# Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics

Fan Chen, Yiqing Lin, Zhenjie Ren, Songbo Wang

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Fan Chen, Yiqing Lin, Zhenjie Ren, Songbo Wang. Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics. Electronic Journal of Probability, 2024, 29 (none), pp.article no. 17. 10.1214/24-EJP1079 . hal-04508391

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# Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics

Fan Chen, Yiqing Lin, Zhenjie Ren, Songbo Wang

► To cite this version:

Fan Chen, Yiqing Lin, Zhenjie Ren, Songbo Wang. Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics. Electronic Journal of Probability, 2024, 29 (none), pp.article no. 17. 10.1214/24-EJP1079 . hal-04508391

HAL Id: hal-04508391

https://ip-paris.hal.science/hal-04508391v1

Submitted on 18 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

Electron. J. Probab. 29 (2024), article no. 17, 1–43.
ISSN: 1083-6489 https://doi.org/10.1214/24-EJP1079

# Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics*†

Fan Chen‡

Yiqing Lin§

Zhenjie Ren¶

Songbo Wang||

## Abstract

We study the kinetic mean field Langevin dynamics under the functional convexity assumption of the mean field energy functional. Using hypocoercivity, we first establish the exponential convergence of the mean field dynamics and then show the corresponding *N*-particle system converges exponentially in a rate uniform in *N* modulo a small error. Finally we study the short-time regularization effects of the dynamics and prove its uniform-in-time propagation of chaos property in both the Wasserstein and entropic sense. Our results can be applied to the training of two-layer neural networks with momentum and we include the numerical experiments.

**Keywords:** Langevin diffusion; mean field interaction; convergence to equilibrium; hypocoercivity; propagation of chaos; logarithmic Sobolev inequality.

**MSC2020 subject classifications:** Primary 60J60; 60K35, Secondary 35B40; 35H10; 35Q83; 35Q84.

Submitted to EJP on July 27, 2023, final version accepted on January 8, 2024.
Supersedes arXiv:2307.02168.

## 1 Introduction

Training neural networks by momentum gradient descent has proven to be effective in various applications [38, 22, 35]. However, despite their excellent performance, the theoretical understanding of those algorithms remains elusive. Recently, extensive researches have been conducted to model the loss minimization of neural networks as a mean field optimization problem [28, 8, 34, 18], with most characterizing gradient descent algorithms as *overdamped mean field Langevin* (MFL) dynamics. In this paper,

<footnote>*The second named author's research is supported by NSFC under the project No. 12371473. The third named author's research is supported by Finance For Energy Market Research Centre.</footnote>

<footnote>†The authors would like to thank an anonymous reviewer for pointing out an error in a previous version of the paper.</footnote>

<footnote>‡School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China.</footnote>

<footnote>E-mail: alexnwish@sjtu.edu.cn</footnote>

<footnote>§MOE-LSC, Shanghai Jiao Tong University, Shanghai, China. E-mail: yiqing.lin@sjtu.edu.cn</footnote>

<footnote>¶CEREMADE, Université Paris-Dauphine, PSL, Paris, France. E-mail: ren@ceremade.dauphine.fr</footnote>

<footnote>||CMAP, École polytechnique, IP Paris, Palaiseau, France. E-mail: songbo.wang@polytechnique.edu</footnote>
<figure><img src="image_2.png" /></figure>

Electron. J. Probab. 29 (2024), article no. 17, 1-43.
ISSN: 1083-6489 https://doi.org/10.1214/24-EJP1079

# Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics*†

Fan Chen‡

Yiqing Lin§

Zhenjie Ren¶

Songbo Wang||

## Abstract

We study the kinetic mean field Langevin dynamics under the functional convexity assumption of the mean field energy functional. Using hypocoercivity, we first establish the exponential convergence of the mean field dynamics and then show the corresponding *N*-particle system converges exponentially in a rate uniform in *N* modulo a small error. Finally we study the short-time regularization effects of the dynamics and prove its uniform-in-time propagation of chaos property in both the Wasserstein and entropic sense. Our results can be applied to the training of two-layer neural networks with momentum and we include the numerical experiments.

**Keywords:** Langevin diffusion; mean field interaction; convergence to equilibrium; hypocoercivity; propagation of chaos; logarithmic Sobolev inequality.

**MSC2020 subject classifications:** Primary 60J60; 60K35, Secondary 35B40; 35H10; 35Q83; 35Q84.

Submitted to EJP on July 27, 2023, final version accepted on January 8, 2024.
Supersedes arXiv:2307.02168.

## 1 Introduction

Training neural networks by momentum gradient descent has proven to be effective in various applications [38, 22, 35]. However, despite their excellent performance, the theoretical understanding of those algorithms remains elusive. Recently, extensive researches have been conducted to model the loss minimization of neural networks as a mean field optimization problem [28, 8, 34, 18], with most characterizing gradient descent algorithms as *overdamped mean field Langevin* (MFL) dynamics. In this paper,

<footnote>*The second named author's research is supported by NSFC under the project No. 12371473. The third named author's research is supported by Finance For Energy Market Research Centre.</footnote>

<footnote>†The authors would like to thank an anonymous reviewer for pointing out an error in a previous version of the paper.</footnote>

<footnote>‡School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China.</footnote>

<footnote>E-mail: alexnwish@sjtu.edu.cn</footnote>

<footnote>§MOE-LSC, Shanghai Jiao Tong University, Shanghai, China. E-mail: yiqing.lin@sjtu.edu.cn</footnote>

<footnote>¶CEREMADE, Université Paris-Dauphine, PSL, Paris, France. E-mail: ren@ceremade.dauphine.fr</footnote>

<footnote>||CMAP, École polytechnique, IP Paris, Palaiseau, France. E-mail: songbo.wang@polytechnique.edu</footnote>
we will focus on kinetic dynamics instead, which corresponds to momentum gradient descent in the context of machine learning [30, 23]. Classical studies, such as [40, 27], have explored the exponential convergence of linear kinetic Langevin dynamics based on hypocoercivity and functional inequalities. The kinetic MFL dynamics is studied in [21] to model the momentum gradient descent for the training of neural networks and its convergence to the unique invariant measure is proven without a quantitative rate. The present work studies both the quantitative long-time behavior of the kinetic MFL dynamics and its uniform-in-time propagation of chaos (POC) property, under a functional convexity assumption, and we aim to provide a theoretical justification for the momentum algorithm's efficiency in practice.

## 1.1 Settings and main results

We give an informal preview of our settings and main results in this section. Let $F: P_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional and denote by $D_m F: P_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ its intrinsic derivative. We aim to investigate the long-time behavior of the kinetic MFL defined by

$$dX_t = V_t dt, \\ dV_t = -V_t dt - D_m F(m_t^x, X_t) dt + \sqrt{2} dW_t, \quad \text{where } m_t^x = \text{Law}(X_t),$$

and its associated N-particle system defined by

$$dX_t^i = V_t^i dt \\ dV_t^i = -V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sqrt{2} dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^{N} \delta_{X_t^j}.$$

Here $W_t$, $(W_t^i)_{i=1}^N$ are independent d-dimensional Brownian motions. Denote $m_t = \text{Law}(X_t, V_t)$ and $m_t^N = \text{Law}(X_t^1, \dots, X_t^N, V_t^1, \dots, V_t^N)$, and we suppose the initial conditions $m_0$ and $m_0^N$ have finite second moments. We wish to show the convergence $m_t^N \to m_t^{\otimes N}$ when $N \to +\infty$ in a uniform-in-t way.

We assume

* the mean field functional $F$ is convex in the functional sense;

* its intrinsic derivative $(m, x) \mapsto D_m F(m, x)$ is jointly Lipschitz with respect to the $L^1$-Wasserstein distance.

* for every measure $m \in P_2(\mathbb{R}^d)$, the probability measure proportional to $\exp(-\frac{\delta F}{\delta m}(m, x))dx$ satisfy a logarithmic Sobolev inequality (LSI) with a constant uniform in $m$.

* its second and third-order functional derivatives satisfy certain bounds.

Under these assumptions, we are able to obtain

* when $t \to +\infty$, the mean field flow $m_t$ converges exponentially to the mean field invariant measure $m_\infty$;

* when $t \to +\infty$, the N-particle flow $m_t^N$ converges approximately to the N-tensorized mean field invariant measure $m_\infty^{\otimes N}$, with an exponential rate uniform in $N$;

* if $\frac{1}{N}W_2^2(m_0^N, m_0^{\otimes N}) \to 0$ when $N \to +\infty$, then $\sup_{t \ge 0} \frac{1}{N}W_2^2(m_t^N, m_t^{\otimes N}) \to 0$ and $\sup_{t \ge 1} \frac{1}{N}H(m_t^N|m_t^{\otimes N}) \to 0$ when $N \to +\infty$.

