiltonian \(H(x,t,p)= -\langle p,v(x,t,p) \rangle - L(x,t,v(x,t,p))\) is Hölder continuous on \(B_R\).
One can easily check that \(H_p(x,t,p)= -v(x,t,p)\), which proves that \(H_p\) is Hölder continuous on \(B_R\). Finally, differentiating relation (10) with respect to x and p, we obtain that
$$\begin{aligned} D_x v(x,t,p)=&- L_{vv}(x,t,v(x,t,p))^{-1} L_{vx}(x,t,v(x,t,p)) \\ D_p v(x,t,p)=&- L_{vv}(x,t,v(x,t,p))^{-1}. \end{aligned}$$
We deduce then with assumption (A6) that \(D_x v(x,t,p)\) and \(D_p v(x,t,p)\) (and thus \(H_{px}\) and \(H_{pp}\)) are Hölder continuous on \(B_R\), as was to be proved. \(\square \)
An example of coupling term We finish this section with an example of a mapping f satisfying the regularity assumptions (A5) and (A7). Let \(\varphi \in L^\infty (\mathbb {R}^d)\) be a given Lipschitz continuous mapping, with modulus \(C_1\). Let us set \(C_2= \Vert \varphi \Vert _{L^\infty (\mathbb {R}^d)}\). Let \(K :Q \times [-C_2,C_2] \rightarrow \mathbb {R}\) be a measurable mapping satisfying the following assumptions:
  1. 1.

    The mapping \(x \in \mathbb {T}^d\mapsto K(x,0,0)\) lies in \(L^1(\mathbb {T}^d)\).

     
  2. 2.
    There exist a mapping \(C_3 \in L^1(\mathbb {T}^d)\) and \(\alpha \in (0,1)\) such that for a.e. \(x \in \mathbb {T}^d\), for all \(t_1\) and \(t_2 \in [0,T]\) and for all \(w_1\) and \(w_2 \in [-C_2,C_2]\),
    $$\begin{aligned} |K(x,t_2,w_2)-K(x,t_1,w_1)| \le C_3(x) \big ( |t_2-t_1|^\alpha + |w_2 - w_1|^\alpha \big ). \end{aligned}$$
     
Let us set \(\tilde{\varphi }(x) := \varphi (-x)\). We identify \(m\in L^\infty (\mathbb {T}^d)\) with its extension by 0 over \(\mathbb {R}^d\) so that the convolution product below is well-defined:
$$\begin{aligned} m*\varphi (x) :=\int _{\mathbb {R}^d} m(x-y) \varphi (y) \; \mathrm {d} y, \;\; x \in \mathbb {T}^d. \end{aligned}$$
(11)
We keep in mind that \(m*\varphi \) is a function over \(\mathbb {T}^d\). Then
$$\begin{aligned} \Vert m*\varphi \Vert _{L^\infty (\mathbb {T}^d)} \le \Vert \varphi \Vert _{L^\infty (\mathbb {T}^d)} = C_2, \quad \text {for all }m\in \mathcal {D}_1(\mathbb {T}^d). \end{aligned}$$
(12)
In a similar way we can define
$$\begin{aligned} f_K(x,t,m)= (K(\cdot ,t,m * \varphi (\cdot )) * \tilde{\varphi })(x), \end{aligned}$$
(13)
and we have that
$$\begin{aligned}&\Vert f_K(x,t,m) \Vert _{L^\infty (\mathbb {T}^d)} \le \Vert K(\cdot ,t,m * \varphi ) \Vert _{L^1(\mathbb {T}^d)} \Vert \tilde{\varphi } \Vert _{L^\infty (\mathbb {T}^d)} \nonumber \\&\quad \le ( \Vert K(\cdot ,0,0) \Vert _{L^1(\mathbb {T}^d)} + \Vert C_3 \Vert _{L^1(\mathbb {T}^d)}(T^\alpha + \Vert {\varphi } \Vert _{L^\infty (\mathbb {T}^d)}) ) \Vert \tilde{\varphi } \Vert _{L^\infty (\mathbb {T}^d)}. \end{aligned}$$
(14)
The specific structure of \(f_K\) is actually motivated by the fact that under an additional monotonicity assumption, \(f_K\) derives from a potential (as proved in [4, Example 1.1]). For the moment, we have the following regularity result.

Lemma 2

The above mapping \(f_K\) satisfies assumptions (A5) and (A7).

Proof

Assumption (A5) follows from (14). We next prove (A7). Let \((x_1,t_1)\) and \((x_2,t_2) \in Q\), let \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\). Then
$$\begin{aligned}&|f_K(x_2,t_2,m_2)-f_K(x_1,t_2,m_2)| \\&\quad \le \Vert K(\cdot ,t_2,m_2 * \varphi (\cdot )) \Vert _{L^\infty (\mathbb {T}^d)} \Vert \varphi (x_2-\cdot )- \varphi (x_1-\cdot ) \Vert _{L^\infty (\mathbb {T}^d)} \\&\quad \le C_1 C | x_2 - x_1 |. \end{aligned}$$
Also,
$$\begin{aligned}&|f_K(x_1,t_2,m_2)-f_K(x_1,t_1,m_1)| \\&\quad \le \Vert K(\cdot ,t_2,m_2 * \varphi (\cdot ))-K(\cdot ,t_1,m_1 * \varphi (\cdot )) \Vert _{L^1(\mathbb {T}^d)} \Vert \varphi \Vert _{L^\infty (\mathbb {T}^d)}\\&\quad \le C_2 \Vert C_3 \Vert _{L^1(\mathbb {T}^d)} \big ( |t_2-t_1|^\alpha + \Vert (m_2 - m_1) * \varphi \Vert _{L^\infty (\mathbb {T}^d)}^\alpha \big ). \end{aligned}$$
Finally, we have \(\Vert (m_2 - m_1) * \varphi \Vert _{L^\infty (\mathbb {T}^d)} \le \Vert m_2 - m_1 \Vert _{L^\infty (\mathbb {T}^d)} \Vert \varphi \Vert _{L^\infty (\mathbb {T}^d)}\) and thus, assumption (A7) follows. \(\square \)

3 Main Result and General Approach

Theorem 1

There exists \(\alpha \in (0,1)\) such that (MFGC) has a classical solution (umvP), with
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rl} m \in &{} \mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q), \\ u \in &{} \mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q), \\ P \in &{} \mathcal {C}^\alpha (0,T;\mathbb {R}^k), \\ v \in &{} \mathcal {C}^{\alpha }(Q,\mathbb {R}^d), \ D_x v \in \mathcal {C}^{\alpha }(Q,\mathbb {R}^{d \times d}). \end{array} \end{array}\right. } \end{aligned}$$
(15)

The result is obtained with the Leray–Schauder theorem, recalled below.

Theorem 2

(Leray–Schauder) Let X be a Banach space and let \(\mathcal {T} :X \times [0,1] \rightarrow X\) be a continuous and compact mapping. Let \(x_0 \in X\). Assume that \(\mathcal {T}(x,0)=x_0\) for all \(x \in X\) and assume there exists \(C>0\) such that \(\Vert x \Vert _X < C\) for all \((x,\tau ) \in X \times [0,1]\) such that \(\mathcal {T} (x,\tau ) = x\). Then, there exists \(x \in X\) such that \(\mathcal {T}(x,1) = x\).

A proof of the theorem can be found in [12, Theorem 11.6], for \(x_0=0\). The extension to a general value of \(x_0\) can be easily obtained with a translation argument that we do not detail. The application of the Leray–Schauder theorem and the construction of \(\mathcal {T}\) will be detailed in Sect. 7. Let us mention that the set of fixed points of \(\mathcal {T}(\cdot ,\tau )\), for \(\tau \in [0,1]\), will coincide with the set of solutions of the following parametrization of (MFGC):
Of course, (MFGC\(_\tau \)) corresponds to (MFGC) for \(\tau = 1\). Let us introduce the spaces X and \(X'\), used for the formulation of the fixed-point equation:
$$\begin{aligned} X := \big ( W^{2,1,p}(Q) \big )^2, \quad X' := X \times {L^\infty (Q,\mathbb {R}^d) \times L^\infty (0,T;\mathbb {R}^k)}. \end{aligned}$$
The HJB equation (i) and the Fokker–Planck equation (ii) are classically understood in the viscosity and weak sense, respectively. However, due to the choice of the solution spaces, we may interpret these equations as equalities in \(L^p(Q)\): in particular, if \(u \in W^{2,1,p}(Q)\) and \(P \in L^\infty (0,T;\mathbb {R}^k)\), we have that \(\nabla u \in L^\infty (Q;\mathbb {R}^d)\) (by Lemma 12), and thus \(H(\nabla u + \phi ^\intercal P(t)) \in L^\infty (Q)\). A first and important step of our analysis is the construction of auxiliary mappings allowing to express v and P as functions of m and u. These mappings cannot be obtained in a straightforward way, since in (iii), P depends on v and in (iv), v depends on P.

Lemma 3

Let \(\tau \in [0,1]\), let \((m,v) \in W^{2,1,p}(Q) \times L^\infty (Q,\mathbb {R}^d)\) be a weak solution to the Fokker–Planck equation \(\partial _t m - \sigma \varDelta m + \tau \text {div }(vm)= 0\), \(m(\cdot ,0)= m_0(\cdot )\). Then \(m \ge 0\) and for all \(t \in [0,T]\), \(\int _{\mathbb {T}^d} m(x,t) \; \mathrm {d} x = 1\).

Proof

Multiply (MFGC\(_\tau \))(ii) by \(\mu (x,t):=\min (0,m(x,t))\). Use \(\nabla \mu (x,t)=\mathbf{1}_{\{m(x,t)<0\}} \nabla m(x,t)\), so that integrating (by parts) over \(Q_t:=\mathbb {T}^d\times (0,t)\), since v is essentially bounded, we get that
$$\begin{aligned}&\frac{1}{2} \int _{\mathbb {T}^d} \mu (x,t)^2 \; \mathrm {d} x + \sigma \iint _{Q_t} | \nabla \mu (x,s)|^2 \; \mathrm {d} x \; \mathrm {d} s = \tau \iint _{Q_t} \langle v, \nabla \mu \rangle m \; \mathrm {d} x \; \mathrm {d} s \\&\quad = \tau \iint _{Q_t} \langle v,\nabla \mu \rangle \mu \; \mathrm {d} x \; \mathrm {d} s \le C \iint _{Q_t} |\mu |^2 \; \mathrm {d} x \; \mathrm {d} s + \sigma \iint _{Q_t} | \nabla \mu |^2 \; \mathrm {d} x \; \mathrm {d} s, \end{aligned}$$
so that after cancellation of the contribution of \(\nabla \mu \), we obtain, applying Gronwall’s lemma to \(a(t):=\int _{\mathbb {T}^d} \mu (x,t)^2\), that \(a(t)=0\) for all t which means that m is non-negative. Moreover, for all \(t \in [0,T]\),
$$\begin{aligned} \int _{\mathbb {T}^d} m(x,t) \; \mathrm {d} x = \int _{\mathbb {T}^d} m(x,0) \; \mathrm {d} x + \iint _{Q_t} \sigma {\varDelta m} - \tau \text {div}(v m) \; \mathrm {d} x \; \mathrm {d} s. \end{aligned}$$
Integrating by parts the double integral we see that it is equal to 0, and we conclude by noting that \(\int _{\mathbb {T}^d} m(x,0) \; \mathrm {d} x = \int _{\mathbb {T}^d} m_0(x) \; \mathrm {d} x = 1\). \(\square \)

4 Potential Formulation

In this section, we first establish a potential formulation of the mean field game problem (MFGC\(_\tau \)), that is to say, we prove that for \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)), \((m_\tau ,v_\tau )\) is a solution to an optimal control problem. We prove then that for all t, \(v_\tau (\cdot ,t)\) is the unique solution of some optimization problem, which will enable us to construct the announced auxiliary mappings.

