# Relaxed Lagrangian Approach to ResNet Type Mean Field Game Problems

Cristian Mendico, Kaizhi Wang, Yuchen Xu

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Cristian Mendico, Kaizhi Wang, Yuchen Xu. Relaxed Lagrangian Approach to ResNet Type Mean Field Game Problems. 2024. hal-04818880

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Distributed under a Creative Commons Attribution 4.0 International License
# Relaxed Lagrangian Approach to ResNet Type Mean Field Game Problems

Cristian Mendico * Kaizhi Wang † Yuchen Xu ‡

December 2, 2024

## Abstract

This paper addresses the existence of equilibria for ResNet type Mean Field Games problems. We introduce a relaxed Lagrangian approach on the Wasserstein space to handle the non-convex optimization problem of mean field type. We prove the existence of new relaxed Nash equilibria and we show that our existence result encompasses the classical Mean Field Games problem's existence result under convex data conditions.

**Keywords.** Mean Field Games, Mean Field Control, Residual Neural Networks, Relaxed Lagrangian Approach.

**Mathematics Subject Classification (2020).** 49N80 - 49N90 - 68T07 - 93C10.

## 1 Introduction

The motivation of this work can be found in the recent development of deep learning (DL) and the use of techniques from optimal control and from mean field theory. Indeed, given a set of training data $(\xi, \zeta) \in \Omega$ distributed according to $\mathcal{M} \in \mathcal{P}(\Omega)$ and given an activation function $\varphi$, we construct a neural network having as input $z \in \mathbb{R}^d$ and the output

$$ \frac{1}{n} \sum_{i=1}^{n} \beta_{t,i} \varphi(\alpha_{t,i} z + \rho_{t,i} \zeta_t) = \int_{\mathbb{R}^d} \beta \varphi(\alpha z + \rho \zeta_t) \nu_{t,n}(d\alpha, d\beta, d\rho), \quad (1.1) $$

where $t$ denotes the index of the layer, $(\alpha, \beta, \rho)$ are the data of the neural network and

$$ \nu_{t,n} = \frac{1}{n} \sum_{i=1}^{n} \delta_{\{\alpha_{t,i}, \beta_{t,i}, \rho_{t,i}\}}. $$

Thus, setting $a = (\alpha, \beta, \rho)$ and $\psi(z, a, \zeta_t) = \beta\varphi(\alpha z + \rho\zeta_t)$, the architecture of the neural network can be written as

$$ X_{t_{k+1}}^{\nu_n, \xi, \zeta} = X_{t_k}^{\nu_n, \xi, \zeta} + \frac{\Delta t}{n} \sum_{i=1}^{n} \psi(X_{t_k}^{\nu_n, \xi, \zeta}, a_{t_k}^i, \zeta_{t_k}), $$

<footnote>*Institut de Mathématique de Bourgogne, UMR 5584 CNRS, Université Bourgogne, 21000 Dijon, France; e-mail: cristian.mendico@u-bourgogne.fr</footnote>

<footnote>†School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China; e-mail: kzwang@sjtu.edu.cn</footnote>

<footnote>‡School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China; e-mail: math_rain@sjtu.edu.cn</footnote>
where $\Delta t = t_{k+1} - t_k$. Therefore, as $\Delta t \downarrow 0$ and $n \uparrow \infty$ (heuristically) we obtain

$$X_t^{\nu, \xi, \zeta} = \xi + \int_0^t \int \psi(X_r^{\nu, \xi, \zeta}, a, \zeta_r) \nu_r da dr$$

and the training procedure of such a model can be recast as an optimization problem on the Wasserstein space

$$\min_{\nu} \int_{\Omega} g(\xi, \zeta, \nu, X_T^{\nu, \xi, \zeta}) \mathcal{M}(d\xi, d\zeta). \qquad (1.2)$$

So, looking at the minimization problem (1.4), in this work we aim to extend the possibility of studying neural network architectures by means of the relaxed Lagrangian approach to the case of mean field state equation modeling, for instance, congestion of data or sparsity constraint.

The theory of Mean Field Games (MFG) deals with the analysis of multi-agent dynamical systems involving an infinite number of rational, indistinguishable and uncooperative players. This theory was proposed by Lasry and Lions [25, 26, 27] and Huang, Malhamé and Caines [22] independently. The classical MFG Problem is formulated by a backward Hamilton-Jacobi equation and a forward Fokker-Planck equation modeling the time evolution of the players' optimal strategies, on the one hand, and the evolution of players in space according to such strategies, that is,

$$\left\{
\begin{array}{ll}
-\partial_t u(t, x) + H(x, Du(t, x), m(t)) = 0, & (t, x) \in [0, T] \times \Omega \\
\partial_t m(t) - \operatorname{div}(m(t) D_p H(x, Du(t, x), m(t))) = 0, & (t, x) \in [0, T] \times \Omega \\
u(T, x) = u_T(x), & \text{for } x \in \Omega, \quad m(0) = m_0.
\end{array}
\right.
\qquad (1.3)$$

In particular, following the classical notion of Nash equilibria, we say that the game (1.3) reaches a mean field equilibrium when all strategies and the mean field distribution are consistent, and no agent can unilaterally improve their outcome. From a variational viewpoint, such a notion of the equilibrium is equivalent to look at the minimizers of the functional

$$\int_{0}^{T} L(\gamma(t), \dot{\gamma}(t), m(t)) dt + u_{T}(\gamma(T))$$

as a function of the probability distribution $m \in C([0, T], \mathcal{P}(\Omega))$, where $L$ is the Legendre transform of $H$. Such a problem is classically solved by using the so-called Lagrangian approach, see for instance [7, 9, 19, 30]. However, the convexity assumption of the Lagrangian function, with respect to the velocity of the trajectories, is needed even if this is not always the case when considering some real life applications such as biological modeling, pricing dynamics, multi-agent reinforcement learning and deep learning ([13, 17, 18, 20, 28, 29, 32]). In this work, we address such an issue by generalizing the Lagrangian approach to non-convex mean field optimization problems taking inspiration from the well-known relaxed control approach [4, 5, 6]. Furthermore, we extend the analysis to the case of a general nonlinear controlled dynamics of the form

$$\dot{\gamma}(t) = f(\gamma(t), \alpha(t), m(t)),$$

that is, the state equation directly depends on the distribution of players. In case of stochastic systems, such an analysis has been developed in [2, 12, 13, 14, 15]. However, to the best of the authors' knowledge, no direct results are available for the first-order case. Hence, resuming we aim to solve the problem

$$\min_{\alpha: [0,T] \to \mathbb{R}^n} \int_0^T L(\gamma(t), \alpha(t), m(t)) dt + u_T(\gamma(T)) \quad \text{subject to} \quad \dot{\gamma}(t) = f(\gamma(t), \alpha(t), m(t)), \quad (1.4)$$
where the Lagrangian $L: \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d) \to \mathbb{R}$ fails to be convex with respect to the control variable. The relaxation method, we propose here, transforms the non-convex optimal control problem into an optimal control problem over probability measures on the control space. Such an approach was first proposed by Laker [24], where the author dealt with the existence of MFG equilibria for McKean-Vlasov type problems. Then, in [18] the authors used the relaxed Lagrangian approach to derive the Hamilton-Jacobi-Bellman equation and the Pontryagin optimality conditions for resNet type MFG problems. Following this, in [23] the problem of finding Pontryagin optimality conditions for bayesian neural type MFG problems was addressed. Notably, both the Lagrangian and the state equation in the above manuscripts are not directly related with the distribution of all players. In particular, once the Nash equilibria of the game is found we have, consequently, constructed the optimal neural network architecture for the data we are interested in.

The rest of this paper is organized as follows. In Section 2, we introduce the model we are interested in, the assumptions and the main results of the paper. Section 3 is devoted to recalling well-known results from measure theory which will be used throughout this paper. In Section 4, we describe the relaxation procedure in the case of the Lagrangian approach to resNet type MFG and we prove the existence of relaxed MFG equilibria. Finally, in Section 5, we show that such relaxed equilibria posses some suitable structure and that under the classical convexity assumption the new equilibria coincide with the MFG equilibria in the classical sense.

## Notations

We list some notations that will be used in this paper as follows.

* For any set $A$, denote by $\mathcal{B}(A)$ the family of Borel subsets of $A$, by $\mathcal{P}(A)$ the family of Borel probability measures on $A$, by $\mathcal{M}(A)$ the family of Borel measures on $A$. The support of a measure $P \in \mathcal{M}(A)$, denoted by $\text{spt}(P)$, is a closed set defined by

$$ \text{spt}(P) := \{x \in A \mid P(V_x) > 0 \text{ for every open neighborhood } V_x \text{ of } x\}. $$

* Denote by $\mathbb{R}$ the set of real numbers, by $\mathbb{R}^-$ the set of negative real numbers, by $\mathbb{Z}$ the set of integers, by $\mathbb{N}$ the set of positive integers. For any $n \in \mathbb{N}$, denote by $\mathbb{R}^n$ the $n$-dimensional real Euclidean space, by $\langle \cdot, \cdot \rangle$ the Euclidean scalar product in $\mathbb{R}^n$, by $| \cdot |$ the usual norm in $\mathbb{R}^n$. Denote by $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ the standard $n$-dimensional flat torus. Let $\mathcal{P}(\mathbb{T}^d)$ be endowed with the narrowly convergence. It is convenient to put a metric, i.e., the Kantorovich-Rubinstein distance $d_1$ on $\mathcal{P}(\mathbb{T}^d)$.

* Fix a constant $T > 0$. For any absolutely continuous map $\gamma : [0, T] \to \mathbb{T}^d$, denote by $\|\gamma\|_{\infty}$ the uniform norm of the map $\gamma$, i.e.,

$$ \|\gamma\|_{\infty} = \sup_{t \in [0, T]} |\gamma(t)|. $$

Denote by $\mathcal{L}_{[0,T]}$ the Lebesgue measure on $[0, T]$. For any constant $1 \le \alpha < +\infty$, denote by $\|\gamma\|_{L_{\alpha}(\mathcal{L}_{[0,T]})}$ the $L_{\alpha}$ norm of the map $\gamma$ with respect to the measure $\mathcal{L}_{[0,T]}$, i.e.,

$$ \|\gamma\|_{L_{\alpha}(\mathcal{L}_{[0,T]})} = \left( \int_{0}^{T} |\gamma(t)|^{\alpha} dt \right)^{\frac{1}{\alpha}}. $$
* Let $\Gamma_T$ be a set of absolutely continuous maps $\gamma : [0, T] \to \mathbb{T}^d$. For any $t \in [0, T]$, denote by $e_t$ the evaluation map, i.e.,

$$e_t(\gamma) = \gamma(t), \quad \forall \gamma \in \Gamma_T.$$

* The function $\omega : \mathbb{R}^+ \to \mathbb{R}^+$ is a modulus function if it is a nondecreasing and upper semi-continuous function such that $\lim_{r \to 0+} \omega(r) = 0$.

* Let A be a set. For any subset $B \subset A$, denote by $\mathbf{1}_B : A \to \{0, 1\}$ the indicator function of B, i.e.,

$$\mathbf{1}_B(x) = \begin{cases} 1 & x \in B, \\ 0 & x \notin B. \end{cases}$$

Let $x \in A$. Denote by $\delta_{\{x\}}$ the Dirac mass at point $x$.