## 1.2 Related works

We give in this section a short review of the recent progresses in the long-time behavior and the uniform-in-time propagation of chaos property of McKean-Vlasov dynamics, with an emphasis on kinetic ones. We refer readers to [5, 6] for a more comprehensive review of propagation of chaos.
we will focus on kinetic dynamics instead, which corresponds to momentum gradient descent in the context of machine learning [30, 23]. Classical studies, such as [40, 27], have explored the exponential convergence of linear kinetic Langevin dynamics based on hypocoercivity and functional inequalities. The kinetic MFL dynamics is studied in [21] to model the momentum gradient descent for the training of neural networks and its convergence to the unique invariant measure is proven without a quantitative rate. The present work studies both the quantitative long-time behavior of the kinetic MFL dynamics and its uniform-in-time propagation of chaos (POC) property, under a functional convexity assumption, and we aim to provide a theoretical justification for the momentum algorithm's efficiency in practice.

## 1.1 Settings and main results

We give an informal preview of our settings and main results in this section. Let $F: P_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional and denote by $D_m F: P_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ its intrinsic derivative. We aim to investigate the long-time behavior of the kinetic MFL defined by

$$dX_t = V_t dt, \\ dV_t = -V_t dt - D_m F(m_t^x, X_t) dt + \sqrt{2} dW_t, \quad \text{where } m_t^x = \text{Law}(X_t),$$

and its associated N-particle system defined by

$$dX_t^i = V_t^i dt \\ dV_t^i = -V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sqrt{2} dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^{N} \delta_{X_t^j}.$$

Here $W_t$, $(W_t^i)_{i=1}^N$ are independent d-dimensional Brownian motions. Denote $m_t = \text{Law}(X_t, V_t)$ and $m_t^N = \text{Law}(X_t^1, \dots, X_t^N, V_t^1, \dots, V_t^N)$, and we suppose the initial conditions $m_0$ and $m_0^N$ have finite second moments. We wish to show the convergence $m_t^N \to m_t^{\otimes N}$ when $N \to +\infty$ in a uniform-in-t way.

We assume

* the mean field functional $F$ is convex in the functional sense;

* its intrinsic derivative $(m, x) \mapsto D_m F(m, x)$ is jointly Lipschitz with respect to the $L^1$-Wasserstein distance.

* for every measure $m \in P_2(\mathbb{R}^d)$, the probability measure proportional to $\exp(-\frac{\delta F}{\delta m}(m, x)) dx$ satisfy a logarithmic Sobolev inequality (LSI) with a constant uniform in $m$.

* its second and third-order functional derivatives satisfy certain bounds.

Under these assumptions, we are able to obtain

* when $t \to +\infty$, the mean field flow $m_t$ converges exponentially to the mean field invariant measure $m_\infty$;

* when $t \to +\infty$, the N-particle flow $m_t^N$ converges approximately to the N-tensorized mean field invariant measure $m_\infty^{\otimes N}$, with an exponential rate uniform in $N$;

* if $\frac{1}{N} W_2^2(m_0^N, m_0^{\otimes N}) \to 0$ when $N \to +\infty$, then $\sup_{t \ge 0} \frac{1}{N} W_2^2(m_t^N, m_t^{\otimes N}) \to 0$ and $\sup_{t \ge 1} \frac{1}{N} H(m_t^N | m_t^{\otimes N}) \to 0$ when $N \to +\infty$.

## 1.2 Related works

We give in this section a short review of the recent progresses in the long-time behavior and the uniform-in-time propagation of chaos property of McKean-Vlasov dynamics, with an emphasis on kinetic ones. We refer readers to [5, 6] for a more comprehensive review of propagation of chaos.
**Coupling approaches.** The coupling approach involves constructing a joint probability of the mean field and $N$-particle systems to allow comparisons between them. The synchronous coupling method is employed in [3] and the uniform-in-time POC is shown by assuming the strong monotonicity of the drift and the smallness of the mean field interaction. The strong monotonicity is then relaxed by the reflection coupling method in [11] and we refer readers to [36, 14, 21] for further developments. Let us remark that the synchronous coupling gives often sharp contraction rates under strong convexity assumptions, while the reflection coupling allows us to treat dynamics of more general type but gives far-from-sharp contraction rates.

**Functional approaches.** Another approach to uniform-in-time POC is the functional one, and this is also the major approach of this paper. In this situation in order to study the long-time behaviors and propagation of chaos properties, we construct appropriate (Lyapunov) functionals and investigate the change of their values along the dynamics. The relative entropy is used as the functional in [29] and its follow-up work [16] to study kinetic McKean-Vlasov dynamics with regular interactions. It is worth noting that the relative entropy approach has been successful in handling singular interactions, thanks to the groundbreaking work of Pierre-Emmanuel Jabin and Zhenfu Wang [20], and we refer the readers to [13, 9, 33] for recent developments. However, we are not aware of any works using the relative entropy functional (or its modifications) to study kinetic diffusions with singular interactions.

**Comparison to [7].** The present paper is a continuation of our previous work [7], where the overdamped version of mean field Langevin dynamics is studied, and they share a number of key features. We show the exponential convergence of the particle system using the same componentwise decomposition of Fisher information and the same componentwise log-Sobolev inequality. The uniform-in-time propagation of chaos property for both dynamics is then obtained by combining the exponential convergence of the mean field and particle flow. This paper is also different from [7] in a number of aspects. First, as the dynamics is generated by a hypoelliptic operator instead of an elliptic one, we use hypocoercivity to recover the exponential convergence. Second, since we are not able to show hypercontractivity of the kinetic dynamics (let alone reverse hypercontractivity), we prove the entropic propagation of chaos by studying its short-time regularization effects. In this way we no longer restrict the initial condition of the mean field dynamics, but as a trade-off we require a higher-order regularity in measure of the energy functional. Finally, following in a remark in [7], we use an approximation argument to remove the condition on the higher-order spatial derivatives in this work.

## 1.3 Main contributions

**Hypocoercivity for mean field systems.** We extend the studies of the linear Fokker-Planck equation in [40] to the dynamics with general (but always regular) mean field interactions. In particular, we do not suppose the interaction is in form of a two-body potential, which stands in contrast with [40, Theorem 56] and [29, 16]. Moreover, in hypocoercive computations, we find that the contributions from the mean field interaction can always be dominated by the “diagonal” terms in the Fisher information, already present in the case of linear dynamics. Hence using the convexity of energy, we are able to derive the hypocoercivity without restrictions on the size of the interaction. Furthermore, our assumptions imply a uniform-in-$N$ bound on the operator norm of the second-order derivatives of the effective potential driving the $N$-particle system, and the
**Coupling approaches.** The coupling approach involves constructing a joint probability of the mean field and $N$-particle systems to allow comparisons between them. The synchronous coupling method is employed in [3] and the uniform-in-time POC is shown by assuming the strong monotonicity of the drift and the smallness of the mean field interaction. The strong monotonicity is then relaxed by the reflection coupling method in [11] and we refer readers to [36, 14, 21] for further developments. Let us remark that the synchronous coupling gives often sharp contraction rates under strong convexity assumptions, while the reflection coupling allows us to treat dynamics of more general type but gives far-from-sharp contraction rates.

**Functional approaches.** Another approach to uniform-in-time POC is the functional one, and this is also the major approach of this paper. In this situation in order to study the long-time behaviors and propagation of chaos properties, we construct appropriate (Lyapunov) functionals and investigate the change of their values along the dynamics. The relative entropy is used as the functional in [29] and its follow-up work [16] to study kinetic McKean-Vlasov dynamics with regular interactions. It is worth noting that the relative entropy approach has been successful in handling singular interactions, thanks to the groundbreaking work of Pierre-Emmanuel Jabin and Zhenfu Wang [20], and we refer the readers to [13, 9, 33] for recent developments. However, we are not aware of any works using the relative entropy functional (or its modifications) to study kinetic diffusions with singular interactions.