Let us introduce the cost functional \(B :W^{2,1,p}(Q) \times L^\infty (Q,\mathbb {R}^d) \times L^\infty (Q) \rightarrow \mathbb {R}\), defined by
$$\begin{aligned}&B (m,v;\tilde{f}) = \iint _Q \big ( L ( x,t, v(x,t)) + \tilde{f}(x,t) \big ) m(x,t) \; \mathrm {d} x \; \mathrm {d} t \nonumber \\&\qquad + \int _0^T \varPhi \Big (t, \int _{\mathbb {T}^d} \phi (x,t) v(x,t) m(x,t) \; \mathrm {d} x \Big ) \, \mathrm {d} t + \int _{\mathbb {T}^d} g(x) m(x,T) \; \mathrm {d} x. \end{aligned}$$
(16)
We have the following result.

Lemma 4

For all \(\tau \in [0,1]\) and \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)), the pair \((m_\tau ,v_\tau )\) is the solution to the following optimization problem:
$$\begin{aligned} \min _{\begin{array}{c} m \in W^{2,1,p}(Q) \\ v \in L^\infty (Q,\mathbb {R}^d) \end{array}} \ B(m,v;\tilde{f}_\tau ), \ \text {s.t.: } {\left\{ \begin{array}{ll} \begin{array}{rl} \partial _t m - \sigma \varDelta m + \tau \mathrm {div} (vm) &{}= 0, \\ m(x,0) &{}= m_0(x), \end{array} \end{array}\right. } \end{aligned}$$
(17)
where \(\tilde{f}_\tau (x,t)= f(x,t,m_\tau (t))\).

Remark 1

Let us emphasize that the above optimal control problem is only an incomplete potential formulation, since the term \(\tilde{f}_\tau \) still depends on \(m_\tau \).

Proof

(Lemma 4) Let us consider the case where \(\tau \in (0,1]\). Let \((m,v) \in W^{2,1,p}(Q) \times L^\infty (Q,\mathbb {R}^d)\) be a feasible pair, i.e., it satisfies the constraint in (17). For all \((x,t) \in Q\), we have \(v_\tau = - H_p(\nabla u_\tau + \phi ^\intercal P_\tau )\). Therefore, by (5) and (6), we have that
$$\begin{aligned} L (v)&\ge - H(\nabla u_\tau + \phi ^\intercal P_\tau ) - \left\langle \nabla u_\tau + \phi ^\intercal P_\tau , v \right\rangle , \\ L (v_\tau )&= - H(\nabla u_\tau + \phi ^\intercal P_\tau ) - \left\langle \nabla u_\tau + \phi ^\intercal P_\tau , v_\tau \right\rangle , \end{aligned}$$
for all \((x,t) \in Q\). Moreover, by Lemma 3, \(m \ge 0\) and \(m_\tau \ge 0\). Therefore,
$$\begin{aligned}&L (v) m - L (v_\tau ) m_\tau \nonumber \\&\quad \ge - H(\nabla u_\tau + \phi ^\intercal P_\tau ) (m - m_\tau ) -\left\langle \nabla u_\tau + \phi ^\intercal P_\tau , vm - v_\tau m_\tau \right\rangle . \end{aligned}$$
(18)
Using (i) (MFGC\(_\tau \)), we obtain
$$\begin{aligned}&L (v) m - L (v_\tau ) m_\tau \\&\quad \ge \frac{1}{\tau }(-\partial _t u_\tau -\sigma \varDelta u_\tau - \tau \tilde{f}_\tau ) ( m - m_\tau ) - \left\langle \nabla u_\tau + \phi ^\intercal P_\tau , v m - v_\tau m_\tau \right\rangle . \end{aligned}$$
After integration with respect to x, we obtain that for all t,
$$\begin{aligned}&\int _{\mathbb {T}^d} (L (v) m - L (v_\tau ) m_\tau ) + \tilde{f}_\tau (m-m_\tau ) \; \mathrm {d} x \nonumber \\&\quad \ge \frac{1}{\tau } \int _{\mathbb {T}^d} (-\partial _t u_\tau -\sigma \varDelta u_\tau ) ( m - m_\tau ) \; \mathrm {d} x - \int _{\mathbb {T}^d} \left\langle \nabla u_\tau , v m - v_\tau m_\tau \right\rangle \; \mathrm {d} x \\&\qquad - \langle P_\tau , {\textstyle \int }\phi (v m - v_\tau m_\tau ) \rangle . \end{aligned}$$
We obtain with the convexity of \(\varPhi \) and (iii) (MFGC\(_\tau \)) that
$$\begin{aligned} \varPhi ({\textstyle \int }\phi m v) - \varPhi ({\textstyle \int }\phi v_\tau m_\tau )&\ge \langle \varPsi ({\textstyle \int }\phi m_\tau v_\tau ), {\textstyle \int }\phi (v m - v_\tau m_\tau ) \rangle \nonumber \\&= \langle P_\tau , {\textstyle \int }\phi (m v - m_\tau v_\tau ) \rangle . \end{aligned}$$
(19)
Using the previous calculations to bound \(B(m,v;\tilde{f}_\tau )-B(m_\tau ,v_\tau ;\tilde{f}_\tau )\) from below, we observe that the term \(\langle P_\tau , {\textstyle \int }\phi (m - m_\tau v_\tau ) \rangle \) cancels out and obtain
$$\begin{aligned}&B (m,v;\tilde{f}_\tau ) - B (m_\tau ,v_\tau ;\tilde{f}_\tau ) \\&\qquad \ge \iint _Q \frac{1}{\tau }(-\partial _t u_\tau -\sigma \varDelta u_\tau ) {(m-m_\tau )} - \left\langle \nabla u_\tau , m v - m_\tau v_\tau \right\rangle \; \mathrm {d} x \; \mathrm {d} t \\&\qquad \qquad + \int _{\mathbb {T}^d} g(x) (m(x,T) - m_\tau (x,T) ) \; \mathrm {d} x. \end{aligned}$$
Integrating by parts and using (ii) (MFGC\(_\tau \)), we finally obtain that
$$\begin{aligned} B (m,v;\tilde{f}_\tau ) - B (m_\tau ,v_\tau ;\tilde{f}_\tau ) \ge \frac{1}{\tau } \int _{\mathbb {T}^d} u_\tau (0,x)( m(0,x)-m_\tau (0,x) )\; \mathrm {d} x = 0, \end{aligned}$$
as was to be proved. We do not detail the proof for the case \(\tau =0\), which is actually simpler. Indeed, for \(\tau =0\), the solution to the Fokker–Planck equation is independent of v and thus \(m= m_\tau \) in the above calculations. \(\square \)

We have proved that the pair \((m_\tau ,v_\tau )\) is the solution to an optimal control problem. Therefore, for all t, \(v_\tau (\cdot ,t)\) minimizes the Hamiltonian associated with problem (17). Let us introduce some notation, in order to exploit this property. For \(m \in \mathcal {D}_1(\mathbb {T}^d)\), we denote by \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\) the Hilbert space of measurable mappings \(v :\mathbb {T}^d\rightarrow \mathbb {R}^d\) such that \(\int _{\mathbb {T}^d} |v|^2 m < \infty \), equipped with the scalar product \(\int _{\mathbb {T}^d} \langle v_1,v_2 \rangle m\). An element of \(L_m^2(\mathbb {T}^d)\) is an equivalent class of functions equal m-almost everywhere. Note that \(L^\infty (\mathbb {T}^d) \subset L_m^2(\mathbb {T}^d)\).

For \(t \in [0,T]\), \(m \in \mathcal {D}_1(\mathbb {T}^d)\), and \(w \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\), we consider the mapping
$$\begin{aligned} v \in L_m^2(\mathbb {T}^d, \mathbb {R}^d) \mapsto J(v;t,m,w):= \varPhi \big ( t, {\textstyle \int }\phi v m \big ) + \int _{\mathbb {T}^d} \big ( L(v)+ \langle w,v \rangle \big ) m \; \mathrm {d} x. \end{aligned}$$
Combining inequalities (18) and (19) (with \(m=m_\tau \)), we directly obtain that for all \(t \in [0,T]\), for all \(v \in L_{m}^2(\mathbb {T}^d,\mathbb {R}^d)\) with \(m=m_\tau (\cdot ,t)\),
$$\begin{aligned} J \big ( v;t,m_\tau (t),\nabla u_\tau (t) \big ) \ge J \big ( v_\tau (t);t,m_\tau (t),\nabla u_\tau (t) \big ). \end{aligned}$$
The following lemma will enable us to express \(P_\tau (t)\) and \(v_\tau (\cdot ,t)\) as functions of \(m_\tau (\cdot ,t)\) and \(u_\tau (\cdot ,t)\). The key idea is, roughly speaking, to prove the existence and uniqueness of a minimizer to \(J(\cdot ;t,m,w)\).

Lemma 5

For all \(t \in [0,T]\), for all \(m \in \mathcal {D}_1(\mathbb {T}^d)\), for all \(R > 0\), and for all \(w \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\) such that \(\Vert w \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le R\), there exists a unique pair \((v,P) \in L^\infty (\mathbb {T}^d,\mathbb {R}^d) \times \mathbb {R}^k\), such that
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rl} v(x) &{}= - H_p (x,t, w(x) + \phi (x,t)^\intercal P), \quad \text {for a.e. }x \in \mathbb {T}^d, \\ P &{}= \varPsi (t, {\textstyle \int }\phi v m). \end{array} \end{array}\right. } \end{aligned}$$
(20)
The pair (vP) is then denoted \((\mathbf {v}(t,m,w),\mathbf {P}(t,m,w))\). Moreover, we have
$$\begin{aligned} \Vert \mathbf {v}(t,m,w) \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le C \quad \text {and} \quad |\mathbf {P}(t,m,w)| \le C, \end{aligned}$$
(21)
where the constant C is independent of t, m, and w (but depends on R).