* For any sets $N_1, N_2$ and any $P \in \mathcal{P}(N_1 \times N_2)$, denote by $\pi_1 : N_1 \times N_2 \to N_1$ the canonical projection of the first variable. Define the push-forward measure $\pi_1\sharp P \in \mathcal{P}(N_1)$ by

$$\pi_1\sharp P(B) := P(\pi_1^{-1}(B)), \quad \forall B \in \mathcal{B}(N_1).$$

Let a probability measure $\eta \in \mathcal{P}(N_1)$ and a family of probability measures $(\nu_x)_{x \in N_1} \subset \mathcal{P}(N_2)$. Denote $P = \eta \otimes \nu_x$ if these measures satisfy

$$\int_{N_1 \times N_2} g(x, y)P(dx, dy) = \int_{N_1} \int_{N_2} g(x, y)\nu_x(dy)\eta(dx), \quad \forall \text{Borel map } g : N_1 \times N_2 \to [0, +\infty].$$

Similarly, let a probability measure $\nu \in \mathcal{P}(N_2)$ and a family of probability measures $(\eta_y)_{y \in N_2} \subset \mathcal{P}(N_1)$. Denote $P = \eta_y \otimes \nu$ if these measures satisfy

$$\int_{N_1 \times N_2} g(x, y)P(dx, dy) = \int_{N_2} \int_{N_1} g(x, y)\eta_y(dx)\nu(dy), \quad \forall \text{Borel map } g : N_1 \times N_2 \to [0, +\infty].$$

* For metric spaces $(A, d_A)$ and $(B, d_B)$, denote by $AC(A, B)$ the space of absolutely continuous maps from $A$ to $B$, by $C(A, B)$ the space of continuous maps from $A$ to $B$. Denote by $C(A)$ the space of continuous functions on $A$. Denote by $C_b(A)$ the space of bounded and continuous functions on $A$. Denote by $C^k(A)$ the space of $k$-times continuously differentiable functions on $A$.

# 2 Assumptions and main results

## 2.1 Assumptions

From now on, fix three constants $p, q, p' \in \mathbb{R}$ with $q > p \ge 1$ and $q \ge p' \ge 1$.

Let $L(x, u, \nu) : \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d) \to \mathbb{R}$ be a Lagrangian satisfying the following assumptions.

(L1) For any $\nu \in \mathcal{P}(\mathbb{T}^d)$, the function $(x, u) \mapsto L(x, u, \nu)$ is of class $C^2$.

(L2) There is a modulus function $\omega : [0, +\infty) \to [0, +\infty)$ such that

$$|L(x, u, \nu_1) - L(x, u, \nu_2)| \le \omega(d_1(\nu_1, \nu_2)), \quad \forall (x, u) \in \mathbb{T}^d \times \mathbb{R}^n, \forall \nu_1, \nu_2 \in \mathcal{P}(\mathbb{T}^d).$$
(L3) There are constants $C_1, C_2, C_3 \in \mathbb{R}$ such that

$$C_2 + C_3 |u|^{p'} \le L(x, u, \nu) \le C_1 (1 + |u|^{p'}), \quad \forall (x, u, \nu) \in \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d).$$

(L4) There are constants $C_4 \in \mathbb{R}$ such that

$$\left| \frac{\partial L}{\partial x}(x, u, \nu) \right| \le C_4 (1 + |u|^q), \quad \forall (x, u, \nu) \in \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d).$$

Let $f: \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d) \to \mathbb{R}^d$ be a continuous map satisfying the following assumptions.

(F1) There is a constant $C > 0$ such that

$$|f(x, u, \nu)| \le C (1 + |x| + |u|^p), \quad \forall (x, u, \nu) \in \mathbb{T}^d \times \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d).$$

(F2) For any $(u, \nu) \in \mathbb{R}^n \times \mathcal{P}(\mathbb{T}^d)$, the map $x \mapsto f(x, u, \nu)$ is Lipschitz continuous and

$$\mathrm{Lip}_x(f) := \sup_{\substack{u \in \mathbb{R}^n, \nu \in \mathcal{P}(\mathbb{T}^d) \\ x_1 \neq x_2 \in \mathbb{T}^d}} \frac{|f(x_1, u, \nu) - f(x_2, u, \nu)|}{|x_1 - x_2|} < +\infty.$$

(F3) For any $(x, u) \in \mathbb{T}^d \times \mathbb{R}^n$, the map $\nu \mapsto f(x, u, \nu)$ is Lipschitz continuous and

$$\mathrm{Lip}_{\nu}(f) := \sup_{\substack{x \in \mathbb{T}^d, u \in \mathbb{R}^n \\ \nu_1 \neq \nu_2 \in \mathcal{P}(\mathbb{T}^d)}} \frac{|f(x, u, \nu_1) - f(x, u, \nu_2)|}{d_1(\nu_1, \nu_2)} < +\infty.$$

Fix a time horizon $T > 0$ and the initial distribution of all players $m_0 \in \mathcal{P}(\mathbb{T}^d)$. Denote by $\mathcal{P}_U$ the set of all measures $\mu \in \mathcal{M}([0, T] \times \mathbb{R}^n)$ such that there exists a family of probability measures $(\mu_t)_{t \in [0, T]} \subset \mathcal{P}(\mathbb{R}^n)$ satisfying

$$\int_0^T \int_{\mathbb{R}^n} g(t, u) \mu(dt, du) = \int_0^T \int_{\mathbb{R}^n} g(t, u) \mu_t(du) dt, \quad \forall \text{Borel function } g: [0, T] \times \mathbb{R}^n \to [0, +\infty],$$

that is,

$$\mathcal{P}_U := \left\{ \mu \in \mathcal{M}([0, T] \times \mathbb{R}^n) \mid \exists (\mu_t)_{t \in [0, T]} \subset \mathcal{P}(\mathbb{R}^n), \text{s.t. } \mu = \mathcal{L}_{[0, T]} \otimes \mu_t \right\}.$$

Then, the state equation satisfied by each player has the form

$$\left\{
\begin{array}{ll}
\dot{\gamma}(t) = \int_{\mathbb{R}^n} f(\gamma(t), u, m(t)) \mu_t(du), & t \in [0, T] \\
\gamma(0) = x, & x \in \mathbb{T}^d
\end{array}
\right.
\qquad (2.1)$$

where $\mu \in \mathcal{P}_U$ represents the strategy and $m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$ is the distribution of all players. Since the initial distribution of all players is fixed, $m(0)$ should equal $m_0$. Every player aims to find the most appropriate strategy $\mu^* \in \mathcal{P}_U$ such that the cost function, which is defined by

$$J^m(\gamma, \mu) = \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \mu(dt, du),$$
attains the minimum.

Define

$$
\Gamma_T := \left\{ \gamma \in AC([0, T], \mathbb{T}^d) \mid \begin{array}{l} \exists (x, \mu) \in \mathbb{T}^d \times \mathcal{P}_U, \exists m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d)) \text{ with } m(0) = m_0, \\ \text{s.t. } \gamma \text{ is the solution of the state equation (2.1)} \end{array} \right\}
$$

and we endow $\Gamma_T$ with the topology induced by the uniform norm $\|\cdot\|_{\infty}$. Let $m \in \mathcal{C}([0,T], \mathcal{P}(\mathbb{T}^d))$ be the map such that $m(0) = m_0$ and let $P \in \mathcal{P}(\Gamma_T \times \mathcal{P}_U)$ be a probability measure such that any state-measure pair $(\gamma, \mu) \in \text{spt}(P)$ satisfies

$$
\dot{\gamma}(t) = \int_{\mathbb{R}^n} f(\gamma(t), u, m(t)) \mu_t(du), \quad \forall t \in [0, T]. \tag{2.2}
$$

The cost functional that players aim to minimize is given by

$$
J(m, P) := \int_{\Gamma_T \times \mathcal{P}_U} J^m(\gamma, \mu) P(d\gamma, d\mu),
$$

that is, we consider the following resNet type MFG problem:

$$
\inf_P J \left( \left( e_t \sharp \pi_1 \sharp P \right)_{t \in [0, T]}, P \right),
$$

where the infimum is taken over all the measures $P \in \mathcal{P}(\Gamma_T \times \mathcal{P}_U)$ such that any state-measure pair $(\gamma, \mu) \in \text{spt}(P)$ satisfies (2.2).

2.2 Main results

Let $R$ be a positive constant. Define the subset $\mathcal{P}_U^R \subset \mathcal{P}_U$ by

$$
\mathcal{P}_U^R := \left\{ \mu \in \mathcal{P}_U \mid \int_0^T \int_{\mathbb{R}^n} |u|^q \mu(dt, du) \le R \right\},
$$

where $q$ is the positive constant defined as in Section 2.1. Define the metric $\tilde{d}_1$ on $\mathcal{P}_U$ by

$$
\tilde{d}_1(\mu_1, \mu_2) := d_1\left(\frac{\mu_1}{T}, \frac{\mu_2}{T}\right), \quad \forall \mu_1, \mu_2 \in \mathcal{P}_U.
$$

Note that $\mathcal{P}_U^R$ is a compact subset of $\mathcal{P}_U$ with respect to the $\tilde{d}_1$-topology.

For any $(x, \mu) \in \mathbb{T}^d \times \mathcal{P}_U^R$ and any $m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$ such that $m(0) = m_0$, we denote
by $\gamma$ the solution of the state equation (2.1). By Lemma 4.1 and Lemma 4.2, there exist positive
constants $K_1$ and $K_2$, independent of $x$, $\mu$ and $m$, such that

$$
\|\gamma\|_{\infty} \leq K_1, \quad \|\dot{\gamma}\|_{L_{q/p}(\mathcal{L}_{[0,T]})} \leq K_2.
$$

Since $q > p \ge 1$, there exists a positive constant $r > 1$ such that $p/q + 1/r = 1$. Define the subset $\mathcal{M}_r \subset \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$ by the family of $1/r$-Hölder-continuous maps with Hölder seminorm $K_2$, i.e.,

$$
\mathcal{M}_r := \left\{ m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d)) \mid \sup_{t_1 \neq t_2 \in [0, T]} \frac{d_1(m(t_1), m(t_2))}{|t_1 - t_2|^{1/r}} \le K_2, m(0) = m_0 \right\}.
$$
Define the metric $\hat{d}_1$ on $C([0, T], \mathcal{P}(\mathbb{T}^d))$ by

$$ \hat{d}_1(m^1, m^2) := \sup_{t \in [0, T]} d_1(m^1(t), m^2(t)), \quad \forall m^1, m^2 \in C([0, T], \mathcal{P}(\mathbb{T}^d)). $$

By Lemma 4.3, $\mathcal{M}_r$ is a compact subset of $C([0, T], \mathcal{P}(\mathbb{T}^d))$ with respect to the $\hat{d}_1$-topology.

Define the subset $\Gamma_T^R \subset \Gamma_T$ by

$$ \Gamma_T^R := \left\{ \gamma \in AC([0, T], \mathbb{T}^d) \mid \begin{array}{l} \exists (x, \mu, m) \in \mathbb{T}^d \times \mathcal{P}_U^R \times \mathcal{M}_r, \\ \text{s.t. } \gamma \text{ is the solution of the state equation (2.1)}. \end{array} \right\}. $$

Let $\Gamma_T^R$ also be endowed with the uniform norm $\| \cdot \|_{\infty}$. By Lemma 4.4, $\Gamma_T^R$ is a compact subset of $\Gamma_T$ with respect to the $\| \cdot \|_{\infty}$-topology. We denote by $\gamma = \gamma(\cdot; x, \mu, m) \in \Gamma_T^R$ the solution of the state equation (2.1) with the initial value $x \in \mathbb{T}^d$, the measure $\mu \in \mathcal{P}_U^R$ and the function $m \in \mathcal{M}_r$.

**Definition 2.1.** Let $m \in \mathcal{M}_r$. Define $\mathcal{P}_R(m)$ by the set of probability measures $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ satisfying the following conditions.

(1) $e_0\sharp(\pi_1\sharp P) = m_0$.

(2) For any $(t, v) \in [0, T] \times \mathbb{R}^d$ and any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$,

$$ \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) = 0. \quad (2.3) $$

For any $m \in \mathcal{M}_r$ and any $P \in \mathcal{P}_R(m)$, define the resNet type MFG functional by

$$ J(m, P) := \int_{\Gamma_T^R \times \mathcal{P}_U^R} J^m(\gamma, \mu) P(d\gamma, d\mu), $$

where

$$ J^m(\gamma, \mu) := \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \mu(dt, du). $$

Define the associated minimization problem by

$$ R^*(m) := \underset{P \in \mathcal{P}_R(m)}{\operatorname{argmin}} \{J(m, P)\}, \quad \forall m \in \mathcal{M}_r. $$

**Definition 2.2.** Given $m_0 \in \mathcal{P}(\mathbb{T}^d)$, we say that $P$ is a relaxed MFG equilibrium for $m_0$ if

$$ P \in R^* \left( (e_{t\sharp}(\pi_1\sharp P))_{t \in [0, T]} \right) \text{ and } e_{0\sharp}(\pi_1\sharp P) = m_0. $$

The relaxed MFG equilibrium is well defined since $(e_{t\sharp}(\pi_1\sharp P))_{t \in [0, T]} \in \mathcal{M}_r$ for any $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$. See Remark 4.2 for details.