**Comparison to [7].** The present paper is a continuation of our previous work [7], where the overdamped version of mean field Langevin dynamics is studied, and they share a number of key features. We show the exponential convergence of the particle system using the same componentwise decomposition of Fisher information and the same componentwise log-Sobolev inequality. The uniform-in-time propagation of chaos property for both dynamics is then obtained by combining the exponential convergence of the mean field and particle flow. This paper is also different from [7] in a number of aspects. First, as the dynamics is generated by a hypoelliptic operator instead of an elliptic one, we use hypocoercivity to recover the exponential convergence. Second, since we are not able to show hypercontractivity of the kinetic dynamics (let alone reverse hypercontractivity), we prove the entropic propagation of chaos by studying its short-time regularization effects. In this way we no longer restrict the initial condition of the mean field dynamics, but as a trade-off we require a higher-order regularity in measure of the energy functional. Finally, following in a remark in [7], we use an approximation argument to remove the condition on the higher-order spatial derivatives in this work.

## 1.3 Main contributions

**Hypocoercivity for mean field systems.** We extend the studies of the linear Fokker-Planck equation in [40] to the dynamics with general (but always regular) mean field interactions. In particular, we do not suppose the interaction is in form of a two-body potential, which stands in contrast with [40, Theorem 56] and [29, 16]. Moreover, in hypocoercive computations, we find that the contributions from the mean field interaction can always be dominated by the “diagonal” terms in the Fisher information, already present in the case of linear dynamics. Hence using the convexity of energy, we are able to derive the hypocoercivity without restrictions on the size of the interaction. Furthermore, our assumptions imply a uniform-in-$N$ bound on the operator norm of the second-order derivatives of the effective potential driving the $N$-particle system, and the
entropic hypocoercivity is consequently uniform in $N$. This is different from the situation of $L^2$-hypocoercivity, where the condition given by Villani [40, (7.3)] gives dimension-dependent constants and therefore is unsuitable for studies of particle systems, as remarked in [15]. Finally, let us mention that we derive the entropic hypocoercivity under minimal regularity assumptions, made possible by our approximation argument (of functions *and* of mean field functionals) and the calculus in Wasserstein space developed in [2].

**Regularization in short time.** We obtain two short-time regularization results for the kinetic mean field dynamics. The first, from Wasserstein to entropy, is a consequence of the logarithmic Harnack's inequality, obtained by applying the coupling by change of measure method of Panpan Ren and Feng-Yu Wang in [32] to the mean field and $N$-particle diffusions. We remark also that very recently a similar inequality ([19, (3.13)]) is proved for the propagation of chaos of non-degenerate McKean-Vlasov diffusions. The second regularization, from entropy to Fisher information, is obtained by adapting Hérau's functional in [40] to our mean field setting and follows from the same hypocoercive computations as we prove the convergence of the mean field flow. We stress that although much stronger regularization phenomena are present, for example from measure initial values to $L^p$ for every $p > 1$ and to $H^k$ for every $k \ge 1$, our results have the advantage of growing at most linearly in dimension, making them suitable for studying the $N$-particle systems under the limit $N \to +\infty$.

**Propagation of chaos.** Finally, using the exponential convergence and the short-time regularizations, we derive the propagation of chaos for the kinetic MFL, i.e. bounds on the distances between the particle system and the mean field system. In particular, the initial value of the both systems can be arbitrary measures of finite second moments without any further regularity constraints. Moreover, the error terms do not have any dimension-dependence. It is noteworthy that our approach allows us to not rely on a uniform-in-time log-Sobolev inequality for the mean field flow, and also that the dynamics considered are realized on the whole space instead of the torus, standing in contrast with previous works, e.g. [13, 24, 10].

## 1.4 Notations

Let $d$ be a positive integer and $x, v$ be elements of $\mathbb{R}^d$. We denote the Euclidean norm of $x$ and $v$ by $|x|$ and $|v|$ respectively. The letter $z = (x, v)$ then denotes an element of $\mathbb{R}^{2d}$ with its Euclidean norm denoted by $|z|^2 = |x|^2 + |v|^2$. For a $d \times d$ real matrix $M$, we denote by $|M|_{\text{op}}$ its operator norm with respect to the Euclidean metric of $\mathbb{R}^d$. Let $p \ge 1$. Define $\mathcal{P}_p(\mathbb{R}^d)$ to be the space of probabilities on $\mathbb{R}^d$ of finite $p$-moment, i.e. $\mathcal{P}_p(\mathbb{R}^d) = \{m \in \mathcal{P}(\mathbb{R}^d) : \int |x|^p m(dx) < +\infty\}$. We denote the $L^p$-Wasserstein metric by $W_p$ and refer readers to [2, Chapter 7] for its definition and elementary properties.

Let $F : \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional. Denote by $\frac{\delta F}{\delta m} : \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}$ its linear functional derivative and by $D_m F = \nabla \frac{\delta F}{\delta m} : \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ its intrinsic derivative, if they exist. The definition of linear functional derivative on $\mathcal{P}_2(\mathbb{R}^d)$ can be found in [4, Definition 5.43].

Let $X$ and $Y$ be random variables. We denote the distribution of $X$ by $\text{Law}(X)$ and say $X \sim m$ if $m = \text{Law}(X)$. We also say $X \stackrel{d}{=} Y$ if $\text{Law}(X) = \text{Law}(Y)$.

The set of couplings between probabilities $\mu$ and $\nu$ is denoted by $\Pi(\mu, \nu)$.

Let $N \ge 2$ be integer. The bold letters $x_N = (x^1, \dots, x^N)$, $v_N = (v^1, \dots, v^N)$ denote respectively $N$-tuples of elements in $\mathbb{R}^d$ and $z_N = (z^1, \dots, z^N)$ an $N$-tuple of elements in $\mathbb{R}^{2d}$. We omit the subscript $N$ when there are no ambiguities. Given $x_N = (x^1, \dots, x^N) \in$
entropic hypocoercivity is consequently uniform in $N$. This is different from the situation of $L^2$-hypocoercivity, where the condition given by Villani [40, (7.3)] gives dimension-dependent constants and therefore is unsuitable for studies of particle systems, as remarked in [15]. Finally, let us mention that we derive the entropic hypocoercivity under minimal regularity assumptions, made possible by our approximation argument (of functions *and* of mean field functionals) and the calculus in Wasserstein space developed in [2].

**Regularization in short time.** We obtain two short-time regularization results for the kinetic mean field dynamics. The first, from Wasserstein to entropy, is a consequence of the logarithmic Harnack's inequality, obtained by applying the coupling by change of measure method of Panpan Ren and Feng-Yu Wang in [32] to the mean field and $N$-particle diffusions. We remark also that very recently a similar inequality ([19, (3.13)]) is proved for the propagation of chaos of non-degenerate McKean-Vlasov diffusions. The second regularization, from entropy to Fisher information, is obtained by adapting Hérau's functional in [40] to our mean field setting and follows from the same hypocoercive computations as we prove the convergence of the mean field flow. We stress that although much stronger regularization phenomena are present, for example from measure initial values to $L^p$ for every $p > 1$ and to $H^k$ for every $k \ge 1$, our results have the advantage of growing at most linearly in dimension, making them suitable for studying the $N$-particle systems under the limit $N \to +\infty$.

**Propagation of chaos.** Finally, using the exponential convergence and the short-time regularizations, we derive the propagation of chaos for the kinetic MFL, i.e. bounds on the distances between the particle system and the mean field system. In particular, the initial value of the both systems can be arbitrary measures of finite second moments without any further regularity constraints. Moreover, the error terms do not have any dimension-dependence. It is noteworthy that our approach allows us to not rely on a uniform-in-time log-Sobolev inequality for the mean field flow, and also that the dynamics considered are realized on the whole space instead of the torus, standing in contrast with previous works, e.g. [13, 24, 10].

## 1.4 Notations

Let $d$ be a positive integer and $x, v$ be elements of $\mathbb{R}^d$. We denote the Euclidean norm of $x$ and $v$ by $|x|$ and $|v|$ respectively. The letter $z = (x, v)$ then denotes an element of $\mathbb{R}^{2d}$ with its Euclidean norm denoted by $|z|^2 = |x|^2 + |v|^2$. For a $d \times d$ real matrix $M$, we denote by $|M|_{\text{op}}$ its operator norm with respect to the Euclidean metric of $\mathbb{R}^d$. Let $p \ge 1$. Define $\mathcal{P}_p(\mathbb{R}^d)$ to be the space of probabilities on $\mathbb{R}^d$ of finite $p$-moment, i.e. $\mathcal{P}_p(\mathbb{R}^d) = \{m \in \mathcal{P}(\mathbb{R}^d) : \int |x|^p m(dx) < +\infty\}$. We denote the $L^p$-Wasserstein metric by $W_p$ and refer readers to [2, Chapter 7] for its definition and elementary properties.