Proof

If the pair (vP) satifies (20), then
$$\begin{aligned} v= - H_p ( w + \phi ^\intercal \varPsi ({\textstyle \int }\phi v m) ) \quad \text {a.e. on }\mathbb {T}^d. \end{aligned}$$
(22)
One can easily check that for proving the existence and uniqueness of a pair (vP) satisfying (20), it is sufficient to prove the existence and uniqueness of \(v \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\) satisfying (22). For future reference, let us observe that by (6), relation (22) is equivalent to
$$\begin{aligned} \phi ^\intercal (x,t) \varPsi (t,{\textstyle \int }\phi v m) + L_v(x,t,v(x)) + w(x)= 0, \quad \text {for a.e. }x \in \mathbb {T}^d. \end{aligned}$$
(23)
Step 1 existence and uniqueness of a minimizer of \(J(\cdot ;t,m,w)\).
In view of (A1), \(v \mapsto \int _{\mathbb {T}^d} L(v)m \; \mathrm {d} x\) is strongly convex over \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\). Since the sum of a l.s.c. convex function and of a l.s.c. strongly convex function is l.s.c. and strongly convex, so is the function \(J(\cdot ;t,m,w)\). Thus, it possesses a unique minimizer \(\bar{v}\) in \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\). We obtain
$$\begin{aligned} C\Vert \bar{v}\Vert ^2_{L_m^2(\mathbb {T}^d,\mathbb {R}^d)} - C \le J(\bar{v};t,m,w)) \le J(0;t,m,w)) =C, \end{aligned}$$
(24)
so that \(\Vert \bar{v}\Vert ^2_{L_m^2(\mathbb {T}^d,\mathbb {R}^d)} \le C\), with C independent of t, m, and w, but depending on R, as all constants C used in the proof.

Step 2 existence of \(\mathbf {v}(t,m,w)\) and a priori bound.

One can check that the mapping \(\delta v \in L^\infty (\mathbb {T}^d,\mathbb {R}^d) \mapsto J(\bar{v}+ \delta v;t,m,w)\) is differentiable. Since \(\bar{v}\) is optimal, the derivative of the above mapping is null at \(\delta v=0\) and thus
$$\begin{aligned} \big [ \phi ^\intercal (x,t) \varPsi (t,{\textstyle \int }\phi (x')\bar{v}(x')m(x')\, \mathrm {d} x') + L_v(x,t,\bar{v}(x)) + w(x) \big ] m(x)= 0, \end{aligned}$$
for a.e. \(x \in \mathbb {T}^d\). Using then the equivalence of (22) and (23), we obtain that
$$\begin{aligned} m(x) > 0 \Longrightarrow \bar{v}(x)= - H_p \big (x,t,w(x) + \phi ^\intercal (x,t) \varPsi (t, {\textstyle \int }\phi (x',t) \bar{v}(x')m(x') \, \mathrm {d} x') \big ), \end{aligned}$$
for a.e. \(x \in \mathbb {T}^d\). Consider now the measurable function v defined by
$$\begin{aligned} v(x)= - H_p \big (x,t,w(x) + \phi ^\intercal (x,t) \varPsi (t,{\textstyle \int }\phi (x',t) \bar{v}(x') m(x')\, \mathrm {d} x'\big ), \end{aligned}$$
for a.e. \(x \in \mathbb {T}^d\). The two functions v and \(\bar{v}\) may not be equal for a.e. x if \(m(x)=0\) on a subset of \(\mathbb {T}^d\) of non-zero measure. Still they are equal in \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\), which ensures in particular that \({\textstyle \int }\phi (x',t) \bar{v}(x') m(x') \, \mathrm {d} x' = {\textstyle \int }\phi (x',t) {v}(x') m(x') \, \mathrm {d} x'\) and finally that v satisfies (22) and lies in \(L^\infty (\mathbb {T}^d,\mathbb {R}^d)\), as a consequence of the continuity of \(H_p\) (proved in Lemma 1). We also have that \(\Vert \bar{v} \Vert _{L_m^2(\mathbb {T}^d,\mathbb {R}^d)}= \Vert v \Vert _{L_m^2(\mathbb {T}^d,\mathbb {R}^d)} \le C\), by (24). Using the Cauchy–Schwarz inequality and assumption (A6), we obtain that \(|{\textstyle \int }\phi v m| \le C\). We obtain then with assumption (A4) that for \(P= \varPsi ( {\textstyle \int }\phi v m)\), we have \(|P| \le C\). Using assumption (A6) and the continuity of \(H_p\), we finally obtain that \(\Vert v \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le C\). Thus the bound (21) is satisfied.

Step 3 uniqueness of \(\mathbf {v}(t,m,w)\).

Let \(v_1\) and \(v_2 \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\) satisfy (22). Then \(DJ(v_i;t,m,w)= 0\), proving that \(v_1\) and \(v_2\) are minimizers of \(J(\cdot ;t,m,w)\) and thus are equal in \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\). Therefore \({\textstyle \int }\phi (x',t) v_1(x') m(x') \; \mathrm {d} x' = {\textstyle \int }\phi (x',t) v_2(x') m(x') \; \mathrm {d} x'\) and finally that \(v_1= v_2\), by (22). \(\square \)

5 Regularity Results for the Auxiliary Mappings

We provide in this section some regularity results for the mappings \(\mathbf {v}\) and \(\mathbf {P}\). We begin by proving that \(\mathbf {P}(\cdot ,\cdot ,\cdot )\) is locally Hölder continuous. For this purpose, we perform a stability analysis of the optimality condition (23).

Lemma 6

Let \(t_1\) and \(t_2 \in [0,T]\), let \(w_1\) and \(w_2 \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\), let \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\). Let \(R>0\) be such that \(\Vert w_i \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le R\), for \(i=1,2\). Then, there exist a constant \(C>0\) and an exponent \(\alpha \in (0,1)\), both independent of \(t_1\), \(t_2\), \(w_1\), \(w_2\), \(m_1\), and \(m_2\) but depending on R, such that
$$\begin{aligned}&| \mathbf {P}(t_2,m_2,w_2)-\mathbf {P}(t_1,m_1,w_1) | \nonumber \\&\quad \le C \big ( |t_2-t_1|^{\alpha } + \Vert w_2 - w_1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)}^{\alpha } + \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d)}^{\alpha } \big ). \end{aligned}$$
(25)