The first main result is stated as follows.

**Theorem 1.** Assume (L1)-(L4) and (F1)-(F3). There exists at least one relaxed MFG equilibrium.
For any $(x, m) \in \mathbb{T}^d \times \mathcal{M}_r$ and any $u : [0, T] \to \mathbb{R}^n$ such that $\int_0^T |u(t)|^q dt \le R$, denote by $\gamma_x^u := \gamma_x^u(\cdot; x, \mathcal{L}_{[0,T]} \otimes \delta_{u(t)}, m)$ the solution of the state equation (2.1). Define a probability measure $\eta^u \in \mathcal{P}(\Gamma_T^R)$ by

$$ \eta^u(A) := \int_{\mathbb{T}^d} \delta_{\{\gamma_x^u\}}(A)m_0(dx), \quad \forall A \in \mathcal{B}(\Gamma_T^R). \qquad (2.4) $$

Then, the probability measure $P^u := \eta^u \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u(t)}\}}$ belongs to the set $\mathcal{P}_R(m)$ by Lemma 5.1.

**Definition 2.3.** Given $m_0 \in \mathcal{P}(\mathbb{T}^d)$, we say that $P$ is a strict relaxed MFG equilibrium for $m_0$ if it satisfies:

(i) $P$ is a relaxed MFG equilibrium for $m_0$.

(ii) There exists a function $u : [0, T] \to \mathbb{R}^n$ such that $\int_0^T |u(t)|^q dt \le R$ and

$$ P = \eta^u \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u(t)}\}}, $$

where $\eta^u$ is defined as in (2.4).

The second main result is stated as follows and, as a consequence, we immediately obtain that Theorem 1 encompasses the classical existence result for MFG equilibria.

**Theorem 2.** Let $m \in \mathcal{M}_r$ and $P \in R^*(m)$. For any $(t, x) \in [0, T] \times \mathbb{T}^d$, the set

$$ \mathcal{L}(t, x) := \left\{ (\lambda, w, v) \in \mathbb{R}^{d+2} \mid \exists u \in \mathbb{R}^n, \text{s.t. } \lambda \ge L(x, u, m(t)), w \ge |u|^q, v = f(x, u, m(t)) \right\} $$

is convex. Then the following statements hold.

(1) There exists a function $u^* : [0, T] \to \mathbb{R}^n$, such that the probability measure $P_0 := \eta^{u^*} \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u^*(t)}\}} \in R^*(m)$, where $u^*$ is the optimal control for

$$ \inf_u \int_{\mathbb{T}^d} \int_0^T L(\gamma_x^u(t), u(t), m(t)) dt m_0(dx), $$

subject to the state equation

$$ \left\{ \begin{array}{ll} \dot{\gamma}_x^u(t) = f(\gamma_x^u(t), u(t), m(t)), & t \in [0, T] \\ \gamma_x^u(0) = x, & x \in \mathbb{T}^d \end{array} \right. \qquad (2.5) $$

where the infimum is taken over all the functions $u : [0, T] \to \mathbb{R}^n$ such that $\int_0^T |u(t)|^q dt \le R$.

(2) There exists at least one strict relaxed MFG equilibrium.

**Remark 2.1.** In synthesis, according to Theorem 1 and Theorem 2, we have that all the information of the neural network architecture described in (1.1) and (1.2) is captured by the relaxed MFG equilibrium. In particular, it is on the second marginal of a relaxed MFG equilibrium that we can find the parameters of the neural network that minimize the suitable loss function with respect to the analyzed data.
## 3 Known facts on measure theory

In this section, we recall some results from measure theory that will be useful in this paper. Throughout this section, the space $(M, \rho)$ is a separable metric space.

For a sequence $\{\mu_n\}_{n=1}^\infty \subset \mathcal{P}(M)$, $\mu_n$ narrowly converges to some $\mu \in \mathcal{P}(M)$ as $n \to \infty$ if

$$ \lim_{n \to \infty} \int_M f(x) \mu_n(dx) = \int_M f(x) \mu(dx), \quad \forall f \in C_b(M). $$

For each $p \in [1, +\infty)$, the Wasserstein space of order $p$ is defined by

$$ \mathcal{P}_p(M) := \left\{ \mu \in \mathcal{P}(M) \mid \int_M \rho^p(x_0, x) \mu(dx) < +\infty \right\}, $$

where $x_0 \in M$ is an arbitrary point. The Monge-Kantorovich distance on $\mathcal{P}_p(M)$ is defined by

$$ d_p(\mu_1, \mu_2) := \inf_{\eta \in \Pi(\mu_1, \mu_2)} \left( \int_{M^2} \rho^p(x, y) \eta(dx, dy) \right)^{1/p}, \quad \forall \mu_1, \mu_2 \in \mathcal{P}_p(M), $$

where $\Pi(\mu_1, \mu_2)$ is the set of Borel probability measures on $M^2$ such that $\eta(A \times M) = \mu_1(A)$ and $\eta(M \times A) = \mu_2(A)$ for any Borel set $A \in \mathcal{B}(M)$.

As for the distance $d_1$, which is often called Kantorovich-Rubinstein distance, can be characterized by a useful duality formula (see, for instance, [11]) as

$$ d_1(\mu_1, \mu_2) = \sup \left\{ \int_M g(x) \mu_1(dx) - \int_M g(x) \mu_2(dx) \right\}, \quad \forall \mu_1, \mu_2 \in \mathcal{P}_1(M), $$

where the supremum is taken over all 1-Lipschitz functions $g : M \to \mathbb{R}$.

We now recall the relations between the narrowly convergence and the $d_p$-convergence. See [1, Theorem 7.1.5] and [31, Theorem 7.12] for examples.

**Proposition 3.1.** If a sequence of measures $\{\mu_i\}_{i=1}^\infty \subset \mathcal{P}_p(M)$ converges to some $\mu \in \mathcal{P}_p(M)$ in $d_p$-topology, then $\mu_n$ converges to $\mu$ narrowly. Conversely, if $\bigcup_{i=1}^\infty \text{spt}\{\mu_i\}$ is contained in a compact subset of $M$ and $\mu_i$ converges to $\mu$ narrowly, then $\mu_i$ converges to $\mu$ in $d_p$-topology.

**Proposition 3.2.** Let $\mu \in \mathcal{P}_p(M)$ and $\{\mu_i\}_{i=1}^\infty \subset \mathcal{P}_p(M)$. The following statements are equivalent.

(1) $d_p(\mu_i, \mu) \to 0$.

(2) $\mu_i$ converges to $\mu$ narrowly. There exists $x_0 \in M$ such that

$$ \limsup_{r \to +\infty} \sup_{i \in \mathbb{N}} \int_{\{x | \rho^p(x, x_0) \ge r\}} \rho^p(x, x_0) \mu_i(dx) = 0. $$

(3) For each continuous function $f : M \to \mathbb{R}$ satisfies that there exist $x_0 \in M$ and $c > 0$ such that

$$ |f(x)| \le c(1 + \rho^p(x, x_0)), \quad \forall x \in M, $$

we have

$$ \lim_{i \to \infty} \int_M f(x) \mu_i(dx) = \int_M f(x) \mu(dx). $$
The next theorem reveals the relation between the narrowly convergence and the almost surely convergence. See, for instance, [3, Theorem 6.7].

**Theorem 3** (Skorokhod's Representation Theorem). Let {$\mu_i$}$_{i=1}^\infty \subset \mathcal{P}(M)$ be a sequence of probability measures such that $\mu_i$ converges to some $\mu \in \mathcal{P}(M)$ narrowly. There exist M-valued random variables {$X_i$}$_{i=1}^\infty$ and $X$, whose distributions are {$\mu_i$}$_{i=1}^\infty$ and $\mu$ respectively. Then, $X_i$ converges to $X$ $\mu$-almost surely.

Based on Theorem 3, we can easily derive Corollary 3.1.

**Corollary 3.1.** Let {$\mu_i$}$_{i=1}^\infty \subset \mathcal{P}(M)$ be a sequence of probability measures such that $\mu_i$ converges to some $\mu \in \mathcal{P}(M)$ narrowly. For any lower semi-continuous and bounded function $g: M \to \mathbb{R}$, we have

$$ \liminf_{i \to \infty} \int_M g(x) \mu_i(dx) \geq \int_M g(x) \mu(dx). $$

Finally, we recall the disintegration theorem. See, for instance, [11, Theorem 8.5].

**Theorem 4** (Disintegration Theorem). Let $X$ and $Y$ be Radon separable metric spaces. Let $\mu$ be a Borel probability measure on $X$. Let $\pi: X \to Y$ be a Borel map. Define $\nu = \pi\sharp\mu \in \mathcal{P}(Y)$. Then, there exists a $\mu$-almost everywhere uniquely determined family of Borel probability measures {$\nu_y$}$_{y \in Y} \subset \mathcal{P}(X)$ such that

$$ \nu_y(X \setminus \pi^{-1}(y)) = 0, \quad \text{for } \mu-\text{ a.e. } y \in Y, \\ \int_X g(x) \mu(dx) = \int_Y \left( \int_{\pi^{-1}(y)} g(x) \nu_y(dx) \right) \nu(dy) $$

for every Borel map $g: X \to [0, +\infty]$.

# 4 Existence of relaxed MFG equilibria

## 4.1 Proof of the properties of $\mathcal{P}_R(m)$

**Lemma 4.1.** There exists a constant $K_1 > 0$ such that for any $(x, \mu) \in \mathbb{T}^d \times \mathcal{P}_U^R$ and any $m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$, the solution $\gamma$ of the state equation (2.1) satisfies

$$ \|\gamma\|_{\infty} \leq K_1. $$

*Proof.* By Hölder's inequality, for any $\mu \in \mathcal{P}_U^R$ and any $t \in [0, T]$, we have

$$ \int_0^t \int_{\mathbb{R}^n} |u|^p \mu(ds, du) \le \left( \int_0^t \int_{\mathbb{R}^n} |u|^q \mu(ds, du) \right)^{\frac{p}{q}} T^{\frac{1}{r}} \le R^{\frac{p}{q}} T^{\frac{1}{r}}, $$

where the constants $p, q$ and $r$ are defined as in Section 2. Define a constant $R' := R^{\frac{p}{q}} T^{\frac{1}{r}}$. Since $\mathbb{T}^d$ is compact, there exists a constant $\hat{C} > 0$ such that $\sup_{x \in \mathbb{T}^d} |x| \le \hat{C}$.

Since $\gamma$ is the solution of the state equation (2.1), it can be represented as

$$ \gamma(t) = x + \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du). $$
By (F1),

$$
\begin{align*}
|\gamma(t)| &\le |x| + \int_0^t \int_{\mathbb{R}^n} |f(\gamma(s), u, m(s))| \mu(ds, du) \\
&\le |x| + C \int_0^t \int_{\mathbb{R}^n} (1 + |\gamma(s)| + |u|^p) \mu(ds, du) \\
&\le \hat{C} + CT + C \int_0^t |\gamma(s)| ds + CR'.
\end{align*}
$$

By Gronwall’s inequality,

$$
|\gamma(t)| \leq (\hat{C} + CT + CR') e^{CT}.
$$

Let the constant $K_1$ equal to $(\hat{C} + CT + CR') e^{CT}$. The proof is complete.