Let $F: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional. Denote by $\frac{\delta F}{\delta m} : \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}$ its linear functional derivative and by $D_m F = \nabla \frac{\delta F}{\delta m} : \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ its intrinsic derivative, if they exist. The definition of linear functional derivative on $\mathcal{P}_2(\mathbb{R}^d)$ can be found in [4, Definition 5.43].

Let $X$ and $Y$ be random variables. We denote the distribution of $X$ by $\text{Law}(X)$ and say $X \sim m$ if $m = \text{Law}(X)$. We also say $X \stackrel{d}{=} Y$ if $\text{Law}(X) = \text{Law}(Y)$.

The set of couplings between probabilities $\mu$ and $\nu$ is denoted by $\Pi(\mu, \nu)$.

Let $N \ge 2$ be integer. The bold letters $\mathbf{x}_N = (x^1, \dots, x^N)$, $\mathbf{v}_N = (v^1, \dots, v^N)$ denote respectively $N$-tuples of elements in $\mathbb{R}^d$ and $\mathbf{z}_N = (z^1, \dots, z^N)$ an $N$-tuple of elements in $\mathbb{R}^{2d}$. We omit the subscript $N$ when there are no ambiguities. Given $\mathbf{x}_N = (x^1, \dots, x^N) \in$
$\mathbb{R}^{dN}$, we denote the corresponding empirical measure by

$$\mu_{\mathbf{x}_N} = \frac{1}{N} \sum_{i=1}^{N} \delta_{x_i}.$$

For $i = 1, \dots, N$, we define $-i = \{1, \dots, N\} \setminus \{i\}$, that is, the complementary index set, and we denote the empirical measures formed by the $N-1$ points $(x_j)_{j \neq i}$ by

$$\mu_{\mathbf{x}_N^{-i}} = \frac{1}{N-1} \sum_{j \neq i} \delta_{x_j}.$$

For an $\mathbb{R}^{dN}$-valued random variable $\mathbf{X}_N = (X^i)_{i=1}^N$, we can form the random empirical measures $\mu_{\mathbf{X}_N}, \mu_{\mathbf{X}_N^{-i}}$.

Let $I \subset \{1, \dots, N\}$ and $J = \{1, \dots, N\} \setminus I$ be the complementary index set. Let $Z$ be an $\mathbb{R}^{2dN}$-valued random variable and $m^N$ be its distribution, belonging to $\mathcal{P}(\mathbb{R}^{2dN})$. We denote the marginal and the (regular) conditional distributions of $m^N$ by

$$m^{N,I} = \text{Law}(Z^i)_{i \in I}, \\ m^{N,I|J}(\mathbf{z}^J) = \text{Law}((Z^i)_{i \in I} | Z_t^j = z^j, j \in J),$$

where the latter is defined $m^{N,J}$-almost surely and $\mathbf{z}^J$ denotes the tuple $(z^j)_{j \in J}$. We identify $i$ with the singleton $\{i\}$ when working with indices.

Whenever a measure $m \in \mathcal{P}(\mathbb{R}^d)$ has a density with respect to the $d$-dimensional Lebesgue measure, we denote its density function by $m$ equally. The relative $H(\cdot|\cdot)$ between probabilities are always well defined and the absolute entropy $H(\cdot)$ is also well defined if the measure in the argument has finite second moment. If a measure $m \in \mathcal{P}(\mathbb{R}^d)$ has distributional derivatives $Dm$ representable by a finite Borel measure and $Dm$ is absolutely continuous with respect to $m$, we define its Fisher information by

$$I(m) = \int \left| \frac{Dm}{m} \right|^2 m,$$

where $\frac{Dm}{m}$ is the Radon-Nikodým derivative. Otherwise we set $I(m) = +\infty$. One can verify that $I(m)$ is finite only if $m \in W^{1,1}(\mathbb{R}^d)^1$, and in this case $I(m) = \int \frac{|\nabla m|^2}{m} \nabla m$ being the weak derivatives in $L^1(\mathbb{R}^d; \mathbb{R}^d)$. The Fisher information defined in this way corresponds to the functional considered in [1, (2.26)]. If $m$ is a measure on $\mathbb{R}^d$ having finite Fisher information, and if $\gamma$ is another measure on $\mathbb{R}^d$ having weakly differentiable density with respect to the Lebesgue, we define the relative Fisher information by

$$I(m|\gamma) = \int \left| \frac{\nabla m}{m} - \frac{\nabla \gamma}{\gamma} \right|^2 m.$$

For nonnegative functions $f: \mathbb{R}^d \to [0, +\infty)$ we define its entropy by

$$\text{Ent}_m f = \mathbb{E}_m[f \log f] - \mathbb{E}_m[f] \log \mathbb{E}_m[f],$$

which is well defined in $[0, +\infty]$ by Jensen's inequality.

<footnote n="1">¹We sketch the proof here. Suppose $m$ has finite Fisher information. Set $m^n = m \star \rho^n$ for a mollifying sequence $(\rho^n)_{n \in \mathbb{N}}$. Then we have $\|m^n\|_{W^{1,1}} \le C$ for all $n \in \mathbb{N}$. By Gagliardo-Nirenberg, $m^n$ is uniformly bounded in $L^p$ for some $p > 1$, so upon an extraction of subsequence, $(m^n)_{n \in \mathbb{N}}$ converges to some $m' \in L^p$ weakly. But $m^n \to m$ in $\mathcal{P}$. The two limits coincide, i.e. $m = m'$. Hence $m$ has density with respect to the Lebesgue and so does $Dm$.</footnote>
$\mathbb{R}^{dN}$, we denote the corresponding empirical measure by

$$ \mu_{\mathbf{x}_N} = \frac{1}{N} \sum_{i=1}^{N} \delta_{x_i}. $$

For $i = 1, \dots, N$, we define $-i = \{1, \dots, N\} \setminus \{i\}$, that is, the complementary index set, and we denote the empirical measures formed by the $N-1$ points $(x_j)_{j \neq i}$ by

$$ \mu_{\mathbf{x}_N^{-i}} = \frac{1}{N-1} \sum_{j \neq i} \delta_{x_j}. $$

For an $\mathbb{R}^{dN}$-valued random variable $\mathbf{X}_N = (X^i)_{i=1}^N$, we can form the random empirical measures $\mu_{\mathbf{X}_N}, \mu_{\mathbf{X}_N^{-i}}$.

Let $I \subset \{1, \dots, N\}$ and $J = \{1, \dots, N\} \setminus I$ be the complementary index set. Let $Z$ be an $\mathbb{R}^{2dN}$-valued random variable and $m^N$ be its distribution, belonging to $\mathcal{P}(\mathbb{R}^{2dN})$. We denote the marginal and the (regular) conditional distributions of $m^N$ by

$$ \begin{aligned} m^{N,I} &= \mathrm{Law}(Z^i)_{i \in I}, \\ m^{N,I|J}(\mathbf{z}^J) &= \mathrm{Law}((Z^i)_{i \in I} | Z_t^j = z^j, j \in J), \end{aligned} $$

where the latter is defined $m^{N,J}$-almost surely and $\mathbf{z}^J$ denotes the tuple $(z^j)_{j \in J}$. We identify $i$ with the singleton $\{i\}$ when working with indices.

Whenever a measure $m \in \mathcal{P}(\mathbb{R}^d)$ has a density with respect to the $d$-dimensional Lebesgue measure, we denote its density function by $m$ equally. The relative $H(\cdot|\cdot)$ between probabilities are always well defined and the absolute entropy $H(\cdot)$ is also well defined if the measure in the argument has finite second moment. If a measure $m \in \mathcal{P}(\mathbb{R}^d)$ has distributional derivatives $Dm$ representable by a finite Borel measure and $Dm$ is absolutely continuous with respect to $m$, we define its Fisher information by

$$ I(m) = \int \left| \frac{Dm}{m} \right|^2 m, $$

where $\frac{Dm}{m}$ is the Radon-Nikodým derivative. Otherwise we set $I(m) = +\infty$. One can verify that $I(m)$ is finite only if $m \in W^{1,1}(\mathbb{R}^d)^1$, and in this case $I(m) = \int \frac{|\nabla m|^2}{m} \cdot \nabla m$ being the weak derivatives in $L^1(\mathbb{R}^d; \mathbb{R}^d)$. The Fisher information defined in this way corresponds to the functional considered in [1, (2.26)]. If $m$ is a measure on $\mathbb{R}^d$ having finite Fisher information, and if $\gamma$ is another measure on $\mathbb{R}^d$ having weakly differentiable density with respect to the Lebesgue, we define the relative Fisher information by

$$ I(m|\gamma) = \int \left| \frac{\nabla m}{m} - \frac{\nabla \gamma}{\gamma} \right|^2 m. $$

For nonnegative functions $f: \mathbb{R}^d \to [0, +\infty)$ we define its entropy by

$$ \mathrm{Ent}_m f = \mathrm{E}_m[f \log f] - \mathrm{E}_m[f] \log \mathrm{E}_m[f], $$

which is well defined in $[0, +\infty]$ by Jensen's inequality.