Proof

Note that all constants \(C>0\) and all exponents \(\alpha \in (0,1)\) involved below are independent of \(t_1\), \(t_2\), \(w_1\), \(w_2\), \(m_1\), and \(m_2\). They are also independent of \(x \in \mathbb {T}^d\) and \(\varepsilon > 0\). For \(i=1,2\), we set \(v_i= \mathbf {v}(t_i,m_i,w_i)\) and \(\phi _i= \phi (\cdot ,t_i) \in L^\infty (\mathbb {T}^d)\). By (21), we have
$$\begin{aligned} \Vert v_i \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le C. \end{aligned}$$
(26)
By the optimality condition (23), we have that
$$\begin{aligned} \phi _i^\intercal \varPsi \big ( t_i, {\textstyle \int }\phi _i v_i m_i \big ) + L_v (t_i,v_i) + w_i= 0, \quad \text {for a.e. }x \in \mathbb {T}^d. \end{aligned}$$
(27)
Consider the difference of (27) for \(i=2\) with (27) for \(i=1\). Integrating with respect to x the scalar product of the obtained difference with \(v_2m_2 - v_1m_1\), we obtain that \((a_1) + (a_2) + (a_3)= 0\), where
$$\begin{aligned} (a_1)=&\int _{\mathbb {T}^d} \big \langle \phi _2^\intercal \varPsi (t_2, {\textstyle \int }\phi _2 v_2 m_2) - \phi _1^\intercal \varPsi (t_1, {\textstyle \int }\phi _1 v_1 m_1) ,v_2 m_2 - v_1 m_1 \big \rangle \; \mathrm {d} x, \\ (a_2)=&\int _{\mathbb {T}^d} \langle L_v(t_2,v_2) - L_v(t_1,v_1), v_2 m_2 - v_1 m_1 \rangle \; \mathrm {d} x, \\ (a_3)=&\int _{\mathbb {T}^d} \langle w_2 - w_1, v_2 m_2 - v_1 m_1 \rangle \; \mathrm {d} x. \end{aligned}$$
We look for a lower estimate of these three terms. Let us mention that the term \(v_2 m_2 - v_1 m_1\), appearing in the three terms, will be estimated only at the end.
Estimation from below of\((a_1)\). We have \((a_1)= (a_{11}) + (a_{12})\), where
$$\begin{aligned} (a_{11})=&\int _{\mathbb {T}^d} \big \langle \phi _2^\intercal \varPsi (t_2,{\textstyle \int }\phi _2 v_2 m_2) - \phi _1^\intercal \varPsi (t_1, {\textstyle \int }\phi _1 v_2 m_2), v_2 m_2 - v_1 m_1 \rangle \; \mathrm {d} x \\ (a_{12})=&\int _{\mathbb {T}^d} \big \langle \phi _1^\intercal \varPsi (t_1, {\textstyle \int }\phi _1 v_2 m_2) - \phi _1^\intercal \varPsi (t_1, {\textstyle \int }\phi _1 v_1 m_1 ), v_2 m_2 - v_1 m_1 \big \rangle \; \mathrm {d} x. \end{aligned}$$
By monotonicity of \(\varPsi \), we have that
$$\begin{aligned} (a_{12})= \Big \langle \varPsi (t_1, {\textstyle \int }\phi _1 v_2 m_2) - \varPsi (t_1, {\textstyle \int }\phi _1 v_1 m_1 ), \int _{\mathbb {T}^d} \phi _1 v_2 m_2 - \phi _1 v_1 m_1 \; \mathrm {d} x \Big \rangle \ge 0. \end{aligned}$$
Let us consider \((a_{11})\). We set
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rl} \varPsi _i &{}= \varPsi (t_i, {\textstyle \int }\phi _i v_2 m_2), \quad \text {for }i=1,2, \\ \xi (x) &{}= \phi _2(x)^\intercal \varPsi _2 - \phi _1(x)^\intercal \varPsi _1, \end{array} \end{array}\right. } \end{aligned}$$
so that \((a_{11})= \int _{\mathbb {T}^d} \langle \xi , v_2 m_2 - v_1 m_1 \rangle \, \mathrm {d} x\). Using assumption (A6), one can check that \(|\varPsi _i | \le C\) and that \(\big | \varPsi _2 - \varPsi _1 \big | \le C|t_2- t_1|^\alpha \). Since \(\xi = (\phi _2-\phi _1)^\intercal \varPsi _2 + \phi _1^\intercal (\varPsi _2-\varPsi _1)\), we obtain with assumption (A6) again that
$$\begin{aligned} \Vert \xi \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \le C \big ( |\varPsi _2-\varPsi _1| + \Vert \phi _2- \phi _1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^{k \times d})} \big ) \le C |t_2-t_1|^\alpha \end{aligned}$$
and further with Young’s inequality that
$$\begin{aligned} |(a_{11})| \le \frac{C}{\varepsilon } |t_2-t_1|^\alpha + \frac{\varepsilon }{2} \Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)}^2. \end{aligned}$$
Estimation from below of\((a_2)\). We have \((a_2)= (a_{21}) + (a_{22}) + (a_{23})\), where
$$\begin{aligned} (a_{21})=&\int _{\mathbb {T}^d} \langle L_v(t_2,v_2)- L_v(t_1,v_2), v_2 m_2 - v_1 m_1 \rangle \; \mathrm {d} x \\ (a_{22})=&\int _{\mathbb {T}^d} \langle L_v(t_1,v_2)- L_v(t_1,v_1), v_2(m_2-m_1) \rangle \; \mathrm {d} x \\ (a_{23}) =&\int _{\mathbb {T}^d} \langle L_v(t_1,v_2)- L_v(t_1,v_1), (v_2-v_1)m_1 \rangle \; \mathrm {d} x. \end{aligned}$$
As a consequence of (26), assumption (A6), and Young’s inequality, we have
$$\begin{aligned} |(a_{21})|&\le \frac{1}{2 \varepsilon } \Vert L_v(t_2,v_2(\cdot ))-L_v(t_1,v_2(\cdot )) \Vert _{L^\infty (\mathbb {T}^d)}^2 + \frac{\varepsilon }{2} \Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)}^2 \\ \le \,&\frac{C}{\varepsilon } |t_2 -t_1 |^\alpha + \frac{\varepsilon }{2} \Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)}^2. \end{aligned}$$
By (26) and assumption (A6), \(| L_v(t_1,x,v_i(x)) | \le C\), therefore
$$\begin{aligned} |(a_{22})| \le C \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)}. \end{aligned}$$
Finally, since \(m_1 \ge 0\) and by assumption (A1), we have
$$\begin{aligned} (a_{23}) \ge \frac{1}{C} \int _{\mathbb {T}^d} |v_2-v_1|^2 m_1 \; \mathrm {d} x. \end{aligned}$$
Estimation from below of\((a_3)\). Using (29) and Young’s inequality, we obtain that
$$\begin{aligned} |(a_{3})| \le \frac{1}{2\varepsilon } \Vert w_2-w_1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)}^2 + \frac{\varepsilon }{2} \Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)}^2. \end{aligned}$$
Conclusion. We have proved that
$$\begin{aligned}&\frac{1}{C} \int _{\mathbb {T}^d} |v_2 -v_1|^2 m_1 \; \mathrm {d} x \le (a_{23}) = (a_2) - (a_{21}) - (a_{22}) \nonumber \\&\quad = -(a_{1}) - (a_{21}) - (a_{22}) - (a_3) \nonumber \\&\quad \le -(a_{11}) - (a_{21}) - (a_{22}) - (a_3) \nonumber \\&\quad \le \frac{C}{\varepsilon } |t_2 -t_1 |^\alpha + \frac{1}{2\varepsilon } \Vert w_2 - w_1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)}^2 \nonumber \\&\qquad + C \Vert m_2- m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)} + \frac{3}{2} \varepsilon \Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d;\mathbb {R}^d)}^2. \end{aligned}$$
(28)
Let us estimate \(\Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d;\mathbb {R}^d)}\). Using the Cauchy–Schwarz inequality, we obtain that
$$\begin{aligned}&\Vert v_2 m_2 - v_1 m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)} \le \Vert v_2(m_2-m_1) \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)} + \Vert (v_2-v_1) m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)} \nonumber \\&\quad \le C \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d,\mathbb {R}^d)} + \Big ( \int _{\mathbb {T}^d} |v_2- v_1 |^2 m_1 \; \mathrm {d} x \Big )^{1/2}. \end{aligned}$$
(29)
Injecting this inequality in (28) and taking \(\varepsilon = {1}/{3C}\), we obtain that
$$\begin{aligned} \int _{\mathbb {T}^d} |v_2 -v_1|^2 m_1 \le C \Big ( |t_2 - t_1|^\alpha + \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d)} + \Vert w_2 - w_1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)}^2 \Big ). \end{aligned}$$
(30)
Let us prove (25). We have
$$\begin{aligned}&\int _{\mathbb {T}^d} \phi _2 v_2 m_2 \; \mathrm {d} x - \int _{\mathbb {T}^d} \phi _1 v_1 m_1 \; \mathrm {d} x = \int _{\mathbb {T}^d} (\phi _2-\phi _1) v_2 m_2 \; \mathrm {d} x \\&\qquad + \int _{\mathbb {T}^d} \phi _1 v_2 (m_2-m_1) \; \mathrm {d} x + \int _{\mathbb {T}^d} \phi _1 (v_2-v_1) m_1 \; \mathrm {d} x. \end{aligned}$$
Therefore, using assumption (A6) and (30), we obtain that
$$\begin{aligned}&\Big | \int _{\mathbb {T}^d} \phi _2 v_2 m_2 \; \mathrm {d} x - \int _{\mathbb {T}^d} \phi _1 v_1 m_1 \; \mathrm {d} x \Big | \\&\quad \le C \Big ( \Vert \phi _2-\phi _1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^{k\times d})} + \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d)} + \Big ( \int _{\mathbb {T}^d} |v_2-v_1|^2 m_1 \Big )^{1/2} \Big ) \\&\quad \le C \Big ( |t_2-t_1|^{\alpha } + \Vert m_2 - m_1 \Vert _{L^1(\mathbb {T}^d)}^{1/2} + \Vert w_2 - w_1 \Vert _{L^\infty (\mathbb {T}^d,\mathbb {R}^d)} \Big ). \end{aligned}$$
Inequality (25) follows, using assumption (A6). The lemma is proved. \(\square \)
Given \(m \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\) and \(w \in L^\infty (Q)\), we consider the Nemytskii operators associated with \(\mathbf {v}\) and \(\mathbf {P}\), that we still denote by \(\mathbf {v}\) and \(\mathbf {P}\) without risk of confusion:
$$\begin{aligned}&\mathbf {v}(m,w) \in L^\infty (Q,\mathbb {R}^d),&\mathbf {v}(m,w)(x,t) = \mathbf {v}(t,m(\cdot ,t),w(\cdot ,t))(x), \\&\mathbf {P}(m,w) \in L^\infty (0,T;\mathbb {R}^k),&\mathbf {P}(m,w)(t) = \mathbf {P}(t,m(\cdot ,t),w(\cdot ,t)), \end{aligned}$$
for all \((x,t) \in Q\). We use now Lemma 6 to prove regularity properties of the Nemytskii operators \(\mathbf {v}\) and \(\mathbf {P}\). We recall that \(X = \big ( W^{2,1,p}(Q) \big )^2\).

Lemma 7

For all \(R>0\), the mapping
$$\begin{aligned}&(m,w) \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d)) \times B \big ( L^\infty (Q,\mathbb {R}^d),R) \nonumber \\&\qquad \qquad \mapsto \mathbf {P}(m,w) \in L^\infty (0,T;\mathbb {R}^k) \end{aligned}$$
(31)
and the mapping
$$\begin{aligned}&(u,m) \in B(W^{2,1,p}(Q),R) \times L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d)) \nonumber \\&\qquad \qquad \mapsto \mathbf {v}(m,\nabla u) \in L^\infty (Q,\mathbb {R}^d) \cap L^p(0,T;W^{1,p}(\mathbb {T}^d)) \end{aligned}$$
(32)
are both Hölder continuous, that is, there exist \(\alpha \in (0,1)\) and \(C>0\) such that
$$\begin{aligned}&\Vert \mathbf {P}(m_2,w_2)-\mathbf {P}(m_1,w_1) \Vert _{L^\infty (0,T;\mathbb {R}^k)} \\&\quad \le C \big ( \Vert m_2 - m_1 \Vert _{L^\infty (Q)}^\alpha + \Vert w_2 - w_1 \Vert _{L^\infty (Q)}^\alpha \big ),\\&\Vert \mathbf {v}(m_2,\nabla u_2)-\mathbf {v}(m_1,\nabla u_1) \Vert _{L^\infty (Q,\mathbb {R}^d) \cap L^p(0,T;W^{1,p}(\mathbb {T}^d))} \\&\quad \le C \big ( \Vert u_2 - u_1 \Vert _{W^{2,1,p}(Q)}^\alpha + \Vert m_2 - m_1 \Vert _{L^\infty (Q)}^\alpha \big ), \end{aligned}$$
for all \(m_1\) and \(m_2 \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), for all \(w_1\) and \(w_2 \in B(L^\infty (Q,\mathbb {R}^d),R)\), and for all \(u_1\) and \(u_2\) in \(B(W^{2,1,p}(Q),R)\).

Proof

The Hölder continuity of the first mapping is a direct consequence of Lemma 6. As a consequence, the mapping
$$\begin{aligned}&(u,m) \in B(W^{2,1,p}(Q),R) \times L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d)) \\&\qquad \qquad \qquad \mapsto \nabla u + \phi ^\intercal \mathbf {P}(m,\nabla u) \in L^\infty (Q,\mathbb {R}^d) \end{aligned}$$
is Hölder continuous. Using then the relations
$$\begin{aligned} \begin{array}{rl} \mathbf {v}(m,\nabla u) &{}= - H_p(\nabla u + \phi ^\intercal \mathbf {P}(m,\nabla u)), \\ D_x \mathbf {v}(m,\nabla u) &{}= - H_{px}(\nabla u + \phi ^\intercal \mathbf {P}(m,\nabla u)) \\ &{} \quad - H_{pp}(\nabla u + \phi ^\intercal \mathbf {P}(m,\nabla u))(\nabla ^2 u + D\phi ^\intercal \mathbf {P}(m,\nabla u)), \end{array} \end{aligned}$$
(33)
and the Hölder continuity of \(H_p\), \(H_{px}\), and \(H_{pp}\) on bounded sets (Lemma 1), we obtain that the second mapping is Hölder continuous. \(\square \)

Remark 2

As a consequence of Lemma 7, the images of the mappings given by (31) and (32) are bounded. This fact will be used in the steps 3 and 5 of the proof of Proposition 1.

Lemma 8

Let \(R>0\) and \(\beta \in (0,1)\). Then, there exists \(\alpha \in (0,1)\) and \(C>0\) such that for all \(u \in B(W^{2,1,p}(Q),R)\) and for all \(m \in B(\mathcal {C}^{\beta }(Q),R) \cap L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), \(\Vert \mathbf {P}(m,\nabla u) \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).