<figure><img src="image_3.png" /></figure>

**Remark 4.1.** For any $(t, x, \mu) \in [0, T] \times \mathbb{T}^d \times \mathcal{P}_U^R$ and any $m \in C([0, T], \mathcal{P}(\mathbb{T}^d))$, denote by $\gamma$ the related solution of the state equation (2.1). We have

$$
\int_0^t \int_{\mathbb{R}^n} |u|^p \mu(ds, du) \leq R_q^{\frac{p}{q}} T^{\frac{1}{r}}
$$

and

$$
\int_0^t \int_{\mathbb{R}^n} |f(\gamma(s), u, m(s))| \mu(ds, du) \le CT + CTK_1 + CR_q^{\frac{p}{q}} T^{\frac{1}{r}}.
$$

**Lemma 4.2.** There exists a constant $K_2 > 0$ such that for any $(x, \mu) \in \mathbb{T}^d \times \mathcal{P}_U^R$ and any $m \in C([0, T], \mathcal{P}(\mathbb{T}^d))$, the solution $\gamma$ of the state equation (2.1) satisfies

$$
\|\dot{\gamma}\|_{L_{q/p}(\mathcal{L}_{[0,T]})} \leq K_2.
$$

*Proof.* Since $\gamma$ is the solution of the state equation (2.1), we have

$$
\begin{align*}
\int_0^T |\dot{\gamma}(t)|^q dt &= \int_0^T \left| \int_{\mathbb{R}^n} f(\gamma(t), u, m(t)) \mu_t(du) \right|^q dt \\
&\le \int_0^T \int_{\mathbb{R}^n} |f(\gamma(t), u, m(t))|^q \mu_t(du) dt \\
&\le \int_0^T \int_{\mathbb{R}^n} C^q (1 + |\gamma(t)| + |u|^p)^q \mu_t(du) dt \\
&\le \int_0^T \int_{\mathbb{R}^n} C^q (1 + K_1 + |u|^p)^q \mu_t(du) dt \\
&\le 2^q (1 - TC^q)^q (1 + K_1)^q + 2^q (1 - C^q)^q R.
\end{align*}
$$

Here, the second line holds by Jensen's inequality, the third one holds by (F1) and the last one
holds by (4.1) stated as follows. Define the function $h: \mathbb{R}^{+} \to \mathbb{R}^{+}$ by

$$
h(x) := x^{q/p}, \quad \forall x \in \mathbb{R}^{+}.
$$

Since the function $h$ is convex, we have

$$
\left(\frac{a+b}{2}\right)^{\frac{q}{p}} \le \frac{a^{\frac{q}{p}}+b^{\frac{q}{p}}}{2}, \quad \forall a,b \in \mathbb{R}^{+}. \tag{4.1}
$$

Let the constant $K_2$ equal to $(2^{\frac{q}{p}-1}TC^{\frac{q}{p}}(1+K_1)^{\frac{q}{p}} + 2^{\frac{q}{p}-1}C^{\frac{q}{p}}R)^{\frac{p}{q}}$. The proof is complete.

<figure><img src="image_4.png" /></figure>
**Lemma 4.3.** $\mathcal{M}_r$ is a compact subset of $\mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$ with respect to the $\hat{d}_1$-topology, where $\hat{d}_1$ is defined as in Section 2.2.

*Proof.* Since $\mathbb{T}^d$ is compact, $\mathcal{P}(\mathbb{T}^d)$ is also compact with respect to the $d_1$-topology. Thus, $\mathcal{M}_r$ is uniformly bounded. By the definition of $\mathcal{M}_r$, it is uniformly equi-Hölder-continuous. By Ascoli-Arzelà theorem, $\mathcal{M}_r$ is relatively compact. For any sequence $\{m^i\}_{i=1}^\infty \subset \mathcal{M}_r$ that converges to some $m \in \mathcal{C}([0, T], \mathcal{P}(\mathbb{T}^d))$ with respect to the $\hat{d}_1$-topology, it is clear that

$$ \lim_{i \to \infty} d_1(m^i(t), m(t)) = 0, \quad \forall t \in [0, T]. $$

For any $t_1 \neq t_2 \in [0, T]$, we have

$$ d_1(m(t_1), m(t_2)) = d_1(m^i(t_1), m(t_1)) + d_1(m^i(t_1), m^i(t_2)) + d_1(m^i(t_2), m(t_2)). $$

Let $i$ tend to infinity. We obtain that

$$ d_1(m(t_1), m(t_2)) \leq K_2 |t_1 - t_2|^{1/r}. $$

Since $\lim_{i \to \infty} d_1(m^i(0), m(0)) = d_1(m_0, m(0)) = 0$, it follows that $m(0) = m_0$. Thus, we have $m \in \mathcal{M}_r$. Therefore, $\mathcal{M}_r$ is compact with respect to the $\hat{d}_1$-topology. $\square$

**Lemma 4.4.** $\Gamma_T^R$ is a compact subset of $\Gamma_T$ with respect to the uniform norm $\| \cdot \|_\infty$.

*Proof.* By Lemma 4.1, $\Gamma_T^R$ is uniformly bounded. By Lemma 4.2, for any $s \le t \in [0, T]$, we have

$$ \begin{aligned} |\gamma(t) - \gamma(s)| &= \left| \int_s^t \dot{\gamma}(\tau) d\tau \right| \le \int_s^t |\dot{\gamma}(\tau)| d\tau \\ &\le \left( \int_s^t |\dot{\gamma}(\tau)|^{\frac{q}{p}} d\tau \right)^{\frac{p}{q}} \cdot (t-s)^{\frac{1}{r}} \le K_2 \cdot (t-s)^{\frac{1}{r}}, \end{aligned} $$

implying that $\Gamma_T^R$ is uniformly equi-Hölder-continuous. Thus, $\Gamma_T^R$ is relatively compact by Ascoli-Arzelà theorem.

Let $\{\gamma_i\}_{i=1}^\infty \subset \Gamma_T^R$ be a sequence that converges to some $\tilde{\gamma}$ in the uniform norm $\| \cdot \|_\infty$. There exist sequences $\{x_i\}_{i=1}^\infty \subset \mathbb{T}^d$, $\{m^i\}_{i=1}^\infty \subset \mathcal{M}_r$ and $\{\mu^i\}_{i=1}^\infty \subset \mathcal{P}_U^R$ such that

$$ \gamma_i(t) = x_i + \int_0^t \int_{\mathbb{R}^n} f(\gamma_i(s), u, m^i(s)) \mu_s^i(du)ds, \quad \forall t \in [0, T]. $$

Since the sets $\mathbb{T}^d$, $\mathcal{M}_r$ and $\mathcal{P}_U^R$ are compact, there exist $x \in \mathbb{T}^d$, $m \in \mathcal{M}_r$ and $\mu \in \mathcal{P}_U^R$ such that the sequences $\{x_i\}_{i=1}^\infty$, $\{m^i\}_{i=1}^\infty$ and $\{\mu^i\}_{i=1}^\infty$ converge to $x$, $m$ and $\mu$ respectively. Define $\gamma(t) := x + \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu_s(du)ds$. We claim that $\gamma = \tilde{\gamma}$. Indeed,

$$ \begin{aligned} |\gamma_i(t) - \gamma(t)| &\le |x_i - x| + \int_0^t \int_{\mathbb{R}^n} |f(\gamma_i(s), u, m^i(s)) - f(\gamma(s), u, m(s))| \mu_s^i(du)ds \\ &\quad + \int_0^t \int_{\mathbb{R}^n} |f(\gamma(s), u, m(s))| (\mu_s^i - \mu_s)(du)ds \\ &\triangleq I_1 + I_2 + I_3. \end{aligned} $$
First of all, $\lim_{i \to \infty} I_1 = 0$ obviously. By Proposition 3.2, we have $\lim_{i \to \infty} I_3 = 0$. By (F2) and (F3),

$$
\begin{align*}
I_2 &\le \int_0^t \int_{\mathbb{R}^n} |f(\gamma_i(s), u, m^i(s)) - f(\gamma(s), u, m^i(s))| \mu_s^i(du) ds \\
&\quad + \int_0^t \int_{\mathbb{R}^n} |f(\gamma(s), u, m^i(s)) - f(\gamma(s), u, m(s))| \mu_s^i(du) ds \\
&\le \int_0^t \text{Lip}_x(f) |\gamma_i(s) - \gamma(s)| + \text{Lip}_\nu(f) d_1(m^i(s), m(s)) ds \\
&\le \text{Lip}_x(f) \int_0^t |\gamma_i(s) - \gamma(s)| ds + \text{Lip}_\nu(f) T \hat{d}_1(m^i, m).
\end{align*}
$$

Since $\hat{d}_1(m^i, m) \to 0$ as $i \to \infty$, there exists a sequence of constants $\{c_i\}_{i=1}^\infty$ such that $\lim_{i \to \infty} c_i = 0$ and

$$I_1 + I_3 + \text{Lip}_\nu(f)T\hat{d}_1(m^i, m) \le c_i.$$

By Gronwall's inequality,

$$|\gamma_i(t) - \gamma(t)| \le c_i e^{\text{Lip}_x(f)t} \le c_i e^{\text{Lip}_x(f)T}.$$

In conclusion, we have $\lim_{i \to \infty} \|\gamma_i - \gamma\|_\infty = 0$. Furthermore, for any $i \in \mathbb{N}$, we obtain that

$$\|\gamma - \tilde{\gamma}\|_{\infty} \le \|\gamma - \gamma_i\|_{\infty} + \|\gamma_i - \tilde{\gamma}\|_{\infty}.$$

Let $i$ tend to infinity. The limit $\tilde{\gamma}$ of the sequence $\{\gamma_i\}_{i=1}^\infty$ is actually the curve $\gamma$ we defined above. Therefore, $\Gamma_T^R$ is compact. $\square$

**Proposition 4.1.** For each $m \in \mathcal{M}_r$, the set $\mathcal{P}_R(m)$ has the following properties.

(1) $\mathcal{P}_R(m)$ is compact with respect to the narrowly convergence.

(2) $\mathcal{P}_R(m)$ is compact with respect to the $d_1$-topology.

(3) $\mathcal{P}_R(m)$ is non-empty.

(4) For any $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$, the probability measure $P$ satisfies the condition (2.3) if and only if for each state-measure pair $(\gamma, \mu) \in \text{spt}(P)$, the pair satisfies (2.2).

*Proof.* (1) Since $\Gamma_T^R$ and $\mathcal{P}_U^R$ are compact, the set $\mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ is also compact with respect to the narrowly convergence. Let $\{P_i\}_{i=1}^\infty \subset \mathcal{P}_R(m)$ such that $P_i$ converges to some $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ narrowly. For any function $g \in \mathcal{C}_b(\mathbb{T}^d)$, we have

$$
\begin{align*}
\int_{\mathbb{T}^d} g(x) m_0(dx) &= \lim_{i \to \infty} \int_{\mathbb{T}^d} g(x) e_{0\#}(\pi_1 \sharp P_i) (dx) \\
&= \lim_{i \to \infty} \int_{\Gamma_T^R \times \mathcal{P}_U^R} g(\gamma(0)) P_i(d\gamma, d\mu) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} g(\gamma(0)) P(d\gamma, d\mu) \\
&= \int_{\mathbb{T}^d} g(x) e_{0\#}(\pi_1 \sharp P) (dx),
\end{align*}
$$
which indicates that $e_0\sharp(\pi_1\sharp P) = m_0$.

For each $(t, v) \in [0, T] \times \mathbb{R}^d$, define the function $\kappa : \Gamma_T^R \times \mathcal{P}_U^R \to \mathbb{R}$ by

$$ (\gamma, \mu) \mapsto \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle. \quad (4.2) $$

We claim that the function $\kappa$ is bounded and continuous. By Lemma 4.1, the function

$$ (\gamma, \mu) \mapsto \langle v, \gamma(t) - \gamma(0) \rangle \quad (4.3) $$

is bounded. Additionally, by Remark 4.1, the function

$$ (\gamma, \mu) \mapsto \left\langle v, \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle \quad (4.4) $$

is bounded as well. On the other hand, let $\gamma_i \to \gamma$ in $\Gamma_T^R$ and $\mu^i \to \mu$ in $\mathcal{P}_U^R$. It is obviously that the function (4.3) is continuous. Moreover, consider

$$ \begin{aligned} A_1 := & \left| \int_0^t \int_{\mathbb{R}^n} f(\gamma_i(s), u, m(s)) \, \mu^i(ds, du) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right| \\ & \le \int_0^t \int_{\mathbb{R}^n} \left| f(\gamma_i(s), u, m(s)) - f(\gamma(s), u, m(s)) \right| \, \mu^i(du, ds) \\ & \quad + \left| \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu^i(ds, du) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right| \\ & \triangleq A_2 + A_3. \end{aligned} $$

By (F2),

$$ \lim_{i \to \infty} A_2 \le \lim_{i \to \infty} \int_0^t \int_{\mathbb{R}^n} \mathrm{Lip}_x(f) \|\gamma_i - \gamma\|_{\infty} \mu^i(du, ds) \le \lim_{i \to \infty} \mathrm{Lip}_x(f) T \|\gamma_i - \gamma\|_{\infty} = 0. $$

By (F1) and Proposition 3.2, $\lim_{i \to \infty} A_3 = 0$. In conclusion, the function $\kappa$ is both bounded and continuous.