<footnote n="1">¹We sketch the proof here. Suppose $m$ has finite Fisher information. Set $m^n = m \star \rho^n$ for a mollifying sequence $(\rho^n)_{n \in \mathbb{N}}$. Then we have $\|m^n\|_{W^{1,1}} \le C$ for all $n \in \mathbb{N}$. By Gagliardo-Nirenberg, $m^n$ is uniformly bounded in $L^p$ for some $p > 1$, so upon an extraction of subsequence, $(m^n)_{n \in \mathbb{N}}$ converges to some $m' \in L^p$ weakly. But $m^n \to m$ in $\mathcal{P}$. The two limits coincide, i.e. $m = m'$. Hence $m$ has density with respect to the Lebesgue and so does $Dm$.</footnote>
**Organization of paper.** In Section 2, we introduce our assumptions, define the kinetic mean field Langevin and the particle system, and state our main results. We provide in Section 3 an exemplary dynamics modeling neural networks' training and present our numerical experiments. Moving on to the proofs, we first show in Section 4 the exponential convergence of the mean field and particle system dynamics. We then study in Section 5 finite-time propagation of chaos and regularizations of the kinetic MFL before combining all previous results and showing the propagation of chaos theorem in its full form. Finally, several technical results are proved in the appendices.

## 2 Assumptions and main results

**Assumptions.** Let $F: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional. We suppose $F$ is convex in the sense that for every $t \in [0, 1]$ and every $m, m' \in \mathcal{P}_2(\mathbb{R}^d)$,

$$F((1-t)m + tm') \le (1-t)F(m) + tF(m'). \quad (2.1)$$

Suppose also its intrinsic derivative $D_m F: \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ exists and satisfies

$$\forall x, x' \in \mathbb{R}^d, \forall m, m' \in \mathcal{P}_2(\mathbb{R}^d), \\ |D_m F(m, x) - D_m F(m', x')| \le M_{mm}^F W_1(m, m') + M_{mx}^F |x - x'| \quad (2.2)$$

for some constants $M_{mm}^F, M_{mx}^F \ge 0$. For each $m \in \mathcal{P}_2(\mathbb{R}^d)$ we define a probability measure $\Pi^x(m)$ on $\mathbb{R}^d$ by $\Pi^x(m)(dx) \propto \exp(-\frac{\delta F}{\delta m}(m, x))dx$ and suppose $\Pi^x(m)$ satisfies the $\rho^x$-logarithmic Sobolev inequality (LSI), uniformly in $m$, for some $\rho^x > 0$, that is, for every $m \in \mathcal{P}_2(\mathbb{R}^d)$,

$$\forall f \in C_b^1(\mathbb{R}^d), \quad \rho^x \operatorname{Ent}_{\Pi^x(m)}(f^2) \le \operatorname{E}_{\Pi^x(m)}[|\nabla f|^2]. \quad (2.3)$$

Finally for some of the results we suppose additionally that $F$ is third-order differentiable in measure with $\sup_{m \in \mathcal{P}_2(\mathbb{R}^d)} \sup_{x, x' \in \mathbb{R}^d} |D_m^2 F(m, x, x')|_{op} \le M_{mm}^F$ and

$$\forall m, m' \in \mathcal{P}_2(\mathbb{R}^d), \forall x \in \mathbb{R}^d, \\ \left|\iint \left[\nabla_x \frac{\delta^3 F}{\delta m^3}(m', x, x', x') - \nabla_x \frac{\delta^3 F}{\delta m^3}(m', x, x', x'')\right] m(dx')m(dx'')\right| \le M_{mmm}^F \quad (2.4)$$

for some constant $M_{mmm}^F$.

**Definition of $\hat{m}$ and functional inequalities.** For each $m \in \mathcal{P}(\mathbb{R}^{2d})$, we define $\hat{m}$ to be the probability on $\mathbb{R}^{2d}$ satisfying

$$\hat{m}(dxdv) \propto \exp\left(-\frac{\delta F}{\delta m}(m^x, x) - \frac{1}{2}|v|^2\right)dxdv, \quad (2.5)$$

where $m^x$ is the spatial marginal of $m$. Sometimes we will abuse the notation and define for a measure $m^x \in \mathcal{P}_2(\mathbb{R}^d)$, the probability $\overline{m}^x(dxdv) \propto \exp(-\frac{\delta F}{\delta m}(m^x, x) - \frac{1}{2}|v|^2)dxdv$. If $F$ satisfies (2.3) with the LSI constant $\rho^x$, then setting

$$\rho = \rho^x \wedge \frac{1}{2}, \quad (2.6)$$

we have that the $\rho$-LSI holds for $\hat{m}$: for every $f \in C_b^1(\mathbb{R}^{2d})$,

$$\rho \operatorname{Ent}_{\hat{m}}(f^2) \le \operatorname{E}_{\hat{m}}[|\nabla f|^2]. \quad (2.7)$$
**Organization of paper.** In Section 2, we introduce our assumptions, define the kinetic mean field Langevin and the particle system, and state our main results. We provide in Section 3 an exemplary dynamics modeling neural networks' training and present our numerical experiments. Moving on to the proofs, we first show in Section 4 the exponential convergence of the mean field and particle system dynamics. We then study in Section 5 finite-time propagation of chaos and regularizations of the kinetic MFL before combining all previous results and showing the propagation of chaos theorem in its full form. Finally, several technical results are proved in the appendices.

## 2 Assumptions and main results

**Assumptions.** Let $F: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a mean field functional. We suppose $F$ is convex in the sense that for every $t \in [0, 1]$ and every $m, m' \in \mathcal{P}_2(\mathbb{R}^d)$,

$$F((1-t)m + tm') \le (1-t)F(m) + tF(m'). \quad (2.1)$$

Suppose also its intrinsic derivative $D_m F: \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ exists and satisfies

$$\forall x, x' \in \mathbb{R}^d, \forall m, m' \in \mathcal{P}_2(\mathbb{R}^d), \\ |D_m F(m, x) - D_m F(m', x')| \le M_{mm}^F W_1(m, m') + M_{mx}^F |x - x'| \quad (2.2)$$

for some constants $M_{mm}^F, M_{mx}^F \ge 0$. For each $m \in \mathcal{P}_2(\mathbb{R}^d)$ we define a probability measure $\Pi^x(m)$ on $\mathbb{R}^d$ by $\Pi^x(m)(dx) \propto \exp(-\frac{\delta F}{\delta m}(m, x))dx$ and suppose $\Pi^x(m)$ satisfies the $\rho^x$-logarithmic Sobolev inequality (LSI), uniformly in $m$, for some $\rho^x > 0$, that is, for every $m \in \mathcal{P}_2(\mathbb{R}^d)$,

$$\forall f \in C_b^1(\mathbb{R}^d), \quad \rho^x \operatorname{Ent}_{\Pi^x(m)}(f^2) \le \operatorname{E}_{\Pi^x(m)} [|\nabla f|^2]. \quad (2.3)$$

Finally for some of the results we suppose additionally that $F$ is third-order differentiable in measure with $\sup_{m \in \mathcal{P}_2(\mathbb{R}^d)} \sup_{x, x' \in \mathbb{R}^d} |D_m^2 F(m, x, x')|_{op} \le M_{mm}^F$ and

$$\forall m, m' \in \mathcal{P}_2(\mathbb{R}^d), \forall x \in \mathbb{R}^d, \\ \left|\iint \left[\nabla_x \frac{\delta^3 F}{\delta m^3}(m', x, x', x') - \nabla_x \frac{\delta^3 F}{\delta m^3}(m', x, x', x'')\right] m(dx')m(dx'')\right| \le M_{mmm}^F \quad (2.4)$$

for some constant $M_{mmm}^F$.