Proof

We recall that by Lemma 12, \(\Vert \nabla u \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^d)} \le C \Vert u \Vert _{W^{2,1,p}(Q)}\). We obtain then the bound on \(\Vert \mathbf {P}(m,\nabla u) \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)}\) with Lemma 6. \(\square \)

Lemma 9

Let \(R>0\) and \(\beta \in (0,1)\). There exist \(\alpha \in (0,1)\) and \(C>0\) such that for all \(u \in B(\mathcal {C}^{2+\beta ,1+\beta /2}(Q),R)\) and for all \(m \in B(\mathcal {C}^\beta (Q),R) \cap L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\),
$$\begin{aligned} \Vert \mathbf {v}(m,\nabla u) \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^d)} \le C \quad \text {and} \quad \Vert D_x \mathbf {v}(m,\nabla u) \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^{d \times d})} \le C. \end{aligned}$$

Proof

The result follows from relations (33), Lemma 8, and from the Hölder continuity of \(H_p\), \(H_{px}\), and \(H_{pp}\) on bounded sets. \(\square \)

6 A Priori Estimates for Fixed Points

Proposition 1

There exist a constant \(C>0\) and an exponent \(\alpha \in (0,1)\) such that for all \(\tau \in [0,1]\), for all \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)),
$$\begin{aligned} m_\tau \in&\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q),&\Vert m_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C, \\ u_\tau \in&\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q),&\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C, \\ P_\tau \in&\mathcal {C}^\alpha (0,T;\mathbb {R}^k),&\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C, \\ v_\tau \in&\mathcal {C}^{\alpha }(Q,\mathbb {R}^d),&\Vert v_\tau \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^d)} \le C, \\ D_x v_\tau \in&\mathcal {C}^{\alpha }(Q,\mathbb {R}^{d\times d}),&\Vert D_x v_\tau \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^{d \times d})} \le C. \end{aligned}$$

Proof

Let us fix \(\tau \in [0,1]\) and \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)). All constants C and all exponents \(\alpha \in (0,1)\) involved below are independent of \((u_\tau ,m_\tau ,v_\tau ,P_\tau )\) and \(\tau \). Let us recall that \(\tilde{f}_\tau \in L^\infty (Q)\) has been defined in Lemma 4 by \(\tilde{f}_\tau (x,t)= f(x,t,m_\tau (t))\).

Step 1\(\Vert P_\tau \Vert _{L^2(0,T;\mathbb {R}^k)} \le C\).

Let \(v^0=0\) and let \(m^0\) be the solution to \(\partial _t m^0 - \sigma \varDelta m^0= 0\), \(m^0(x,0) = m_0 (x)\). By Lemma 4, \(B(m_\tau ,v_\tau ;\tilde{f}_\tau ) \le B(m^0,v^0;\tilde{f}_\tau )\). Since \(\Vert \phi \Vert _{L^\infty (Q,\mathbb {R}^{k \times d})} \le C\), we have for all \(\varepsilon >0\) and for all \(t \in [0,T]\) that
$$\begin{aligned}&\Big | \int _{\mathbb {T}^d} \phi v_\tau m_\tau \; \mathrm {d} x \Big | \le C \int _{\mathbb {T}^d} |v_\tau | m_\tau \; \mathrm {d} x \\&\quad \le C \Big ( \int _{\mathbb {T}^d} |v_\tau |^2 m_\tau \; \mathrm {d} x \Big )^{1/2} \le \frac{C}{\varepsilon } + C \varepsilon \int _{\mathbb {T}^d} |v_\tau |^2 m_\tau \; \mathrm {d} x, \end{aligned}$$
by the Cauchy–Schwarz inequality and Young’s inequality. The constant C is also independent of \(\varepsilon \). Using then the lower bounds (8) and (9) and assumptions (A5) and (A8), we obtain that
$$\begin{aligned} C&\ge B(m^0,v^0;\tilde{f}_\tau ) \ge B(m_\tau ,v_\tau ;\tilde{f}_\tau ) \\ \ge&\iint _Q \frac{1}{C} | v_\tau |^2 m_\tau \; \mathrm {d} x \; \mathrm {d} t - C \Big | \int _{Q} \phi v_\tau m_\tau \; \mathrm {d} x \; \mathrm {d} t \Big | - C \\&\ge \Big ( \frac{1}{C} - C \varepsilon \Big ) \iint _Q |v_\tau |^2 m_\tau \; \mathrm {d} x \; \mathrm {d} t - C\Big (1 + \frac{1}{\varepsilon } \Big ). \end{aligned}$$
Taking \(\varepsilon = 1 / (2C^2)\), we deduce that \(\iint _Q |v_\tau |^2 m_\tau \, \mathrm {d} x \, \mathrm {d} t \le C\). Using then assumption (A4), the boundedness of \(\phi \), the Cauchy–Schwarz inequality and the estimate obtained previously, we deduce that
$$\begin{aligned} \Vert P_\tau \Vert _{L^2(0,T;\mathbb {R}^k)}&= \int _0^T | \varPsi (t, {\textstyle \int }\phi v_\tau m_\tau ) |^2 \; \mathrm {d} t \le C+C \int _0^T \Big | \int _{\mathbb {T}^d} \phi v_\tau m_\tau \; \mathrm {d} x \Big |^2 \mathrm {d} t \nonumber \\&\le C + C\iint _Q |v_\tau |^2 m_\tau \; \mathrm {d} x \; \mathrm {d} t \le C. \end{aligned}$$
(34)
Step 2\(\Vert u_\tau \Vert _{L^\infty (Q)} \le C\), \(\Vert \nabla u_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\).
The argument is classical. We have that \(u_\tau \) is the unique solution to the HJB equation (i) (MFGC\(_\tau \)). It is therefore the value function associated with the following stochastic optimal control problem:
$$\begin{aligned} u_\tau (x,t) = \tau \Big ( \, \inf _{\alpha \in L_{\mathbb {F}}^2(t,T;\mathbb {R}^d)} J_\tau (x,t,\alpha ) \Big ), \end{aligned}$$
(35)
where \(J_\tau (x,t,\alpha )\) is defined by
$$\begin{aligned} \mathbb {E}\Big [ \int _{t}^{T} \big ( L(X_s,s, \alpha _s) + \left\langle \phi (X_s,s)^\intercal P_\tau (s) , \alpha _s \right\rangle + \tilde{f}_\tau (X_s,s) \big ) \; \mathrm {d} s + g(X_T) \Big ], \end{aligned}$$
and \((X_s)_{s \in [t,T]}\) is the solution to the stochastic dynamic \(\mathrm {d} X_s = \tau \alpha _s \mathrm {d} s + \sqrt{2\sigma } \mathrm {d} B_s, \; X_t = x\). Here, \(L_{\mathbb {F}}^2(t,T;\mathbb {R}^d)\) denotes the set of stochastic processes on (tT), with values in \(\mathbb {R}^d\), adapted to the filtration \(\mathbb {F}\) generated by the Brownian motion \((B_s)_{s \in [0,T]}\), and such that \(\mathbb {E}\big [ \int _t^T |\alpha (s)|^2 \; \mathrm {d} s \big ] < \infty \). Then, the boundedness of \(u_\tau \) from above can be immediately obtained by choosing \(\alpha = 0\) in (35) and using the boundedness of g. We can as well bound \(u_\tau \) from below since for all \((x,s) \in Q\) and for all \(\alpha \in \mathbb {R}^d\), we have
$$\begin{aligned} L(x,s,\alpha ) + \left\langle \phi (x,s)^\intercal P_\tau (s) , \alpha \right\rangle&\ge \frac{1}{C} | \alpha |^2 - \Vert \phi \Vert _{L^\infty (Q,\mathbb {R}^{k \times d})} | P_\tau (s) | | \alpha | -C \\&\ge \frac{1}{C} | \alpha |^2 - C |P_\tau (s)|^2 -C, \end{aligned}$$
for some constant C independent of (xs), \(\alpha \), and \(P_\tau (s)\). We already know from the previous step that \( \Vert P_\tau \Vert _{L^2(0,T;\mathbb {R}^k)} \le C\). So we can conclude that \(u_\tau \) is also bounded from below, and thus \(\Vert u_\tau \Vert _{L^\infty (Q)} \le C\). We also deduce from the above inequality that for all \(\alpha \in L_{\mathbb {F}}^2(t,T;\mathbb {R}^d)\),
$$\begin{aligned} \mathbb {E} \Big [ \int _t^T |\alpha _s|^2 \; \mathrm {d} s \Big ] \le C \big ( J_\tau (x,t,\alpha ) + 1 \big ). \end{aligned}$$
(36)
Let us bound \(\nabla u_\tau \). Choose \(\varepsilon \in (0,1)\). For arbitrary (xt), take an \(\varepsilon \)-optimal stochastic optimal control \({\tilde{\alpha }}\) for (35). We can deduce from the boundedness of the map \(u_\tau \) and inequality (36) that
$$\begin{aligned} \mathbb {E} \Big [ \int _t^T |\tilde{\alpha }_s|^2 \; \mathrm {d} s \Big ] \le C \big ( J_\tau (x,t,\alpha ) + 1 \big ) \le C(u_\tau (x,t)+ \varepsilon + 1) \le C, \end{aligned}$$
(37)
where C is independent of \((\tau ,x,t)\) and \(\varepsilon \). Let \(y \in \mathbb {T}^d\). Set
$$\begin{aligned} \mathrm {d} X_s = \tau {\tilde{\alpha }}_s \mathrm {d} s + \sqrt{2\sigma } \mathrm {d} B_s , \; X_t = x, \quad \text {and} \quad Y_s = X_s - x + y, \end{aligned}$$
(38)
then obviously \(\mathrm {d} Y_s = {\tilde{\alpha }}_s \mathrm {d} s + \sqrt{2\sigma } \mathrm {d} B_s , \; Y_t = y\). We have
$$\begin{aligned} u_\tau (x,t) + \varepsilon&\ge \ \tau \mathbb {E}\Big [ \int _{t}^{T} L(X_s,s, {\tilde{\alpha }}_s) + \left\langle P_\tau (s) , \phi (X_s,s)^\intercal {\tilde{\alpha }}_s \right\rangle \; \mathrm {d} s \\&+ \int _t^T \tilde{f}_\tau (X_s,s) \; \mathrm {d} s + g(X_T) \Big ],\\ u_\tau (y,t)&\le \tau \mathbb {E}\Big [ \int _{t}^{T} \Big ( L(Y_s,s, {\tilde{\alpha }}_s) + \left\langle \phi (Y_s , s)^\intercal P_\tau (s) , {\tilde{\alpha }}_s \right\rangle \; \mathrm {d} s \\&+ \int _t^T \tilde{f}_\tau (Y_s,s) \Big ) \; \mathrm {d} s + g(Y_T) \Big ]. \end{aligned}$$
Therefore, \(u_\tau (y,t) - u_\tau (x,t) \le \varepsilon + |(a)| + |(b)| + |(c)| + |(d)|\), where (a), (b), (c), (d) are given by
$$\begin{aligned} (a)= \,&\tau \mathbb {E} \Big [ \int _t^T L(Y_s,s,\tilde{\alpha }_s) - L(X_s,s,\tilde{\alpha }_s) \; \mathrm {d} s \Big ], \\ (b)=\,&\tau \mathbb {E} \Big [ \int _t^T \big ( \phi (Y_s,s) - \phi (X_s,s) \big )^\intercal P_\tau (s), \tilde{\alpha }_s \rangle \; \mathrm {d} s \Big ] , \\ (c)= \,&\tau \mathbb {E} \big [ g(Y_T)-g(X_T) \big ], \\ (d)= \,&\tau \mathbb {E} \Big [ \int _t^T \big ( \tilde{f}_\tau (Y_s,s) - \tilde{f}(X_s,s) \big ) \; \mathrm {d} s \Big ]. \end{aligned}$$
First, we have
$$\begin{aligned} |(a)| \le \,&\tau \mathbb {E} \Big [ \int _t^T \big | L(Y_s,s, {\tilde{\alpha }}_s) - L(X_s,s,{\tilde{\alpha }}_s) \big | \; \mathrm {d} s \Big ] \\ \le \,&C |y-x| \Big ( 1 + \mathbb {E} \Big [ \int _t^T | \tilde{\alpha }_s |^2 \; \mathrm {d} s \Big ] \Big ) \le C |y-x|, \end{aligned}$$
as a consequence of assumption (A3) and (37). Then, using assumption (A6), (34), and (37), we obtain
$$\begin{aligned} |(b)| \le \,&\tau \mathbb {E} \Big [ \int _t^T |\phi (Y_s , s)-\phi (X_s , s)| |P_\tau (s)| |\tilde{\alpha }(s)| \; \mathrm {d} s \Big ] \\ \le \,&C |y-x| \, \Vert P_\tau \Vert _{L^2(0,T;\mathbb {R}^k)} \, \mathbb {E} \Big [ \int _t^T |\tilde{\alpha }(s)|^2 \; \mathrm {d} s \Big ] \le C |y-x|. \end{aligned}$$
By assumption (A8), \(|(c)| \le \mathbb {E} \big [ |g(Y_T)-g(X_T)| \big ] \le C |y-x|\). Finally, since \(\tilde{f}_\tau \) is a Lipschitz function (by assumption (A7)),
$$\begin{aligned} |(d)| \le \,&\tau \mathbb {E} \Big [ \int _t^T \big | \tilde{f}_\tau (Y_s,s) - \tilde{f}_\tau (X_s,s) \big | \; \mathrm {d} s \Big ] \le \, C |y-x|. \end{aligned}$$
(39)
Letting \(\varepsilon \rightarrow 0\), we obtain that \(u_\tau (y,t) - u_\tau (x,t) \le C |y-x|\). Exchanging x and y, we obtain that \(u_\tau \) is Lipschitz continuous with modulus C and finally that \(\Vert \nabla u_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\).