For any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$, we have

$$ \begin{aligned} & \left| \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle P(d\gamma, d\mu) \right| \\ & \le \int_{\Gamma_T^R \times \mathcal{P}_U^R} \mathbf{1}_N(\gamma, \mu) \left| \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle \right| P(d\gamma, d\mu) \\ & \le \liminf_{i \to \infty} \int_{\Gamma_T^R \times \mathcal{P}_U^R} \mathbf{1}_N(\gamma, \mu) \left| \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle \right| P_i(d\gamma, d\mu) \\ & = \liminf_{i \to \infty} \int_{N^+} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) \\ & \quad - \int_{N^-} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) = 0, \end{aligned} $$

where the sets $N^+$ and $N^-$ are defined by

$$ N^+ := \{(\gamma, \mu) \in N \mid \kappa(\gamma, \mu) > 0\}, $$
$$N^{-} := \{(\gamma, \mu) \in N \mid \kappa(\gamma, \mu) < 0\}.$$

Since the function $\kappa$ is continuous, both $N^+$ and $N^-$ are open sets. In conclusion, $P \in \mathcal{P}_R(m)$, and $\mathcal{P}_R(m)$ is closed. Furthermore, since $\mathcal{P}_R(m) \subset \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ and $\mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ is compact, the set $\mathcal{P}_R(m)$ is also compact with respect to the narrowly convergence.

(2) This point is a direct consequence of the point (1).

(3) Let $\mu_0 = \mathcal{L}_{[0,T]} \otimes \delta_{\{0\}} \in \mathcal{P}_U^R$. The state equation (2.1) is transformed into an autonomous ordinary differential equation

$$\begin{cases} \dot{\gamma}(t) = f(\gamma(t), 0, m(t)), \\ \gamma(0) = x. \end{cases} \qquad (4.5)$$

For any $x \in \mathbb{T}^d$, there exists a unique solution $\gamma_x(t) = \gamma_x(t; x, \mu_0, m)$ of (4.5) by (F2). Define the map $p: \mathbb{T}^d \to \Gamma_T^R$ by

$$x \mapsto p(x) = \gamma_x.$$

Define $P := p_\sharp m_0 \otimes \delta_{\{\mu_0\}} \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$. For any function $g \in C_b(\mathbb{T}^d)$, we have

$$\begin{aligned} \int_{\mathbb{T}^d} g(x) e_0\sharp (\pi_1\sharp P) (dx) &= \int_{\Gamma_T^R} g(\gamma(0)) \pi_1\sharp P (d\gamma) \\ &= \int_{\Gamma_T^R} g(\gamma(0)) p_\sharp m_0 (d\gamma) \\ &= \int_{\mathbb{T}^d} g(p(x)(0)) m_0 (dx) \\ &= \int_{\mathbb{T}^d} g(x) m_0 (dx), \end{aligned}$$

which indicates that $e_0\sharp(\pi_1\sharp P) = m_0$.

For any $(t, v) \in [0, T] \times \mathbb{R}^d$ and any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$,

$$\begin{aligned} & \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) \\ &= \int_{\pi_1(N)} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t f(\gamma(s), 0, m(s)) ds \right\rangle p_\sharp m_0 (d\gamma) \\ &= \int_{p^{-1}(\pi_1(N))} \left\langle v, p(x)(t) - p(x)(0) - \int_0^t f(p(x)(s), 0, m(s)) ds \right\rangle m_0 (dx) \\ &= \int_{p^{-1}(\pi_1(N))} \left\langle v, p(x)(t) - p(x)(0) - \int_0^t \frac{\partial p(x)(s)}{\partial s} ds \right\rangle m_0 (dx) = 0, \end{aligned}$$

where the set $p^{-1}(\pi_1(N))$ is defined by

$$p^{-1}(\pi_1(N)) := \{x \in \mathbb{T}^d \mid p(x) \in \pi_1(N)\}.$$

Above all, we have $P \in \mathcal{P}_R(m)$. Therefore, $\mathcal{P}_R(m)$ is non-empty.

(4) Let $P$ be the probability measure that satisfies the condition (2.3). Suppose there exist $(\gamma_0, \mu^0) \in \text{spt}(P)$ and $t_0 \in [0, T]$ such that

$$\gamma_0(t_0) - \gamma_0(0) \neq \int_0^{t_0} \int_{\mathbb{R}^n} f(\gamma_0(s), u, m(s)) \mu^0(ds, du).$$
Without loss of generality, assume there exists $v_0 \in \mathbb{R}^d$ such that

$$ \left\langle v_0, \gamma_0(t_0) - \gamma_0(0) - \int_0^{t_0} \int_{\mathbb{R}^n} f(\gamma_0(s), u, m(s)) \mu^0(ds, du) \right\rangle > 0. $$

Similar to the proof of the continuity of $\kappa$, the map

$$ (\gamma, \mu) \mapsto \left\langle v_0, \gamma(t_0) - \gamma(0) - \int_0^{t_0} \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle $$

is also continuous. Hence, there exists an open neighborhood $N$ of the point $(\gamma_0, \mu^0)$ such that $P(N) > 0$ and

$$ \int_N \left\langle v_0, \gamma(t_0) - \gamma(0) - \int_0^{t_0} \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) > 0, $$

which contradicts (2.3). Therefore, we have

$$ \gamma(t) - \gamma(0) = \int_{0}^{t} \int_{\mathbb{R}^{n}} f(\gamma(s), u, m(s)) \mu(ds, du), \quad \forall (\gamma, \mu) \in \text{spt}(P), \forall t \in [0, T]. $$

Taking the derivative with respect to $t$ on both sides gives us (2.2).

The other point can be easily proved by definition. $\square$

## 4.2 Proof of the existence of relaxed MFG equilibria

In this section, we prove the existence of relaxed MFG equilibria by Kakutani's theorem.

Define a subset $\mathcal{P}_0(\Gamma_T^R) \subset \mathcal{P}(\Gamma_T^R)$ by

$$ \mathcal{P}_0(\Gamma_T^R) := \{\eta \in \mathcal{P}(\Gamma_T^R) \mid e_0\#\eta = m_0\}. $$

Obviously, $\mathcal{P}_0(\Gamma_T^R)$ is compact with respect to the $d_1$-topology. Define the set-valued map by

$$ E : (\mathcal{P}_0(\Gamma_T^R), d_1) \Rightarrow (\mathcal{P}_0(\Gamma_T^R), d_1), \quad \eta \mapsto E(\eta), $$

where

$$ E(\eta) := \left\{ \pi_1 \# P \mid P \in \mathcal{R}^* \left( (e_t \# \eta)_{t \in [0, T]} \right) \right\}. $$

**Remark 4.2.** Let $\eta \in \mathcal{P}_0(\Gamma_T^R)$. Define the map $m : [0, T] \to \mathcal{P}(\mathbb{T}^d)$ by $m(t) = e_t \# \eta$ for any $t \in [0, T]$. We claim that $m \in \mathcal{M}_r$.

More precisely, it is clear that $m(0) = e_0 \# \eta = m_0$. For any $t_1 \neq t_2 \in [0, T]$ and any 1-Lipschitz function $g : \mathbb{T}^d \to \mathbb{R}$, we obtain that

$$ \begin{aligned} & \left| \int_{\mathbb{T}^d} g(x) e_{t_1} \# \eta(dx) - \int_{\mathbb{T}^d} g(x) e_{t_2} \# \eta(dx) \right| \\ & \leq \int_{\Gamma_T^R} |g(\gamma(t_1)) - g(\gamma(t_2))| \eta(d\gamma) \\ & \leq \int_{\Gamma_T^R} |\gamma(t_1) - \gamma(t_2)| \eta(d\gamma) \\ & \leq K_2 |t_1 - t_2|^{\frac{1}{r}}, \end{aligned} $$
where the last inequality comes from Lemma 4.2. By the arbitrariness of the function g, we have

$$d_1(m(t_1), m(t_2)) \leq K_2 |t_1 - t_2|^{\frac{1}{r}}.$$

Thus, $m \in \mathcal{M}_r$. Therefore, the set-valued map E is well-defined.

**Lemma 4.5.** For each $m \in \mathcal{M}_r$,

(1) the family of functions $\{J^m(\cdot, \mu) : \Gamma_T^R \to \mathbb{R} \mid \mu \in \mathcal{P}_U^R\}$ is uniformly equi-continuous.

(2) Let $\gamma \in \Gamma_T^R$. If $\mu^i$ converges to $\mu$ narrowly, then

$$\liminf_{i \to \infty} J^m(\gamma, \mu^i) \geq J^m(\gamma, \mu). \qquad (4.6)$$

*Proof.* (1) Fix a sequence $\{\gamma_i\}_{i=1}^\infty$ that converges to some $\gamma$ in $\Gamma_T^R$. For any $\mu \in \mathcal{P}_U^R$, we have

$$\begin{aligned} |J^m(\gamma_i, \mu) - J^m(\gamma, \mu)| &\leq \int_0^T \int_{\mathbb{R}^n} |L(\gamma_i(t), u, m(t)) - L(\gamma(t), u, m(t))| \mu(dt, du) \\ &\leq \|\gamma_i - \gamma\|_{\infty} \int_0^T \int_{\mathbb{R}^n} \int_0^1 \left| \frac{\partial L}{\partial x}(\gamma(t) + \lambda(\gamma_i(t) - \gamma(t)), u, m(t)) \right| d\lambda \mu(dt, du). \end{aligned}$$

By (L4), we obtain that

$$\int_0^T \int_{\mathbb{R}^n} \int_0^1 \left| \frac{\partial L}{\partial x}(\gamma(t) + \lambda(\gamma_i(t) - \gamma(t)), u, m(t)) \right| d\lambda \mu(dt, du) \leq C_4 (T + R),$$

which completes the proof.

(2) For any $\epsilon > 0$, we obtain that

$$\begin{aligned} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \mu^i(dt, du) &= \int_0^T \int_{\mathbb{R}^n} \frac{L(\gamma(t), u, m(t))}{1 + \epsilon|u|^{p'}} (1 + \epsilon|u|^{p'}) \mu^i(dt, du) \\ &\geq \int_0^T \int_{\mathbb{R}^n} \frac{L(\gamma(t), u, m(t))}{1 + \epsilon|u|^{p'}} \mu^i(dt, du). \end{aligned}$$

By (L1) and (L3), the map

$$(t, u) \mapsto \frac{L(\gamma(t), u, m(t))}{1 + \epsilon|u|^{p'}}$$

is bounded and continuous. Thus,

$$\liminf_{n \to \infty} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \mu^i(dt, du) \geq \int_0^T \int_{\mathbb{R}^n} \frac{L(\gamma(t), u, m(t))}{1 + \epsilon|u|^{p'}} \mu(dt, du).$$

Let $\epsilon$ tend to 0. The proof is complete.