**Definition of $\hat{m}$ and functional inequalities.** For each $m \in \mathcal{P}(\mathbb{R}^{2d})$, we define $\hat{m}$ to be the probability on $\mathbb{R}^{2d}$ satisfying

$$\hat{m}(dxdv) \propto \exp\left(-\frac{\delta F}{\delta m}(m^x, x) - \frac{1}{2}|v|^2\right)dxdv, \quad (2.5)$$

where $m^x$ is the spatial marginal of $m$. Sometimes we will abuse the notation and define for a measure $m^x \in \mathcal{P}_2(\mathbb{R}^d)$, the probability $\overline{m}^x(dxdv) \propto \exp(-\frac{\delta F}{\delta m}(m^x, x) - \frac{1}{2}|v|^2)dxdv$. If $F$ satisfies (2.3) with the LSI constant $\rho^x$, then setting

$$\rho = \rho^x \wedge \frac{1}{2}, \quad (2.6)$$

we have that the $\rho$-LSI holds for $\hat{m}$: for every $f \in C_b^1(\mathbb{R}^{2d})$,

$$\rho \operatorname{Ent}_{\hat{m}}(f^2) \le \operatorname{E}_{\hat{m}} [|\nabla f|^2]. \quad (2.7)$$
As a consequence, we have the *Poincaré inequality*: for every $f \in C_b^1(\mathbb{R}^{2d})$,

$$2\rho \mathrm{Var}_{\hat{m}}(f) \leq \mathrm{E}_{\hat{m}}[|\nabla f|^2]; \qquad (2.8)$$

and *Talagrand's T$_2$ transport inequality*: for every $\mu \in \mathcal{P}_2(\mathbb{R}^{2d})$,

$$2\rho W_2^2(\mu, \hat{m}) \leq H(\mu|\hat{m}). \qquad (2.9)$$

**Mean field and particle system.** We study the mean field kinetic Langevin dynamics, that is, the following McKean-Vlasov SDE

$$\begin{aligned} dX_t &= V_t dt, \\ dV_t &= -V_t dt - D_m F(m_t^x, X_t) dt + \sqrt{2} dW_t, \quad \text{where } m_t^x = \mathrm{Law}(X_t). \end{aligned} \qquad (2.10)$$

Let $N \geq 2$. The corresponding $N$-particle system is defined by

$$\begin{aligned} dX_t^i &= V_t^i dt \\ dV_t^i &= -V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sqrt{2} dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^{N} \delta_{X_t^j}. \end{aligned} \qquad (2.11)$$

Here $W$ and $W^i$ are standard Brownian motions in $\mathbb{R}^d$, and $(W^i)_{i=1}^N$ are independent. Their marginal distributions $m_t = \mathrm{Law}(X_t)$, $m_t^N = \mathrm{Law}(\mathbf{X}_t) = \mathrm{Law}(X_t^1, \dots, X_t^N)$ solve respectively the Fokker-Planck equations:

$$\partial_t m = \Delta_v m + \nabla_v \cdot (mv) - v \cdot \nabla_x m + D_m F(m_t^x, x) \cdot \nabla_v m, \qquad (2.12)$$

$$\partial_t m^N = \sum_{i=1}^{N} \left( \Delta_{v^i} m^N + \nabla_{v^i} \cdot (m^N v^i) - v^i \cdot \nabla_{x^i} m^N + D_m F(\mu_{\mathbf{x}}, x^i) \cdot \nabla_{v^i} m^N \right), \qquad (2.13)$$

where on the second line $\mu_{\mathbf{x}} := \frac{1}{N} \sum_{i=1}^N \delta_{x^i}$. The mean field equation (2.12) is non-linear while the $N$-particle system equation (2.13) is linear. We will show in Lemma 4.5 the wellposedness of the mean field dynamics (2.12) with initial conditions of finite second moment.

**Remark 2.1.** We have fixed the volatility and the friction constants to simplify the computations. In order to apply our results to the diffusion process defined by

$$\begin{aligned} dX_t &= \alpha V_t dt, \\ dV_t &= -\gamma V_t dt - D_m F(\mathrm{Law}(X_t), X_t) dt + \sigma dW_t, \end{aligned} \qquad (2.14)$$

with $\alpha, \gamma, \sigma > 0$, we introduce the new variables:

$$x' = \frac{(2\gamma^3)^{1/2}}{\alpha\sigma} x, \quad v' = \frac{(2\gamma)^{1/2}}{\sigma} v, \quad t' = \gamma^{-1}t,$$

define $m'$ to be the push-out of measure $m$ under $x \mapsto x'$, and set

$$F'(m') = \left(\frac{2}{\gamma\sigma^2}\right)^{1/2} F(m).$$

Then the stochastic process $t' \mapsto (X'_{t'}, V'_{t'})$ satisfy

$$\begin{aligned} dX'_{t'} &= V'_{t'} dt', \\ dV'_{t'} &= -V'_{t'} dt' - D_m F'(\mathrm{Law}(X'_{t'}), X'_{t'}) dt + \sqrt{2} dW'_{t'}, \end{aligned}$$
As a consequence, we have the *Poincaré inequality*: for every $f \in C_b^1(\mathbb{R}^{2d})$,

$$2\rho \mathrm{Var}_{\hat{m}}(f) \leq \mathbb{E}_{\hat{m}}[|\nabla f|^2]; \qquad (2.8)$$

and *Talagrand's T₂ transport inequality*: for every $\mu \in \mathcal{P}_2(\mathbb{R}^{2d})$,

$$2\rho W_2^2(\mu, \hat{m}) \leq H(\mu|\hat{m}). \qquad (2.9)$$

**Mean field and particle system.** We study the mean field kinetic Langevin dynamics, that is, the following McKean-Vlasov SDE

$$\begin{aligned} dX_t &= V_t dt, \\ dV_t &= -V_t dt - D_m F(m_t^x, X_t) dt + \sqrt{2} dW_t, \quad \text{where } m_t^x = \mathrm{Law}(X_t). \end{aligned} \qquad (2.10)$$

Let $N \geq 2$. The corresponding $N$-particle system is defined by

$$\begin{aligned} dX_t^i &= V_t^i dt \\ dV_t^i &= -V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sqrt{2} dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^N \delta_{X_t^j}. \end{aligned} \qquad (2.11)$$

Here $W$ and $W^i$ are standard Brownian motions in $\mathbb{R}^d$, and $(W^i)_{i=1}^N$ are independent. Their marginal distributions $m_t = \mathrm{Law}(X_t)$, $m_t^N = \mathrm{Law}(\mathbf{X}_t) = \mathrm{Law}(X_t^1, \dots, X_t^N)$ solve respectively the Fokker-Planck equations:

$$\partial_t m = \Delta_v m + \nabla_v \cdot (mv) - v \cdot \nabla_x m + D_m F(m_t^x, x) \cdot \nabla_v m, \qquad (2.12)$$

$$\partial_t m^N = \sum_{i=1}^{N} \left( \Delta_{v^i} m^N + \nabla_{v^i} \cdot (m^N v^i) - v^i \cdot \nabla_{x^i} m^N + D_m F(\mu_{\mathbf{x}}, x^i) \cdot \nabla_{v^i} m^N \right), \qquad (2.13)$$

where on the second line $\mu_{\mathbf{x}} := \frac{1}{N} \sum_{i=1}^N \delta_{x^i}$. The mean field equation (2.12) is non-linear while the $N$-particle system equation (2.13) is linear. We will show in Lemma 4.5 the wellposedness of the mean field dynamics (2.12) with initial conditions of finite second moment.

**Remark 2.1.** We have fixed the volatility and the friction constants to simplify the computations. In order to apply our results to the diffusion process defined by

$$\begin{aligned} dX_t &= \alpha V_t dt, \\ dV_t &= -\gamma V_t dt - D_m F(\mathrm{Law}(X_t), X_t) dt + \sigma dW_t, \end{aligned} \qquad (2.14)$$

with $\alpha, \gamma, \sigma > 0$, we introduce the new variables:

$$x' = \frac{(2\gamma^3)^{1/2}}{\alpha\sigma} x, \quad v' = \frac{(2\gamma)^{1/2}}{\sigma} v, \quad t' = \gamma^{-1}t,$$

define $m'$ to be the push-out of measure $m$ under $x \mapsto x'$, and set

$$F'(m') = \left( \frac{2}{\gamma\sigma^2} \right)^{1/2} F(m).$$

Then the stochastic process $t' \mapsto (X'_{t'}, V'_{t'})$ satisfy

$$\begin{aligned} dX'_{t'} &= V'_{t'} dt', \\ dV'_{t'} &= -V'_{t'} dt' - D_m F'(\mathrm{Law}(X'_{t'}), X'_{t'}) dt + \sqrt{2} dW'_{t'}, \end{aligned}$$
where $W'_{t} := \gamma^{1/2}W_{t}$ is a standard Brownian motion. In the same way we can treat the particle system defined by

$$ 
\begin{aligned}
dX_t^i &= \alpha V_t^i dt \\
dV_t^i &= -\gamma V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sigma dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^{N} \delta_{X_t^j}.
\end{aligned} 
\quad (2.15) $$