Step 3\(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)} \le C\).

By Lemma 3, \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). We have that \(\Vert \nabla u_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\) and \(P_\tau = \mathbf {P}(m_\tau ,\nabla u_\tau )\). The bound on \(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)}\) follows then from Lemma 7 and Remark 2.

Step 4\(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\).

By assumption (A6), \(\phi \) is bounded. We have proved that \(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)} \le C\) and by Lemma 1, H is continuous. Thus, \(\Vert H(\nabla u_\tau + \phi ^\intercal P_\tau ) \Vert _{L^\infty (Q)} \le C\). By assumption (A5), \(\Vert \tau \tilde{f}_\tau \Vert _{L^\infty (Q)} \le C\). It follows that \(u_\tau \), as the solution to the HJB equation (i) (MFGC\(_\tau \)), is the solution to a parabolic equation with bounded coefficients. Thus, by Theorem 6, \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\). We also obtain with Lemma 12 that \(\Vert u_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\) and \(\Vert \nabla u_\tau \Vert _{\mathcal {C}^\alpha (Q,\mathbb {R}^d)} \le C\).

Step 5\(\Vert v_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\), \(\Vert D_x v_\tau \Vert _{L^p(Q,\mathbb {R}^{d \times d})} \le C\).

We have proved that \(v_\tau = \mathbf {v}(m_\tau ,\nabla u_\tau )\) and \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\). The estimate follows directly with Lemma 7 and Remark 2.

Step 6\(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\).

The Fokker–Planck equation can be written in the form of a parabolic equation with coefficients in \(L^p\): \(\partial _t m_{\tau } - \sigma \varDelta m_\tau + \tau \langle v_\tau , \nabla m_\tau \rangle + \tau m_\tau \text {div}(v_\tau )= 0\), since \(\Vert D_x v_\tau \Vert _{L^p(Q,\mathbb {R}^{d \times d})} \le C\). Combining Theorem 4 and Lemma 12, we get that \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\).

Step 7\(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).

We already know that \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\), that \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\), and that \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). Thus Lemma 8 applies and yields that \(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).

Step 8\(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C\).

We have proved that \(\Vert \nabla u_\tau \Vert _{\mathcal {C}^\alpha (Q,\mathbb {R}^d)} \le C\) and \(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\). Moreover, we have assumed that \(\phi \) is Hölder continuous and know that H is Hölder continuous on bounded sets. It follows that \(\Vert H(\nabla u_\tau + \phi ^\intercal P_\tau ) \Vert _{\mathcal {C}^\alpha (Q)} \le C\). It follows from assumption (A7) that \(\tau \tilde{f}_\tau \) is Hölder continuous. Since \(g \in \mathcal {C}^{2+\alpha }(\mathbb {T}^d)\), we finally obtain that \(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C\), by Theorem 7.

Step 9\(\Vert v_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^{d})} \le C\) and \(\Vert D_x v_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^{d \times d})} \le C\).

We have \(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+\alpha /2}(Q)} \le C\) and \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\). Thus Lemma 9 applies and the announced estimates hold true.

Step 10\(\Vert m_\tau \Vert _{\mathcal {C}^{2+ \alpha , 1 + \alpha /2}(Q)} \le C\).

A direct consequence of Step 9 is that \(m_\tau \) is the solution to a parabolic equation with Hölder continuous coefficients. Therefore \(\Vert m_\tau \Vert _{\mathcal {C}^{2+\alpha ,1 + \alpha /2}(Q)} \le C\), by Theorem 7, which concludes the proof of the proposition. \(\square \)

7 Application of the Leray–Schauder Theorem

Proof

(Theorem 1) Step 1 construction of \(\mathcal {T}\).

Let us define the mapping \(\mathcal {T} :X \times [0,1] \rightarrow X\) which is used for the application of the Leray–Schauder theorem. A difficulty is that the auxiliary mappings \(\mathbf {P}\) and \(\mathbf {v}\) are only defined for \(m \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). Therefore we need a kind of projection operator on this set. Note that \(\int _{\mathbb {T}^d} 1 \, \mathrm {d} x = 1\). We consider the mapping
$$\begin{aligned} \rho : m \in L^\infty (Q) \mapsto \rho (m) \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d)), \end{aligned}$$
defined by
$$\begin{aligned} \rho (m)= \frac{m_+(x,t)}{\max ( 1, {\textstyle \int }m_+(y,t) \; \mathrm {d} y )} + 1 - \frac{{\textstyle \int }m_+(y,t) \; \mathrm {d} y}{\max ( 1, {\textstyle \int }m_+(y,t) \; \mathrm {d} y )}, \end{aligned}$$
where \(m_+(x,t)= \max (0,m(x,t))\). For checking that \(\rho (m) \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), we suggest to consider the two cases: \({\textstyle \int }m_+(y,t) \; \mathrm {d} y < 1\) and \({\textstyle \int }m_+(y,t) \; \mathrm {d} y \ge 1\) separetely. The following properties can be easily checked:
  • For all \(m \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), \(\rho (m)=m\).

  • The mapping \(\rho \) is locally Lipschitz continuous, from \(L^\infty (Q)\) to \(L^\infty (Q)\).

  • For all \(\alpha \in (0,1)\), there exists a constant \(C>0\) such that if \(m \in \mathcal {C}^\alpha (Q)\), then \(\rho (m) \in \mathcal {C}^\alpha (Q)\) and \(\Vert \rho (m) \Vert _{\mathcal {C}^{\alpha }(Q)} \le C \Vert m \Vert _{\mathcal {C}^{\alpha }(Q)}\).

For a given \((u,m,\tau ) \in X \times [0,1]\), the pair \((\tilde{u},\tilde{m})= \mathcal {T}(u,m,\tau )\) is defined as follows: \(\tilde{u}\) is the solution to
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{l} -\partial _t \tilde{u} -\sigma \varDelta \tilde{u} + \tau H(\nabla u + \phi ^\intercal \mathbf {P}(\rho (m),\nabla u) ) = \tau f(\rho (m(t))) \quad (x,t) \in Q, \\ \tilde{u}(x,T) = \tau g(x) \quad x \in \mathbb {T}^d, \end{array} \end{array}\right. } \end{aligned}$$
and \(\tilde{m}\) is the solution to
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{l} \partial _t \tilde{m} - \sigma \varDelta \tilde{m} + \tau \mathrm {div} (\mathbf {v}(\rho (m),\nabla {\tilde{u}}) m ) = 0 \quad (x,t) \in Q, \\ \tilde{m}(x,0) = m_0(x) x \in \mathbb {T}^d. \end{array} \end{array}\right. } \end{aligned}$$
It directly follows from the definition of \(\mathcal {T}\) that \(\mathcal {T}(u,m,0)\) is constant, as required by the Leray–Schauder theorem.

Step 2 a priori bound.

Let \(\tau \in [0,1]\) and let \((u_\tau ,m_\tau )\) be such that \((u_\tau ,m_\tau )= \mathcal {T}(u_\tau ,m_\tau ,\tau )\). Then, by Lemma 3, \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). Thus, \(m_\tau = \rho (m_\tau )\) and finally, by Lemma 5, the quadruplet \((u_\tau ,m_\tau ,P_\tau ,v_\tau )\), with \(P_\tau = \mathbf {P}(m_\tau ,\nabla u_\tau )\) and \(v_\tau = \mathbf {v}(m_\tau ,\nabla u_\tau )\), is a solution to (MFGC\(_\tau \)). We directly conclude with Proposition 1 that \(\Vert (u_\tau ,m_\tau ) \Vert _X \le C\), where C is independent of \(\tau \).

Step 3 continuity of \(\mathcal {T}\).

Using the continuity of \(\rho \), Lemma 7, the Hölder continuity of H, and assumption (A7), we obtain that the mappings
$$\begin{aligned}&(u,m) \in X \mapsto H(\nabla u + \phi ^\top \mathbf {P}(\rho (m),\nabla u)) - f(\rho (m)) \in L^\infty (Q), \\&(u,m) \in X \mapsto \text {div}(\mathbf {v}(\rho (m),\nabla u)m) \in L^p(Q) \end{aligned}$$
are continuous. By Theorem 6, the solution to a parabolic equation of the form (51), with b and c null (in \(W^{2,1,p}(Q)\)) is a continuous mapping of the right-hand side (in \(L^p(Q)\)). Thus, \(\tilde{u} \in W^{2,1,p}(Q)\) depends in a continuous way on \(\tau H(\nabla u + \phi ^\top \mathbf {P}(\rho (m),\nabla u))\) and therefore \(\tilde{u}\) depends in a continuous way on \((\tau ,u,m)\) by composition. Again, by Theorem 6, \(\tilde{m} \in W^{2,1,p}(Q)\) depends in a continuous way on \(\tau \text {div}(\mathbf {v}(\rho (m),\nabla \tilde{u})m)\) and therefore depends in a continuous way on \((\tau ,u,m)\).