<figure><img src="image_5.png" /></figure>

For any $x \in \mathbb{T}^d$, denote by $\Gamma_m^*(x)$ the set of curves associated with an optimal control $\mu^*$, i.e.,

$$\Gamma_m^*(x) := \left\{ \gamma^* \in \Gamma_T^R \mid \gamma^* = \gamma^*(\cdot; x, \mu^*, m), J^m(\gamma^*, \mu^*) = \inf_{\mu \in \mathcal{P}_U^R, \gamma = \gamma(\cdot; x, \mu, m)} J^m(\gamma, \mu) \right\}.$$

Define the map $F_m : \mathbb{T}^d \to \mathcal{B}(\Gamma_T^R)$ by

$$F_m(x) := \Gamma_m^*(x), \quad \forall x \in \mathbb{T}^d.$$
**Lemma 4.6.** Let $m \in \mathcal{M}_r$. Let the sequence $\{x_i\}_{i=1}^\infty$ converge to $x$ in $\mathbb{T}^d$. If there exists $\gamma_{x_i} \in F_m(x_i)$ for each $i \in \mathbb{N}$ such that the sequence $\{\gamma_{x_i}\}_{i=1}^\infty$ converges to some $\gamma$ in $\Gamma_T^R$, then $\gamma \in F_m(x)$.

*Proof.* Denote by $\mu_{x_i}$ the measure such that $\gamma_{x_i} = \gamma_{x_i}(\cdot; x_i, \mu_{x_i}, m)$ for any $i \in \mathbb{N}$. Due to the compactness of $\mathcal{P}_U^R$, there exists a measure $\bar{\mu}$ such that $\bar{d}_1(\mu_{x_i}, \bar{\mu})$ converges to 0 as $i \to \infty$. Let $\tilde{\gamma} = \tilde{\gamma}(\cdot; x, \tilde{\mu}, m)$ and $\tilde{\gamma}_{x_i} = \tilde{\gamma}_{x_i}(\cdot; x_i, \tilde{\mu}, m)$ for any $\tilde{\mu} \in \mathcal{P}_U^R$. Since $x_i$ converges to $x$ in $\mathbb{T}^d$, we have $\tilde{\gamma}_{x_i}$ converges to $\tilde{\gamma}$ in $\Gamma_T^R$. Moreover, since $\gamma_{x_i} \in \Gamma_m^*(x_i)$, we obtain that

$$J^m(\gamma_{x_i}, \mu_{x_i}) \leq J^m(\tilde{\gamma}_{x_i}, \tilde{\mu}). \qquad (4.7)$$

Let $i$ tend to infinity. By Lemma 4.5, we have $J^m(\gamma, \bar{\mu}) \leq J^m(\tilde{\gamma}, \tilde{\mu})$.

The only thing left to prove is that $\gamma = \gamma(\cdot; x, \bar{\mu}, m)$, i.e.,

$$\gamma(t) = x + \int_{0}^{t} \int_{\mathbb{R}^{n}} f(\gamma(s), u, m(s)) \bar{\mu}(ds, du).$$

Since $\gamma_{x_i}(t) = x_i + \int_0^t \int_{\mathbb{R}^n} f(\gamma_{x_i}(s), u, m(s)) \mu_{x_i}(ds, du)$, the proof is complete by (F1), (F2) and Proposition 3.2. $\square$

**Proposition 4.2.** For each $m \in \mathcal{M}_r$,

(1) the set $\Gamma_m^*(x)$ is non-empty for any $x \in \mathbb{T}^d$.

(2) the map $F_m$ has a closed graph.

*Proof.* The non-emptiness of $\Gamma_m^*(x)$ can be easily proved by the convexity of $J^m(\gamma, \mu)$ with respect to $\mu$. See [10] for examples. Point (2) is a direct consequence of Lemma 4.6. $\square$

**Proposition 4.3.** For any $\eta \in \mathcal{P}_0(\Gamma_T^R)$, the set $E(\eta)$ is non-empty and convex.

*Proof.* Let $m(t) = e_t^\sharp \eta$ for any $t \in [0, T]$. By Proposition 4.2 and [8, Proposition 9.5], the map $F_m$ is measurable. Thus, there exists a measurable selection $\tilde{\gamma}_x \in \Gamma_m^*(x)$ by [16, Chapter3, Theorem 5.3]. Define the measure $\tilde{\eta} \in \mathcal{P}_0(\Gamma_T^R)$ by

$$\tilde{\eta}(A) := \int_{\mathbb{T}^d} \delta_{\{\tilde{\gamma}_x\}}(A)m_0(dx), \quad \forall A \in \mathcal{B}(\Gamma_T^R).$$

We claim that $\tilde{\eta} \in E(\eta)$. Define the map $h: \{\tilde{\gamma}_x | x \in \mathbb{T}^d\} \to \Gamma_T^R \times \mathcal{P}_U^R$ by

$$\tilde{\gamma}_x \mapsto (\tilde{\gamma}_x, \mu^x), \text{ where } \dot{\tilde{\gamma}}_x(t) = \int_{\mathbb{R}^n} f(\tilde{\gamma}_x(t), u, m(t)) \mu_t^x(du), \quad \forall t \in [0, T].$$

It is clear that $\tilde{\eta} = \pi_1^\sharp(h^\sharp \tilde{\eta})$. More precisely, for any function $g \in \mathcal{C}_b(\Gamma_T^R)$, we have

$$\begin{align*}
\int_{\Gamma_T^R} g(\gamma) \pi_1^\sharp(h^\sharp \tilde{\eta}) (d\gamma) &= \int_{\Gamma_T^R} g(\gamma) \mathbf{1}_{\{\tilde{\gamma}_x | x \in \mathbb{T}^d\}}(\gamma) \tilde{\eta}(d\gamma) \\
&= \int_{\mathbb{T}^d} \int_{\Gamma_T^R} \mathbf{1}_{\{\tilde{\gamma}_x | x \in \mathbb{T}^d\}}(\gamma) g(\gamma) \delta_{\{\tilde{\gamma}_x\}}(d\gamma) m_0(dx) \\
&= \int_{\mathbb{T}^d} \int_{\Gamma_T^R} g(\gamma) \delta_{\{\tilde{\gamma}_x\}}(d\gamma) m_0(dx) \\
&= \int_{\Gamma_T^R} g(\gamma) \tilde{\eta}(d\gamma).
\end{align*}$$
Hence, we only need to verify that $h\sharp\tilde{\eta} \in R^*(m)$. Indeed, we have $h\sharp\tilde{\eta} \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ and $e_0\sharp\tilde{\eta} = m_0$ by definition. For any pair $(\gamma, \mu) \in \text{spt}(h\sharp\tilde{\eta})$, the pair satisfies $\dot{\gamma} = \gamma(\cdot; \gamma(0), \mu, m)$. By Proposition 4.1(4), we obtain that $h\sharp\tilde{\eta} \in \mathcal{P}_R(m)$.

Regarding the minimization property of $h\sharp\tilde{\eta}$, we first define $\Gamma_T^R(m) \subset \Gamma_T^R$ by

$$ \Gamma_T^R(m) := \left\{ \gamma \in \Gamma_T^R \mid \exists (x, \mu) \in \mathbb{T}^d \times \mathcal{P}_U^R, \text{s.t. } \gamma = \gamma(\cdot; x, \mu, m) \right\}. $$

By Proposition 4.1(4), for any $\tilde{P} \in \mathcal{P}_R(m)$, the support of $\tilde{P}$ satisfies that $\text{spt}(\tilde{P}) \subset \Gamma_T^R(m) \times \mathcal{P}_U^R$. Thus, we obtain that

$$
\begin{align*}
J(m, h^{\sharp} \tilde{\eta}) &= \int_{\Gamma_T^R \times \mathcal{P}_U^R} J^m(\gamma, \mu) h^{\sharp} \tilde{\eta}(d\gamma, d\mu) \\
&= \int_{\mathbb{T}^d} \int_{\Gamma_T^R} J^m(h(\gamma)) \delta_{\{\tilde{\gamma}_x\}}(d\gamma) m_0(dx) \\
&= \int_{\mathbb{T}^d} J^m(\tilde{\gamma}_x, \mu^x) m_0(dx) \\
&= \int_{\mathbb{T}^d} J^m(\tilde{\gamma}_x, \mu^x) e_0^{\sharp} (\pi_1^{\sharp} \tilde{P}) (dx) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} J^m(\tilde{\gamma}_{\gamma(0)}, \mu^{\gamma(0)}) \tilde{P}(d\gamma, d\mu) \\
&= \int_{\Gamma_T^R(m) \times \mathcal{P}_U^R} J^m(\tilde{\gamma}_{\gamma(0)}, \mu^{\gamma(0)}) \tilde{P}(d\gamma, d\mu) \\
&\le \int_{\Gamma_T^R(m) \times \mathcal{P}_U^R} J^m(\gamma, \mu) \tilde{P}(d\gamma, d\mu) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} J^m(\gamma, \mu) \tilde{P}(d\gamma, d\mu) = J(m, \tilde{P}).
\end{align*}
$$

Therefore, the measure $\tilde{\eta} \in E(\eta)$, which implies the non-emptiness of $E(\eta)$.

The convexity can be easily proved by definition. $\square$

**Proposition 4.4.** Let $\{\eta_i\}_{i=1}^\infty \subset \mathcal{P}_0(\Gamma_T^R)$ and $\eta \in \mathcal{P}_0(\Gamma_T^R)$ such that $\eta_i$ converges to $\eta$ narrowly. If there exists $\hat{\eta}_i \in E(\eta_i)$ for any $i \in \mathbb{N}$ such that $\hat{\eta}_i$ converges to some $\hat{\eta}$ narrowly, then $\hat{\eta} \in E(\eta)$.

*Proof.* For any $t \in [0, T]$, define $m^i(t) = e_t^{\sharp} \eta_i$ and $m(t) = e_t^{\sharp} \eta$. We first claim that the function $J^{m^i}(\gamma, \mu) : \Gamma_T^R \times \mathcal{P}_U^R \to \mathbb{R}$ uniformly converges to the function $J^m(\gamma, \mu) : \Gamma_T^R \times \mathcal{P}_U^R \to \mathbb{R}$ as $i \to \infty$. By (L2),

$$
\begin{align*}
|J^{m^i}(\gamma, \mu) - J^m(\gamma, \mu)| &\le \int_0^T \int_{\mathbb{R}^n} |L(\gamma(t), u, m^i(t)) - L(\gamma(t), u, m(t))| \mu_t(du)dt \\
&\le \int_0^T \int_{\mathbb{R}^n} \omega(d_1(m^i(t), m(t))) \mu_t(du)dt \\
&= \int_0^T \omega(d_1(m^i(t), m(t))) dt.
\end{align*}
$$
Indeed, since $\eta_i$ converges to $\eta$ narrowly and $\Gamma_T^R$ is compact, we have $d_1(\eta_i, \eta) \to 0$. Furthermore,

$$
\begin{align*}
d_1(m^i(t), m(t)) &= \sup \left\{ \left| \int_{\mathbb{T}^d} g(x)m^i(t)(dx) - \int_{\mathbb{T}^d} g(x)m(t)(dx) \right| \middle| g(x) \text{ is 1-Lip function w.r.t } x \right\} \\
&\le \sup \left\{ \left| \int_{\Gamma_T^R} g(\gamma(t))\eta_i(d\gamma) - \int_{\Gamma_T^R} g(\gamma(t))\eta(d\gamma) \right| \middle| g(\gamma(t)) \text{ is 1-Lip function w.r.t } \gamma \right\} \\
&= d_1(\eta_i, \eta).
\end{align*}
$$

We obtain that

$$
|J^{m^i}(\gamma, \mu) - J^m(\gamma, \mu)| \leq T\omega(d_1(\eta_i, \eta)). \quad (4.8)
$$

Thus, the claim is established. By (4.8), the function $\tilde{P} \mapsto J(m^i, \tilde{P})$ also uniformly converges to the function $\tilde{P} \mapsto J(m, \tilde{P})$ as $i \to \infty$, i.e.,

$$
\lim_{i \to \infty} \sup_{\tilde{P} \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)} \left| J(m^i, \tilde{P}) - J(m, \tilde{P}) \right| = 0. \quad (4.9)
$$