**Free energies and invariant measures.** For measures $m \in \mathcal{P}_2(\mathbb{R}^{2d})$, $m^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, we introduce the mean field and $N$-particle free energies:

$$ \mathcal{F}(m) = F(m^x) + \frac{1}{2} \int |v|^2 m(dx dv) + H(m), \quad (2.16) $$

$$ \mathcal{F}^N(m^N) = \int \left( NF(\mu_{\mathbf{x}}) + \frac{1}{2} |\mathbf{v}|^2 \right) m^N(dx dv) + H(m^N). \quad (2.17) $$

The functionals are well defined with values in $(-\infty, +\infty]$. We will also work with probability measures, $m_\infty \in \mathcal{P}_2(\mathbb{R}^{2d})$ and $m_\infty^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, satisfying

$$ m_\infty(dx dv) \propto \exp\left(-\frac{\delta F}{\delta m}(m_\infty^x, x) - \frac{1}{2}|v|^2\right) dx dv, \quad (2.18) $$

$$ m_\infty^N(dx dv) \propto \exp\left(-NF(\mu_{\mathbf{x}}) - \frac{1}{2}|\mathbf{v}|^2\right) dx dv, \quad (2.19) $$

and having finite exponential moments, i.e., the integrals $\int \exp(\alpha(|x| + |v|))m_\infty(dx dv)$ and $\int \exp(\alpha(|x| + |v|))m_\infty^N(dx dv)$ are finite for every $\alpha \ge 0$. We call $m_\infty$, $m_\infty^N$ invariant measures to the dynamics (2.12) and (2.13) respectively. The existence and uniqueness of the invariant measures are guaranteed by our assumptions (2.1) to (2.3), as will be stated in Lemma 4.1.

**Main results.** Recall that $m_t$ and $m_t^N$ are the respective marginal distributions of the mean field and the $N$-particle system (2.10) and (2.11). We first prove the exponential entropic convergence result for the MFL dynamics (2.10).

**Theorem 2.1 (Entropic convergence of MFL).** Assume $F$ satisfies (2.1) to (2.3). If $m_0$ has finite second moment, finite entropy and finite Fisher information, then there exist constants

$$ C_0 = C_0(M_{mx}^F, M_{mm}^F), \quad \kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F) $$

such that for every $t \ge 0$,

$$ \mathcal{F}(m_t) - \mathcal{F}(m_\infty) \le (\mathcal{F}(m_0) - \mathcal{F}(m_\infty) + C_0 I(m_0|\hat{m}_0))e^{-\kappa t}. \quad (2.20) $$

The proof of the theorem is postponed to Section 4.2. We note that the proof only relies on the $W_2$-Lipschitz continuity of $m \mapsto D_m F(m, x)$, contrary to the $W_1$ one stated in (2.2).

Our second major contribution is the uniform-in-$N$ exponential entropic convergence of the particle systems.

**Theorem 2.2 (Entropic convergence of particle systems).** Assume $F$ satisfies (2.1) to (2.3). If $m_0^N$ has finite second moment, finite entropy and finite Fisher information for some $N \ge 2$, then there exist constants

$$ C_0 = C_0(M_{mx}^F, M_{mm}^F), \quad C_1 = C_1(\rho^x, M_{mx}^F, M_{mm}^F), \quad \kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F) $$
where $W'_{t} := \gamma^{1/2}W_{t}$ is a standard Brownian motion. In the same way we can treat the particle system defined by

$$ 
\begin{aligned}
dX_t^i &= \alpha V_t^i dt \\
dV_t^i &= -\gamma V_t^i dt - D_m F(\mu_{\mathbf{X}_t}, X_t^i) dt + \sigma dW_t^i, \quad \text{where } \mu_{\mathbf{X}_t} = \frac{1}{N} \sum_{j=1}^{N} \delta_{X_t^j}.
\end{aligned} 
\quad (2.15) $$

**Free energies and invariant measures.** For measures $m \in \mathcal{P}_2(\mathbb{R}^{2d})$, $m^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, we introduce the mean field and $N$-particle free energies:

$$ \mathcal{F}(m) = F(m^x) + \frac{1}{2} \int |v|^2 m(dxdv) + H(m), \quad (2.16) $$

$$ \mathcal{F}^N(m^N) = \int \left( NF(\mu_{\mathbf{x}}) + \frac{1}{2} |\mathbf{v}|^2 \right) m^N(d\mathbf{x}d\mathbf{v}) + H(m^N). \quad (2.17) $$

The functionals are well defined with values in $(-\infty, +\infty]$. We will also work with probability measures, $m_\infty \in \mathcal{P}_2(\mathbb{R}^{2d})$ and $m_\infty^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, satisfying

$$ m_\infty(dxdv) \propto \exp\left(-\frac{\delta F}{\delta m}(m_\infty^x, x) - \frac{1}{2}|v|^2\right)dx dv, \quad (2.18) $$

$$ m_\infty^N(d\mathbf{x}d\mathbf{v}) \propto \exp\left(-NF(\mu_{\mathbf{x}}) - \frac{1}{2}|\mathbf{v}|^2\right)d\mathbf{x}d\mathbf{v}, \quad (2.19) $$

and having finite exponential moments, i.e., the integrals $\int \exp(\alpha(|\mathbf{x}| + |\mathbf{v}|))m_\infty(dxdv)$ and $\int \exp(\alpha(|\mathbf{x}| + |\mathbf{v}|))m_\infty^N(dxdv)$ are finite for every $\alpha \ge 0$. We call $m_\infty$, $m_\infty^N$ invariant measures to the dynamics (2.12) and (2.13) respectively. The existence and uniqueness of the invariant measures are guaranteed by our assumptions (2.1) to (2.3), as will be stated in Lemma 4.1.

**Main results.** Recall that $m_t$ and $m_t^N$ are the respective marginal distributions of the mean field and the $N$-particle system (2.10) and (2.11). We first prove the exponential entropic convergence result for the MFL dynamics (2.10).

**Theorem 2.1 (Entropic convergence of MFL).** Assume $F$ satisfies (2.1) to (2.3). If $m_0$ has finite second moment, finite entropy and finite Fisher information, then there exist constants

$$ C_0 = C_0(M_{mx}^F, M_{mm}^F), \quad \kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F) $$

such that for every $t \ge 0$,

$$ \mathcal{F}(m_t) - \mathcal{F}(m_\infty) \le (\mathcal{F}(m_0) - \mathcal{F}(m_\infty) + C_0 I(m_0|\hat{m}_0))e^{-\kappa t}. \quad (2.20) $$

The proof of the theorem is postponed to Section 4.2. We note that the proof only relies on the $W_2$-Lipschitz continuity of $m \mapsto D_m F(m, x)$, contrary to the $W_1$ one stated in (2.2).

Our second major contribution is the uniform-in-$N$ exponential entropic convergence of the particle systems.

**Theorem 2.2 (Entropic convergence of particle systems).** Assume $F$ satisfies (2.1) to (2.3). If $m_0^N$ has finite second moment, finite entropy and finite Fisher information for some $N \ge 2$, then there exist constants

$$ C_0 = C_0(M_{mx}^F, M_{mm}^F), \quad C_1 = C_1(\rho^x, M_{mx}^F, M_{mm}^F), \quad \kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F) $$
such that if $N > C_1/\kappa$, then for every $t \ge 0$,

$$ \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) \le (\mathcal{F}(m_0^N) - N\mathcal{F}(m_\infty) + C_0 I(m_0^N|m_\infty^N))e^{-(\kappa-C_1/N)t} + \frac{C_1 d}{\kappa-C_1/N}. \quad (2.21) $$

The proof of the theorem is postponed to Section 4.3.

**Remark 2.2.** Strictly speaking, the result (2.21) does not imply that the particle systems converge uniformly. We only show $\frac{1}{N}\mathcal{F}^N(m_t^N)$ approaches the mean field minimum $\mathcal{F}(m_\infty)$ uniformly quickly until they are $O(N^{-1})$-close to each other.

**Remark 2.3.** Theorems 2.1 and 2.2 state results concerning the convergence of the respective free energies, which we will also call “convergence of entropy” or “entropic convergence”, since in both cases the differences of free energies are related to relative entropies, as shown in Lemmas 4.2 and 4.3.

We now present the main theorem, which establishes the uniform-in-time propagation of chaos in both the Wasserstein distance and the relative entropy. The results are direct consequences of the exponential convergence in Theorems 2.1 and 2.2 and the regularization phenomena to be studied in Section 5.