Step 4 compactness of \(\mathcal {T}\).

Let \(R>0\), let \((u,m) \in B(X,R)\). We have \(\Vert \rho (m) \Vert _{\mathcal {C}^{\alpha }(Q)} \le C\), where C is independent of (um) (but depends on R). As a consequence of assumption (A7), and since H is Hölder continuous on bounded sets, we have
$$\begin{aligned}&\Vert H(\nabla u + \phi ^\intercal \mathbf {P}(\rho (m),\nabla u))- f(\rho (m)) \Vert _{\mathcal {C}^\alpha (Q)} \le C, \end{aligned}$$
where \(C>0\) and \(\alpha \in (0,1)\) are both independent of (um) (but depend on R). It follows then that \(\Vert u \Vert _{\mathcal {C}^{2+\alpha ,1+\alpha /2}(Q)} \le C\) by Theorem 7. Using Lemma 9, we deduce then that
$$\begin{aligned} \Vert \text {div}(\mathbf {v}(\rho (m),\nabla \tilde{u}) m) \Vert _{\mathcal {C}^\alpha (Q)} \le C, \end{aligned}$$
and finally obtain that \(\Vert m \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C\), by Theorem 7 again. The compactness of \(\mathcal {T}\) follows, since \(\mathcal {C}^{2+\alpha ,1+\alpha /2}(Q)\) is compactly embedded in \(W^{2,1,p}(Q)\), by the Arzelà–Ascoli theorem.

Step 5 Conclusion.

The existence of a fixed point (um) to \(\mathcal {T}(\cdot ,\cdot ,1)\) follows. With the same arguments as those of Step 2, we obtain that \((u,m,\mathbf {P}(m,\nabla u),\mathbf {v}(m,\nabla u))\) is a solution to (MFGC\(_\tau \)) with \(\tau =1\) and that (15) holds, by Proposition 1. \(\square \)

8 Uniqueness and Duality

In this section we prove the uniqueness of the solution (umvP) to (MFGC). We also prove that (Pv) is the solution to a dual problem to (17). Both results are obtained under the following additional monotonicity assumption of f: There exists a measurable mapping \(F(t,m) :[0,T]\times \mathcal {D}_1(\mathbb {T}^d)\rightarrow \mathbb {R}\) such that
$$\begin{aligned} F(t,m_2) - F(t,m_1) \ge \int _{ \mathbb {T}^d} f(x,t,m_1) (m_2(x)-m_1(x)) \; \mathrm {d} x, \end{aligned}$$
(40)
for all \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\) and for a.e. t. Thus, \(F(t,\cdot )\) is a supremum of the exact affine minorants appearing in the above right-hand side, and is therefore a convex function of m.

Remark 3

  1. 1.
    It follows from (40) that f is monotone:
    $$\begin{aligned} \int _{ \mathbb {T}^d} (f(x,t,m_2)- f(x,t,m_2)) (m_2(x) - m_1(x)) \; \mathrm {d} x \ge 0, \end{aligned}$$
    (41)
    for all \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\) and for a.e. t. Conversely, (40) holds true if (41) is satisfied and if F is a primitive of f(., t, .) in the sense that
    $$\begin{aligned} F(t,m_2) - F(t,m_1) = \int _{0}^{1} \int _{\mathbb {T}^d} f(x,t, sm_2 + (1-s)m_1) (m_2(x) - m_1(x)) \; \mathrm {d} s. \end{aligned}$$
    We refer to [5, Proposition 1.2] for a further characterization of functions f deriving from a potential.
     
  2. 2.
    Consider the mapping \(f_K\) proposed in Lemma 2. Assume that for all \((x,t) \in Q\), \(K(x,t,\cdot )\) is non-decreasing and consider the function \(\mathcal {K}\) defined by \(\mathcal {K}(x,t,w) := \int _0^w K(x,t,w') \; \mathrm {d} w'\), for \((x,t,w) \in Q \times [-C_2,C_2]\). Then inequality (40) holds true with \(F_K\) defined by
    $$\begin{aligned} F_K(t,m)= \int _{\mathbb {T}^d} \mathcal {K} (x,t,m * \varphi (x)) \; \mathrm {d} x. \end{aligned}$$
    Indeed, since \(\mathcal {K}\) is convex in its third argument, we have
    $$\begin{aligned}&F_K(t,m_2)-F_K(t,m_1) = \int _{\mathbb {T}^d} \mathcal {K}(x,t,m_2 * \varphi (x)) - \mathcal {K}(x,t,m_1 * \varphi (x)) \; \mathrm {d} x \\&\quad \ge \int _{\mathbb {T}^d} K(x,t,m_1* \varphi (x))((m_2-m_1)*\varphi )(x) \; \mathrm {d} x \\&\quad = \int _{\mathbb {T}^d} (K(\cdot ,t,m_1*\varphi (\cdot ))* \tilde{\varphi })(x) (m_2(x) - m_1(x)) \; \mathrm {d} x \\&\quad = \int _{\mathbb {T}^d} f_K(x,t,m)(m_2(x)-m_1(x)) \; \mathrm {d} x, \end{aligned}$$
    as was to be proved.
     
Without loss of generality, we can assume that \(F(t,m_0)=0\) for a.e. \(t \in (0,T)\). It can then be easily deduced from assumption (A5) and (40) that there exists a constant C such that
$$\begin{aligned} |F(t,m)| \le C, \quad \forall m \in \mathcal {D}_1(\mathbb {T}^d), \text { for a.e. }t \in (0,T). \end{aligned}$$
(42)
Let us consider the potential \(B :W^{2,1,p}(Q) \times L^\infty (Q;\mathbb {R}^k) \rightarrow \mathbb {R}\), defined by
$$\begin{aligned}&B (m,v) = \iint _Q L ( x,t, v(x,t)) m(x,t) \; \mathrm {d} x \; \mathrm {d} t + \int _0^T F(t,m(t)) \; \mathrm {d} t \nonumber \\&\quad + \int _0^T \varPhi \Big (t, \int _{\mathbb {T}^d} \phi (x,t) v(x,t) m(x,t) \; \mathrm {d} x \Big ) \, \mathrm {d} t + \int _{\mathbb {T}^d} g(x) m(x,T) \; \mathrm {d} x. \end{aligned}$$
(43)

Proposition 2

There exists a unique solution \((u,m,v,P) \in X'\) to (MFGC). Moreover, the pair (mv) is the solution to the following optimal control problem
$$\begin{aligned} \min _{\begin{array}{c} \hat{m} \in W^{2,1,p}(Q) \\ \hat{v} \in L^\infty (Q,\mathbb {R}^d) \end{array}} \ B(\hat{m},\hat{v}), \quad \text {s.t.: } {\left\{ \begin{array}{ll} \begin{array}{rl} \partial _t \hat{m} - \sigma \varDelta \hat{m} + \mathrm {div} (\hat{v} \hat{m}) &{}= 0, \\ \hat{m}(x,0) &{}= m_0(x). \end{array} \end{array}\right. } \end{aligned}$$
(44)

Proof

Let \((u,m,v,P) \in X'\) be a solution to (MFGC). Let us prove that (mv) is a solution to (44). Let \((\hat{m},\hat{v})\) be a feasible pair. Denoting \(\tilde{f}(x,t)= f(x,t,m(t))\), we have
$$\begin{aligned}&B(\hat{m},\hat{v})-B(m,v) = \big ( B(\hat{m},\hat{v};\tilde{f})- B(m,v;\tilde{f}) \big ) \\&\quad + \Big ( \int _0^T F(t,\hat{m}(t))- F(t,m(t)) - \int _{\mathbb {T}^d} \tilde{f}(x,t) (\hat{m}(x,t)-m(x,t)) \; \mathrm {d} x \; \mathrm {d} t \Big ). \end{aligned}$$
The two terms in the right-hand side are both nonnegative, as a consequence of Lemma 4 and assumption (40), respectively.
It remains to prove the uniqueness of the solution to (MFGC). Let us prove first a classical property: There exists a constant \(C>0\) such that for all \((x,t) \in Q\), for all \(p \in \mathbb {R}^d\) and for all \(v \in \mathbb {R}^d\),
$$\begin{aligned} H(x,t,p) + L(x,t,v) + \langle p, v \rangle \ge \frac{1}{2C} |v + H_p(x,t,p)|^2. \end{aligned}$$
(45)
Let us set \(\bar{v}= - H_p(x,t,p)\). For a fixed triple (xtp), we have \(H(x,t,p)= - \langle p,\bar{v} \rangle - L(x,t,\bar{v})\). Moreover, \(L_v(x,t,\bar{v})= -p\) and thus by (A1),
$$\begin{aligned} L(x,t,v) \ge L(x,t,\bar{v}) - \langle p, v- \bar{v} \rangle + \frac{1}{2C} |v-\bar{v}|^2. \end{aligned}$$
Inequality (45) follows.
Let \((u_1,m_1,v_1,P_1)\) and \((u_2,m_2,v_2,P_2)\) be two solutions to (MFGC) in \(X'\). We obtain with inequality (45) that
$$\begin{aligned} L (v_2)&\ge - H(\nabla u_1 + \phi ^\intercal P_1) - \left\langle \nabla u_1 + \phi ^\intercal P_1 , v_2 \right\rangle + \frac{1}{2C} |v_2-v_1|^2, \\ L (v_1 )&= - H(\nabla u_1 + \phi ^\intercal P_1 ) - \left\langle \nabla u_1 + \phi ^\intercal P_1 , v_1 \right\rangle . \end{aligned}$$
Proceeding then exactly like in the proof of Lemma 4, we arrive at the following inequality:
$$\begin{aligned} B(m_2,v_2)-B(m_1,v_1) \ge \frac{1}{2C} \iint _Q |v_2 - v_1 |^2 m_2 \; \mathrm {d} x \; \mathrm {d} t. \end{aligned}$$
We also have that \(B(m_1,v_1)-B(m_2,v_2) \ge 0\), thus \(\iint _Q |v_2 - v_1 |^2 m_2 \; \mathrm {d} x \; \mathrm {d} t= 0\). As a consequence, \((v_2-v_1)m_2= 0\), since \(m_2 \ge 0\). We obtain then that
$$\begin{aligned} v_2m_2 - v_1m_1= v_1(m_2-m_1). \end{aligned}$$
(46)
Let us set \(m= m_2- m_1\). Using relation (46), we obtain that m is the solution to the following parabolic equation: \(\partial _t m - \sigma \varDelta m + \text {div}(v_1 m)= 0\), \(m(x,0)=0\). Therefore \(m=0\) and \(m_2= m_1\). We already know that \(v_2m_2= v_1m_2\), we deduce then that \(v_2m_2= v_1m_1\). We obtain further with (iii) that \(P_1= P_2\), then with (i) that \(u_1= u_2\) and finally with (iv) that \(v_1= v_2\), which concludes the proof. \(\square \)
We finish this section with a duality result. For \(\gamma \in L^\infty (\mathbb {T}^d)\), we recall that the convex conjugate of \(F(t,\cdot )\) is defined by
$$\begin{aligned} F^*(t,\gamma )= \sup _{m \in \mathcal {D}_1(\mathbb {T}^d)} \int _{\mathbb {T}^d} \gamma (x) m(x) \; \mathrm {d} x - F(t,m). \end{aligned}$$
It directly follows from the above definition that \(|F^*(t,\gamma )| \le \Vert \gamma \Vert _{L^\infty (\mathbb {T}^d)} + C\), where C is the constant obtained in (42) and thus for \(\gamma \in L^\infty (Q)\), the integral \(\int _0^T F^*(t,\gamma (\cdot ,t)) \; \mathrm {d} t\) is well-defined.
Consider the dual criterion \(D :(u,P,\gamma ) \in W^{2,1,p}(Q) \times L^\infty (0,T;\mathbb {R}^k) \times L^\infty (Q) \mapsto D(u,p,\gamma ) \in \mathbb {R}\cup \{ - \infty \}\), defined by
$$\begin{aligned} D(u,P,\gamma )= \int _{\mathbb {T}^d} u(x,0) m_0(x) \; \mathrm {d} x - \int _0^T \varPhi ^*(t,P(t)) \; \mathrm {d} t - \int _0^T F^*(t,\gamma (t)) \; \mathrm {d} t. \end{aligned}$$
The function \(\varPhi ^*\) is the convex conjugate of \(\varPhi \) with respect to its second argument. Since \(\varPhi (t,0)=0\), we have that \(\varPhi ^*(t,\cdot ) \ge 0\) and thus the first integral is well-defined in \(\mathbb {R}\cup \{ \infty \}\).