For any $i \in \mathbb{N}$, since $\hat{\eta}_i \in E(\eta_i)$, there exists $P_i \in \mathcal{P}_R(m^i)$ such that $\pi_1\sharp P_i = \hat{\eta}_i$. Since $\mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ is compact with respect to the $d_1$-topology, there exists $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ such that $d_1(P_i, P) \to 0$ as $i \to \infty$. It is obviously that $\pi_1\sharp P = \hat{\eta}$. More precisely, for any function $g_1 \in \mathcal{C}_b(\Gamma_T^R)$, we have

$$
\begin{align*}
\int_{\Gamma_T^R} g_1(\gamma) \pi_1 \sharp P(d\gamma) &= \int_{\Gamma_T^R \times \mathcal{P}_U^R} g_1(\gamma) P(d\gamma, d\mu) \\
&= \lim_{i \to \infty} \int_{\Gamma_T^R \times \mathcal{P}_U^R} g_1(\gamma) P_i(d\gamma, d\mu) \\
&= \lim_{i \to \infty} \int_{\Gamma_T^R} g_1(\gamma) \hat{\eta}_i(d\gamma) \\
&= \int_{\Gamma_T^R} g_1(\gamma) \hat{\eta}(d\gamma).
\end{align*}
$$

Moreover, we have $e_0\sharp\pi_1\sharp P = e_0\sharp\hat{\eta} = m_0$ by definition. For any function $g_2 \in \mathcal{C}_b(\mathbb{T}^d)$, we obtain that

$$
\begin{align*}
\int_{\mathbb{T}^d} g_2(x) e_0 \sharp \hat{\eta}(dx) &= \int_{\Gamma_T^R} g_2(\gamma(0)) \hat{\eta}(d\gamma) \\
&= \lim_{i \to \infty} \int_{\Gamma_T^R} g_2(\gamma(0)) \hat{\eta}_i(d\gamma) \\
&= \lim_{i \to \infty} \int_{\mathbb{T}^d} g_2(x) e_0 \sharp \hat{\eta}_i(dx) \\
&= \lim_{i \to \infty} \int_{\mathbb{T}^d} g_2(x) e_0 \sharp \pi_1 \sharp P_i(dx) = \int_{\mathbb{T}^d} g_2(x) m_0(dx).
\end{align*}
$$

For any $(t, v) \in [0, T] \times \mathbb{R}^d$ and any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$, consider

$$
\left| \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m^i(s)) \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) - \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) \right| \le B_1 + B_2,
$$
where

$$B_1 := \left| \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m^i(s)) \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) - \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) \right|,$$

$$B_2 := \left| \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) - \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) \right|.$$

Similar to the proof of Proposition 4.1(1), we have $\lim_{i \to \infty} B_2 = 0$. At the same time, since

$$B_1 = \left| \int_N \left\langle v, \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) - f(\gamma(s), u, m^i(s)) \mu(ds, du) \right\rangle P_i(d\gamma, d\mu) \right| \\ \le \int_N |v| \int_0^t \mathrm{Lip}_\nu(f) d_1(\eta_i, \eta) ds P_i(d\gamma, d\mu) \\ \le |v| \mathrm{Lip}_\nu(f) T d_1(\eta_i, \eta),$$

we have $\lim_{i \to \infty} B_1 = 0$. Since $P_i \in \mathcal{P}_R(m^i)$, we obtain that $P \in \mathcal{P}_R(m)$.

Consider

$$J(m^i, P_i) - J(m, P) = (J(m^i, P_i) - J(m, P_i)) + (J(m, P_i) - J(m, P)) \triangleq B_3 + B_4.$$

By (4.9), $\lim_{i \to \infty} B_3 = 0$. Since the function $(\gamma, \mu) \mapsto J^m(\gamma, \mu)$ is bounded and lower semi-continuous by Lemma 4.5, it follows from Lemma 3.1 that $\liminf_{i \to \infty} B_4 \ge 0$.

In conclusion, for any $\hat{P} \in \mathcal{P}_R(m)$, we have

$$J(m, P) \le \liminf_{i \to \infty} J(m^i, P_i) \le \liminf_{i \to \infty} J(m^i, \hat{P}) = J(m, \hat{P}).$$

Due to the arbitrariness of $\hat{P} \in \mathcal{P}_R(m)$, we complete the proof. $\square$

**Corollary 4.1.** For any $\eta \in \mathcal{P}_0(\Gamma_T^R)$, the set $E(\eta)$ is compact.

*Proof.* By Proposition 4.4, the set-valued map $E$ has a closed graph, which ensures that $E(\eta)$ is a closed set. Furthermore, $E(\eta)$ is contained within the compact set $\mathcal{P}_0(\Gamma_T^R)$. Consequently, $E(\eta)$ is also compact. $\square$

*Proof of Theorem 1.* Combining Proposition 4.3 and Corollary 4.1, the set $E(\eta)$ is non-empty, convex and compact for any $\eta \in \mathcal{P}_0(\Gamma_T^R)$. Furthermore, $\mathcal{P}_0(\Gamma_T^R)$ is non-empty and compact. By Proposition 4.4, the set-valued map $E$ has a closed graph, which indicates the continuity of the map. Consequently, there exists a measure $\bar{\eta} \in \mathcal{P}_0(\Gamma_T^R)$, such that $\bar{\eta} \in E(\bar{\eta})$ by Kakutani's theorem. Thus, there exists $P \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ such that $\bar{\eta} = \pi_1 \sharp P$ and $P \in R^*((e_t \sharp \bar{\eta})_{t \in [0,T]})$. Therefore, the probability measure $P$ belongs to the set $R^*((e_t \sharp \pi_1 \sharp P)_{t \in [0,T]})$, indicating that $P$ is a relaxed MFG equilibrium. $\square$
5 Existence of strict relaxed MFG equilibria

**Lemma 5.1.** Fix $m \in \mathcal{M}_r$. For any $x \in \mathbb{T}^d$ and any $u : [0, T] \to \mathbb{R}^n$ such that $\int_0^T |u(t)|^q dt \le R$, denote by $\gamma_x^u = \gamma_x^u(\cdot; x, \mathcal{L}_{[0,T]} \otimes \delta_{u(t)}, m)$ the solution of the state equation (2.1) when the measure $\mathcal{L}_{[0,T]} \otimes \delta_{u(t)} \in \mathcal{P}_U^R$. Define a probability measure $\eta^u \in \mathcal{P}_0(\Gamma_T^R)$ by

$$ \eta^u(A) := \int_{\mathbb{T}^d} \delta_{\{\gamma_x^u\}}(A) m_0(dx), \quad \forall A \in \mathcal{B}(\Gamma_T^R). $$

Then, the probability measure $P^u := \eta^u \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u(t)}\}}$ belongs to the set $\mathcal{P}_R(m)$.

*Proof.* It is clear that $P^u \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ and $e_0\sharp(\pi_1\sharp P^u) = m_0$. Moreover, for any $(t, v) \in [0, T] \times \mathbb{R}^d$ and any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$, we can deduce that

$$
\begin{align*}
& \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \, \mu(ds, du) \right\rangle P^u (d\gamma, d\mu) \\
&= \int_{\pi_1(N)} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t f(\gamma(s), u(s), m(s)) \, ds \right\rangle \eta^u (d\gamma) \\
&= \int_{\mathbb{T}^d} \int_{\pi_1(N)} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t f(\gamma(s), u(s), m(s)) \, ds \right\rangle \delta_{\{\gamma_x^u\}} (d\gamma) m_0(dx) \\
&= \int_{\{x | \gamma_x^u \in \pi_1(N)\}} \left\langle v, \gamma_x^u(t) - \gamma_x^u(0) - \int_0^t f(\gamma_x^u(s), u(s), m(s)) \, ds \right\rangle m_0(dx) \\
&= \int_{\{x | \gamma_x^u \in \pi_1(N)\}} \left\langle v, \gamma_x^u(t) - \gamma_x^u(0) - \int_0^t \dot{\gamma}_x^u(s) ds \right\rangle m_0(dx) = 0.
\end{align*}
$$

Therefore, the probability measure $P^u$ belongs to the set $\mathcal{P}_R(m)$. $\square$

**Proposition 5.1.** Fix $m \in \mathcal{M}_r$ and $P \in R^*(m)$. There exists a map $\hat{q} : [0, T] \times \mathbb{T}^d \to \mathcal{P}(\mathbb{R}^n)$ such that $\hat{P} := \pi_1\sharp P \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \hat{q}_{t,\gamma(t)}\}} \in R^*(m)$, where we denote $\hat{q}_{t,x} = \hat{q}(t, x)$ for convenience.

*Proof.* Define a measure $\eta$ on $[0, T] \times \mathbb{T}^d \times \mathbb{R}^n$ by

$$ \eta(C) := \frac{1}{T} \int_{\Gamma_T^R \times \mathcal{P}_U^R} \int_0^T \int_{\mathbb{R}^n} \mathbf{1}_C(t, \gamma(t), u) \, \mu(dt, du) P(d\gamma, d\mu), \quad \forall C \in \mathcal{B}([0, T] \times \mathbb{T}^d \times \mathbb{R}^n). $$

Define $\hat{\eta}_t := e_t\sharp(\pi_1\sharp P)$, which is a probability measure on $\mathbb{T}^d$. Define $\eta_{1,2} := \frac{1}{T}\mathcal{L}_{[0,T]} \otimes \hat{\eta}_t(dx)$, which is a probability measure on $[0, T] \times \mathbb{T}^d$. By Theorem 4, construct the map $\hat{q}$ by

$$ \eta(dt, dx, du) = \eta_{1,2}(dt, dx) \otimes \hat{q}_{t,x}(du). $$

Define the measure $\hat{P}$ by $\hat{P} := \pi_1\sharp P \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \hat{q}_{t,\gamma(t)}\}}$.

Define the function $g : \Gamma_T^R \to \mathbb{R}$ by

$$ \gamma \mapsto \int_0^T \int_{\mathbb{R}^n} |u|^q \hat{q}_{t,\gamma(t)}(du) dt, $$
which is a positive measurable function. For any Borel subset $V \subset \Gamma_T^R \times \mathcal{P}_U^R$, there always exists a measure $\mu_V \in \mathcal{P}_U^R$ such that

$$ \int_{\Gamma_T^R \times \mathcal{P}_U^R} \int_0^T \int_{\mathbb{R}^n} \mathbf{1}_V(\gamma, \mu) |u|^q \mu(dt, du) P(d\gamma, d\mu) \\ \leq P(V) \int_0^T \int_{\mathbb{R}^n} |u|^q \mu_V(dt, du) \leq P(V)R. $$

Moreover, we have

$$ \begin{aligned} & \int_{\Gamma_T^R} \mathbf{1}_{\pi_1(V)}(\gamma) g(\gamma) \pi_1 \sharp P(d\gamma) \\ &= \int_{\Gamma_T^R} \int_0^T \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(V)}(\gamma) |u|^q \hat{q}_{t,\gamma(t)}(du) dt \pi_1 \sharp P(d\gamma) \\ &= \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(V)} (e_t^{-1}(x)) |u|^q \hat{q}_{t,x}(du) e_t \sharp (\pi_1 \sharp P)(dx) dt \\ &= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(V)} (e_t^{-1}(x)) |u|^q \hat{q}_{t,x}(du) \eta_{1,2}(dt, dx) \\ &= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(V)} (e_t^{-1}(x)) |u|^q \eta(dt, dx, du) \\ &= \int_{\Gamma_T^R \times \mathcal{P}_U^R} \int_0^T \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(V)}(\gamma) |u|^q \mu(dt, du) P(d\gamma, d\mu) \leq R P(V), \end{aligned} $$

indicating that $g$ is integrable. By Vitali-Carathéodory theorem, for any $\varepsilon > 0$, there exists a lower semi-continuous function $\hat{g} : \Gamma_T^R \to \mathbb{R}$ with $g(\gamma) \leq \hat{g}(\gamma)$ for $\pi_1 \sharp P$-a.e. $\gamma \in \Gamma_T^R$ such that

$$ \int_{\Gamma_T^R} \hat{g}(\gamma) - g(\gamma) \pi_1 \sharp P(d\gamma) \leq \varepsilon. $$

For $\pi_1 \sharp P$-a.e. $\gamma_0 \in \text{spt}(\pi_1 \sharp P)$, there exists a unique measure $\mu^0 \in \mathcal{P}_U^R$ such that the pair $(\gamma_0, \mu^0) \in \text{spt}(P)$ by Proposition 4.1(4). Denote by $V$ the open neighborhood of $(\gamma_0, \mu^0)$, we have $P(V) > 0$ and

$$ \begin{aligned} g(\gamma_0) &\leq \hat{g}(\gamma_0) \leq \liminf_{V \to (\gamma_0, \mu^0)} \frac{1}{P(V)} \int_V \hat{g}(\gamma) P(d\gamma, d\mu) \\ &\leq \liminf_{V \to (\gamma_0, \mu^0)} \frac{1}{P(V)} \int_V g(\gamma) P(d\gamma, d\mu) + \varepsilon \leq R + \varepsilon. \end{aligned} $$

Let $\varepsilon$ tend to 0. We conclude that $\mathcal{L}_{[0,T]} \otimes \hat{q}_{t,\gamma(t)} \in \mathcal{P}_U^R$ for $\pi_1 \sharp P$-a.e. $\gamma \in \text{spt}(\pi_1 \sharp P)$. Let $\mathcal{L}_{[0,T]} \otimes \hat{q}_{t,\gamma(t)}$ equal any measure that belongs to $\mathcal{P}_U^R$ when $\gamma$ belongs to the null set. We can conclude that $\hat{P} \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$ since the difference in the null set has no influence on the following analysis by (F1) and (L3).

It is clear that $e_0 \sharp (\pi_1 \sharp \hat{P}) = e_0 \sharp (\pi_1 \sharp P) = m_0$. Moreover, for any $t \in [0, T]$ and any open set
$N \subset \Gamma_T^R \times \mathcal{P}_U^R$, we can deduce that

$$
\begin{align*}
& \int_{\Gamma_T^R} \int_0^t \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(N)}(\gamma) f(\gamma(s), u, m(s)) \hat{q}_{s,\gamma(s)}(du) \, ds \, \pi_1\sharp P(d\gamma) \\
&= \int_0^t \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(N)}(e_s^{-1}(x)) f(x, u, m(s)) \hat{q}_{s,x}(du) e_{s\sharp}(\pi_1\sharp P)(dx) \, ds \\
&= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(N)}(e_s^{-1}(x)) \mathbf{1}_{[0,t]}(s) f(x, u, m(s)) \hat{q}_{s,x}(du) \eta_{1,2}(ds, dx) \\
&= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(N)}(e_s^{-1}(x)) \mathbf{1}_{[0,t]}(s) f(x, u, m(s)) \eta(ds, dx, du) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} \int_0^T \int_{\mathbb{R}^n} \mathbf{1}_{\pi_1(N)}(\gamma) \mathbf{1}_{[0,t]}(s) f(\gamma(s), u, m(s)) \mu(ds, du) P(d\gamma, d\mu) \\
&= \int_{\pi_1(N) \times \mathcal{P}_U^R} \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) P(d\gamma, d\mu).
\end{align*}
$$

Based on this equality, for any $v \in \mathbb{R}^d$, we have

$$
\begin{align*}
& \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle \hat{P}(d\gamma, d\mu) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} \mathbf{1}_N(\gamma, \mu) \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle \hat{P}(d\gamma, d\mu) \\
&= \int_{\Gamma_T^R} \mathbf{1}_{\pi_1(N)}(\gamma) \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \hat{q}_{s,\gamma(s)}(du) ds \right\rangle \pi_1\sharp P(d\gamma) \\
&= \int_{\pi_1(N) \times \mathcal{P}_U^R} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P(d\gamma, d\mu) = 0.
\end{align*}
$$

Thus, we obtain that $\hat{P} \in \mathcal{P}_R(m)$.

Furthermore,

$$
\begin{align*}
J(m, \hat{P}) &= \int_{\Gamma_T^R} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \hat{q}_{t,\gamma(t)}(du) \, dt \, \pi_1\sharp P(d\gamma) \\
&= \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} L(x, u, m(t)) \hat{q}_{t,x}(du) e_{t\sharp}(\pi_1\sharp P)(dx) \, dt \\
&= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} L(x, u, m(t)) \hat{q}_{t,x}(du) \eta_{1,2}(dt, dx) \\
&= T \int_0^T \int_{\mathbb{T}^d} \int_{\mathbb{R}^n} L(x, u, m(t)) \eta(dt, dx, du) \\
&= \int_{\Gamma_T^R \times \mathcal{P}_U^R} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \mu(dt, du) P(d\gamma, d\mu) \\
&= J(m, P).
\end{align*}
$$

Therefore, we have $\hat{P} \in \mathcal{R}^*(m)$.

□
*Proof of Theorem 2.* (1) Define the map $\hat{q} : [0, T] \times \mathbb{T}^d \to \mathcal{P}(\mathbb{R}^n)$ by Proposition 5.1. By the convexity condition, for any $(t, x) \in [0, T] \times \mathbb{T}^d$, we have

$$ \int_{\mathbb{R}^n} (L(x, u, m(t)), |u|^q, f(x, u, m(t))) \hat{q}_{t,x}(du) \in \mathcal{L}(t, x). $$

By [21, Theorem A.9], there exist negative measurable functions $z_1 : [0, T] \times \mathbb{T}^d \to \mathbb{R}^-$, $z_2 : [0, T] \times \mathbb{T}^d \to \mathbb{R}^-$ and a measurable map $\hat{\alpha} : [0, T] \times \mathbb{T}^d \to \mathbb{R}^n$ such that

$$ L(x, \hat{\alpha}(t, x), m(t)) = \int_{\mathbb{R}^n} L(x, u, m(t)) \hat{q}_{t,x}(du) + z_1(t, x), $$

$$ |\hat{\alpha}(t, x)|^q = \int_{\mathbb{R}^n} |u|^q \hat{q}_{t,x}(du) + z_2(t, x), $$

$$ f(x, \hat{\alpha}(t, x), m(t)) = \int_{\mathbb{R}^n} f(x, u, m(t)) \hat{q}_{t,x}(du). $$

Define the probability measure $P_0 := \pi_1\#P \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{\hat{\alpha}(t,\gamma(t))}\}} \in \mathcal{P}(\Gamma_T^R \times \mathcal{P}_U^R)$. The measure $P_0$ is well-defined since for any $\gamma \in \text{spt}(\pi_1\#P)$, we have

$$ \int_0^T |\hat{\alpha}(t, \gamma(t))|^q dt = \int_0^T \int_{\mathbb{R}^n} |u|^q \hat{q}_{t,\gamma(t)}(du) dt + \int_0^T z_2(t, \gamma(t)) dt \\ \le \int_0^T \int_{\mathbb{R}^n} |u|^q \hat{q}_{t,\gamma(t)}(du) dt \le R. $$

It is clear that $e_0\#(\pi_1\#P_0) = e_0\#(\pi_1\#P) = m_0$. For any $(t, v) \in [0, T] \times \mathbb{R}^d$ and any open set $N \subset \Gamma_T^R \times \mathcal{P}_U^R$, we have

$$ \begin{aligned} & \int_N \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \mu(ds, du) \right\rangle P_0(d\gamma, d\mu) \\ &= \int_{\pi_1(N)} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t f(\gamma(s), \hat{\alpha}(s, \gamma(s)), m(s)) ds \right\rangle \pi_1\#P(d\gamma) \\ &= \int_{\pi_1(N)} \left\langle v, \gamma(t) - \gamma(0) - \int_0^t \int_{\mathbb{R}^n} f(\gamma(s), u, m(s)) \hat{q}_{s,\gamma(s)}(du) ds \right\rangle \pi_1\#P(d\gamma) = 0. \end{aligned} $$

Therefore, we conclude that $P_0 \in \mathcal{P}_R(m)$.

Moreover, we have

$$ \begin{aligned} J(m, P_0) &= \int_{\Gamma_T^R} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \delta_{\hat{\alpha}(t, \gamma(t))} (du) dt \pi_1 \# P(d\gamma) \\ &= \int_{\Gamma_T^R} \int_0^T L(\gamma(t), \hat{\alpha}(t, \gamma(t)), m(t)) dt \pi_1 \# P(d\gamma) \\ &= \int_{\Gamma_T^R} \int_0^T \int_{\mathbb{R}^n} L(\gamma(t), u, m(t)) \hat{q}_{t, \gamma(t)}(du) dt \pi_1 \# P(d\gamma) + \int_{\Gamma_T^R} \int_0^T z_1(t, \gamma(t)) dt \pi_1 \# P(d\gamma) \\ &\le J(m, \hat{P}) = J(m, P). \end{aligned} $$

In conclusion, $P_0 \in \mathcal{R}^*(m)$.
Define the map $u^*(t) := \hat{\alpha}(t, \gamma(t))$. Since the probability measure $P_0 \in R^*(m)$, by Proposition 4.1(4), for any $\gamma \in \text{spt}(\pi_1\sharp P_0)$, we have

$$ \left\{ \begin{array}{l} \dot{\gamma}(t) = f(\gamma(t), u^*(t), m(t)), \\ \gamma(0) \in \text{spt}(m_0). \end{array} \right. $$

Thus, $\text{spt}(\pi_1\sharp P_0) \subset \text{spt}(\eta^{u^*})$. We claim that $\pi_1\sharp P_0 = \pi_1\sharp P = \eta^{u^*}$. For any function $g \in C_b(\Gamma_T^R)$, since $e_0\sharp\pi_1\sharp P_0 = m_0$, we have

$$ \begin{aligned} \int_{\Gamma_T^R} g(\gamma) \pi_1 \sharp P_0 (d\gamma) &= \int_{\Gamma_T^R} g(\gamma) \mathbf{1}_{\{\text{spt}(\pi_1 \sharp P_0)\}} m_0 \circ e_0 (d\gamma) \\ &= \int_{\mathbb{T}^d} g(\gamma^{u^*}) m_0(dx) \\ &= \int_{\Gamma_T^R} g(\gamma) \eta^{u^*} (d\gamma). \end{aligned} $$

Therefore, the probability measure $P_0$ can also be expressed as $P_0 = \eta^{u^*} \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u^*(t)}\}}$.

For any map $u : [0, T] \to \mathbb{R}^n$ such that $\int_0^T |u(t)|^q dt \le R$, we have $P^u = \eta^u \otimes \delta_{\{\mathcal{L}_{[0,T]} \otimes \delta_{u(t)}\}} \in \mathcal{P}_R(m)$ by Lemma 5.1. Thus,

$$ \int_{\mathbb{T}^d} \int_0^T L(\gamma_x^u(t), u(t), m(t)) dt m_0(dx) \ge \int_{\mathbb{T}^d} \int_0^T L(\gamma_x^{u^*}(t), u^*(t), m(t)) dt m_0(dx). $$

(2) By Theorem 1, there exists a relaxed MFG equilibrium $P \in R^*(m)$, where $m(t) = e_t\sharp(\pi_1\sharp P)$ for any $t \in [0, T]$. As a consequence of the point (1) of this theorem, $P_0 \in R^*(m)$. Define $m^0(t) := e_t\sharp(\pi_1\sharp P_0)$ for any $t \in [0, T]$. By the definition of $P_0$, we have $m(t) = m^0(t)$ for any $t \in [0, T]$. Thus, $P_0 \in R^*(m^0)$, which means $P_0$ is also a relaxed MFG equilibrium, and it is strict obviously. $\square$

**Statements and Declarations.** This paper is the authors' original work and has not been published or submitted simultaneously elsewhere.

**Competing Interests.** Authors have no financial interests that are directly related to the work submitted for publication. We have no conflicts of interest to disclose.

**The Data Availability Statement.** No datasets were generated or analysed during the current study.

**Acknowledgements.** Cristian Mendico was partially supported by Istituto Nazionale di Alta Matematica, INdAM-GNAMPA project 2023/2024, by the MIUR Excellence Department Project MatMod@TOV awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C23000330006, and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 "Mean-Field Games: models, theory and computational aspects". Kaizhi Wang is supported by NSFC Grant No. 12171315, 11931016, and Natural Science Foundation of Shanghai No. 22ZR1433100.
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