Lemma 10

Let \((\bar{u},\bar{m},\bar{v},\bar{P})\) be the solution to (MFGC). Let \(\tilde{f}\) be defined by \(\tilde{f}(x,t)= f(x,t,\bar{m}(t))\). Then, \((\bar{u},\bar{P},\tilde{f})\) is a solution to the following problem:
$$\begin{aligned} \max _{\begin{array}{c} u \in W^{2,1,p}(Q) \\ P \in L^\infty (0,T;\mathbb {R}^k) \\ \gamma \in L^\infty (Q) \end{array}} \! \! D(u,P,\gamma ), \ \text {s.t. } {\left\{ \begin{array}{ll} \begin{array}{rl} -\partial _t u - \sigma \varDelta u + H(\nabla u + \phi ^\intercal P) \le &{} \gamma \\ {u}(x,T) \le &{} g(x). \end{array} \end{array}\right. } \end{aligned}$$
(47)
Moreover, for all solutions \((u,P,\gamma )\) to the dual problem, \(P=\bar{P}\). If in addition, \(\gamma = \tilde{f}\) and the above inequalities hold as equalities, then \(u= \bar{u}\).

Proof

For all \(t \in [0,T]\), we have
$$\begin{aligned} - \int _0^T \varPhi ^*(P) \; \mathrm {d} t = \int _0^T \varPhi ({\textstyle \int }\phi \bar{v} \bar{m}) \; \mathrm {d} t - \iint _Q \langle \phi ^{\intercal } P, \bar{v} \bar{m} \rangle \; \mathrm {d} x \; \mathrm {d} t + (a) \end{aligned}$$
(48)
with
$$\begin{aligned} (a)= \int _0^T - \varPhi ^*(P) - \varPhi ({\textstyle \int }\phi \bar{v} \bar{m}) + \langle P, {\textstyle \int }\phi \bar{v} \bar{m} \rangle \; \mathrm {d} t \le 0. \end{aligned}$$
We also have that
$$\begin{aligned} -\int _0^T F^*(t,\gamma (t)) \; \mathrm {d} t + \iint _Q \gamma (x,t) \bar{m}(x,t) \; \mathrm {d} x \; \mathrm {d} t = \int _0^T F(t,\bar{m}(t)) \; \mathrm {d} t + (b), \end{aligned}$$
(49)
where
$$\begin{aligned} (b)= \iint _Q \gamma (x,t)\bar{m}(x,t) \; \mathrm {d} x \; \mathrm {d} t - \int _0^T F(t,\bar{m}(t)) \; \mathrm {d} t - \int _0^T F^*(t,\gamma (t)) \; \mathrm {d} t \le 0. \end{aligned}$$
Integrating by parts (in time), we obtain that
$$\begin{aligned}&\int _{\mathbb {T}^d} {u}(x,0) m_0(x) \; \mathrm {d} x = \iint _Q - \partial _t u \bar{m} - u \partial _t \bar{m} \; \mathrm {d} x \; \mathrm {d} t + \int _{\mathbb {T}^d} u(x,T) \bar{m}(x,T) \; \mathrm {d} x \\&\quad = \ \iint _Q (\sigma \varDelta {u} + \gamma - H(\nabla {u} + \phi ^\intercal {P})) \bar{m} + (-\sigma \varDelta \bar{m} + \text {div}(\bar{v}\bar{m})) {u} \; \mathrm {d} x \; \mathrm {d} t \\&\qquad + \int _{\mathbb {T}^d} g(x) \bar{m}(x,T) \; \mathrm {d} x + (c) + (d), \end{aligned}$$
where
$$\begin{aligned} (c)&= \iint _Q (-\partial _t {u} - \sigma \varDelta {u} + H(\nabla {u} + \phi ^\intercal {P} ) - \gamma ) \bar{m} \; \mathrm {d} x \; \mathrm {d} t \le 0 \\ (d)&= \int _{\mathbb {T}^d} ({u}(x,T)-g(x)) \bar{m}(x,T) \; \mathrm {d} x \le 0. \end{aligned}$$
Integrating by parts (in space), we further obtain that
$$\begin{aligned} \int _{\mathbb {T}^d} {u}(x,0) m_0(x) \; \mathrm {d} x =&\iint _Q \big ( \gamma - H(\nabla {u} + \phi ^\intercal {P}) - \langle \nabla {u}, \bar{v} \rangle \big ) \bar{m} \nonumber \\&+ \int _{\mathbb {T}^d} g(x) \bar{m}(x,T) \; \mathrm {d} x + (c) + (d) \nonumber \\ =&\iint _Q \big ( L(\bar{v}) + \gamma \big ) \bar{m} + \langle \phi ^\intercal \bar{P}, \bar{v} \rangle \bar{m} \; \mathrm {d} x \; \mathrm {d} t \nonumber \\&+ \int _{\mathbb {T}^d} g(x) \bar{m}(x,T) \; \mathrm {d} x + (c) + (d) + (e), \end{aligned}$$
(50)
where
$$\begin{aligned} (e)= \iint _Q \big ( - H(\nabla {u} + \phi ^\intercal {P}) - L(\bar{v}) - \langle \nabla {u} + \phi ^\intercal {P}, \bar{v} \rangle \big ) \bar{m} \; \mathrm {d} x \; \mathrm {d} t \le 0. \end{aligned}$$
Combining (48), (49) and (50) together, we finally obtain that
$$\begin{aligned} D(u,P,\gamma ) =&\int _0^T \varPhi ({\textstyle \int }\phi \bar{v} \bar{m}) \; \mathrm {d} t + \iint _Q L(\bar{v}) \bar{m} \; \mathrm {d} x \; \mathrm {d} t + \int _0^T F(t,m(t)) \; \mathrm {d} t \\&+ \int _{\mathbb {T}^d} g(x) \bar{m}(x,T) \; \mathrm {d} x + (a) + (b) + (c) + (d) + (e) \\ =&B(\bar{m},\bar{v}) + (a) + (b) + (c) + (d) + (e). \end{aligned}$$
The five terms (a), (b), (c), (d), (e) are non-positive and equal to zero if \((u,P,\gamma )=(\bar{u},\bar{P},\tilde{f})\), as can be easily verified. This proves the optimality of \((\bar{u},\bar{P},\tilde{f})\). Moreover, since \(\varPhi \) is differentiable (with gradient \(\varPsi \)), the term (a) is null if and only if \(P(t)= \varPsi ({\textstyle \int }\phi \bar{v} \bar{m})= \bar{P}(t)\), for a.e. \(t \in [0,T]\). Therefore, for all optimal solutions \((u,P,\gamma )\), \(P= \bar{P}\). If moreover \(\gamma = \tilde{f}\) and the inequality constraints in (47) hold as equalities, then (since the HJB equation has a unique solution) \(u= \bar{u}\), which concludes the proof. \(\square \)

Remark 4

It is of interest to check when the density m(xt) is a.e. positive, since this is clearly a necessary condition for the uniqueness of the solution of (44). We note that a sufficient condition for the positivity of m is given in [21, Proposition 3.10].

9 Conclusion

The existence and uniqueness of a classical solution to a mean field game of controls have been demonstrated. A particularly important aspect of the analysis is the fact that the equations (iii) and (iv) (MFGC), encoding the coupling of the agents through the controls, are equivalent to the optimality system of a ‘static’ convex problem. This observation enabled us to eliminate the variables v and P from the coupled system.

The analysis done in this article can be extended in different ways. A more complex interaction between the agents could be considered. For example, it would be possible to replace equations (iii) and (iv) by the following ones:
$$\begin{aligned} \begin{array}{rl} P(t) &{}= \varPsi (t, \int _{\mathbb {T}^d} \varphi (x,t,v(x,t)) m(x,t) \, \mathrm {d} x ) \\ v(x,t) &{}= -H_p(x,t,\nabla u(x,t) D_v\varphi (x,t,v(x,t)^\intercal P(t)), \end{array} \end{aligned}$$
assuming that \(\varphi \) is convex with respect to v and \(\varPsi \ge 0\). For a fixed \(t \in [0,T]\), this system is equivalent to the optimality system associated with the following convex problem:
$$\begin{aligned}&\inf _{v :\mathbb {T}^d\rightarrow \mathbb {R}^d} \varPhi \Big (t,\int _{\mathbb {T}^d} \varphi (x,t,v(x) ) m(x,t) \, \mathrm {d} x \Big ) \\&\quad + \int _{\mathbb {T}^d} \big ( L(v(x)) + \langle \nabla u(x,t), v(x) \rangle ) m(x,t) \; \mathrm {d} x. \end{aligned}$$
Another possibility of extension of our analysis would be to add convex constraints on the control variable.

Future research will aim at exploiting the potential structure of the problem, which can be used to solve it numerically and to prove the convergence of learning procedures, as was done in [5].

Notes

Acknowledgements

Open access funding provided by University of Graz. The authors want to thank an anonymous referee for his useful remarks. The first author acknowledges support from the FiME Lab (Institut Europlace de Finance). The first two authors acknowledges support from the PGMO project “Optimal control of conservation equations”, itself supported by iCODE (IDEX Paris-Saclay) and the Hadamard Mathematics LabEx.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest

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Authors and Affiliations

  1. 1.Inria-Saclay and Ecole PolytechniquePalaiseauFrance
  2. 2.University of GrazGrazAustria

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