**Theorem 2.3 (Wasserstein and entropic propagation of chaos).** Assume $F$ satisfies (2.1) to (2.4). If $m_0$ belongs to $\mathcal{P}_2(\mathbb{R}^d)$ and $m_0^N$ belongs to $\mathcal{P}_2(\mathbb{R}^{dN})$ for some $N \ge 2$, then there exist constants $C_1 = C_1(M_{mx}^F, M_{mm}^F, M_{mmm}^F)$, $C_2 = C_2(\rho^x, M_{mx}^F, M_{mm}^F)$ and $\kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F)$ such that if $N > C_2/\kappa$, then for every $t > 0$,

$$ W_2^2(m_t^N, m_t^{\otimes N}) \le \min \left\{ C_1 W_2^2(m_0^N, m_0^{\otimes N}) e^{C_1 t} + C_1(e^{C_1 t} - 1)(\mathrm{Var} m_0 + d), \frac{C_2 N}{(t \wedge 1)^6} W_2^2(m_0, m_\infty) e^{-\kappa t} + \frac{C_2}{(t \wedge 1)^6} W_2^2(m_0^N, m_\infty^{\otimes N}) e^{-(\kappa - C_2/N)t} + \frac{C_2 d}{\kappa - C_2/N} \right\}; \quad (2.22) $$

moreover, for every $t$ and $s$ such that $s+1 \ge t > s \ge 0$,

$$ H(m_t^N|m_t^{\otimes N}) \le \frac{C_1}{(t-s)^3} W_2^2(m_s^N, m_s^{\otimes N}) + C_1(e^{C_1(t-s)} - 1)(\mathrm{Var} m_s + d). \quad (2.23) $$

The proof of the theorem is postponed to Section 5.4.

**Comments on the assumptions.** Compared to our previous work [7], we have removed the technical assumption that $x \mapsto D_m F(m, x)$ has bounded higher-order derivatives by a mollifying procedure of the mean field functional. However, the spatial Lipschitz constant $M_{mx}^F$, appearing in the assumption (2.2), will contribute to the constants, especially the rate of convergence $\kappa$, in our theorems. Nevertheless, this behavior is expected for kinetic dynamics, as this dependency is already present for the linear Fokker-Planck dynamics in [40]. Finally, we introduce the new condition (2.4) on the second and third-order derivatives in measure of the mean field functional. The condition (2.4) is used to obtain $O(1)$ errors in the propagation of chaos bounds (2.22) and (2.23) in Theorem 2.3, which are stronger than the dimension-dependent errors obtained from the method of Fournier and Guillin [12].
such that if $N > C_1/\kappa$, then for every $t \ge 0$,

$$ \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) \le (\mathcal{F}(m_0^N) - N\mathcal{F}(m_\infty) + C_0 I(m_0^N|m_\infty^N))e^{-(\kappa-C_1/N)t} + \frac{C_1d}{\kappa-C_1/N}. \quad (2.21) $$

The proof of the theorem is postponed to Section 4.3.

**Remark 2.2.** Strictly speaking, the result (2.21) does not imply that the particle systems converge uniformly. We only show $\frac{1}{N}\mathcal{F}^N(m_t^N)$ approaches the mean field minimum $\mathcal{F}(m_\infty)$ uniformly quickly until they are $O(N^{-1})$-close to each other.

**Remark 2.3.** Theorems 2.1 and 2.2 state results concerning the convergence of the respective free energies, which we will also call “convergence of entropy” or “entropic convergence”, since in both cases the differences of free energies are related to relative entropies, as shown in Lemmas 4.2 and 4.3.

We now present the main theorem, which establishes the uniform-in-time propagation of chaos in both the Wasserstein distance and the relative entropy. The results are direct consequences of the exponential convergence in Theorems 2.1 and 2.2 and the regularization phenomena to be studied in Section 5.

**Theorem 2.3 (Wasserstein and entropic propagation of chaos).** Assume $F$ satisfies (2.1) to (2.4). If $m_0$ belongs to $P_2(\mathbb{R}^d)$ and $m_0^N$ belongs to $P_2(\mathbb{R}^{dN})$ for some $N \ge 2$, then there exist constants $C_1 = C_1(M_{mx}^F, M_{mm}^F, M_{mmm}^F)$, $C_2 = C_2(\rho^x, M_{mx}^F, M_{mm}^F)$ and $\kappa = \kappa(\rho^x, M_{mx}^F, M_{mm}^F)$ such that if $N > C_2/\kappa$, then for every $t > 0$,

$$ W_2^2(m_t^N, m_t^{\otimes N}) \le \min \left\{ C_1 W_2^2(m_0^N, m_0^{\otimes N}) e^{C_1 t} + C_1(e^{C_1 t} - 1)(\text{Var } m_0 + d), \frac{C_2 N}{(t \wedge 1)^6} W_2^2(m_0, m_\infty) e^{-\kappa t} + \frac{C_2}{(t \wedge 1)^6} W_2^2(m_0^N, m_\infty^{\otimes N}) e^{-(\kappa - C_2/N)t} + \frac{C_2 d}{\kappa - C_2/N} \right\}; \quad (2.22) $$

moreover, for every $t$ and $s$ such that $s+1 \ge t > s \ge 0$,

$$ H(m_t^N|m_t^{\otimes N}) \le \frac{C_1}{(t-s)^3} W_2^2(m_s^N, m_s^{\otimes N}) + C_1(e^{C_1(t-s)} - 1)(\text{Var } m_s + d). \quad (2.23) $$

The proof of the theorem is postponed to Section 5.4.

**Comments on the assumptions.** Compared to our previous work [7], we have removed the technical assumption that $x \mapsto D_m F(m, x)$ has bounded higher-order derivatives by a mollifying procedure of the mean field functional. However, the spatial Lipschitz constant $M_{mx}^F$, appearing in the assumption (2.2), will contribute to the constants, especially the rate of convergence $\kappa$, in our theorems. Nevertheless, this behavior is expected for kinetic dynamics, as this dependency is already present for the linear Fokker-Planck dynamics in [40]. Finally, we introduce the new condition (2.4) on the second and third-order derivatives in measure of the mean field functional. The condition (2.4) is used to obtain $O(1)$ errors in the propagation of chaos bounds (2.22) and (2.23) in Theorem 2.3, which are stronger than the dimension-dependent errors obtained from the method of Fournier and Guillin [12].
## 3 Application: training neural networks by momentum gradient descent

We have given in Section 3 of our previous work [7] several examples of mean field functionals satisfying conditions (2.1) to (2.3) of our theorems, and the only additional condition that remains to verify is the bound on the higher-order measure derivative (2.4). In the following we will recall the mean field formulation of the loss of two-layer neural networks and its corresponding kinetic dynamics (see [7, Examples 2 and 4]), and verify that it satisfies indeed the additional assumption.

### 3.1 Mean field formulation of neural network

Recall that the structure of a two-layer neural network is determined by its feature map:

$$ \mathbb{R}^d \ni z \mapsto \Phi(\theta; z) := \ell(c)\varphi(a \cdot z + b) \in \mathbb{R}^{d'}, $$

where $\theta := (c, a, b) \in \mathbb{R}^{d'} \times \mathbb{R}^d \times \mathbb{R} := S$ is the parameter of a single neuron, $\varphi: \mathbb{R} \to \mathbb{R}$ is a non-linear activation function satisfying the squashing condition (see [7, (3.4)]), and $\ell: \mathbb{R} \to [-L, L]$ is a truncation function with threshold $L \in (0, +\infty)$. Here the action of the truncation is tensorized: $\ell(c) = \ell(c^1, \dots, c^{d'}) := (\ell(c^1), \dots, \ell(c^{d'}))$ for a $d'$-dimensional vector $c = (c^1, \dots, c^{d'})$. Then given $N$ neurons with respective parameters $\theta^1, \dots, \theta^N$, the associated network's output reads

$$ \mathbb{R}^d \ni z \mapsto \Phi^N(\theta^1, \dots, \theta^N; z) = \frac{1}{N} \sum_{i=1}^N \Phi(\theta^i; z) \in \mathbb{R}. \quad (3.1) $$

Here $z$ should be considered as the input of the network, i.e. the feature, and the value $\Phi^N(\theta^1, \dots, \theta^N; z)$ should correspond to the label. We wish to find the optimal neuron parameters $(\theta_i)_{i=1}^N$ for a possibly unknown distribution of feature-label tuples $\mu \in \mathcal{P}(\mathbb{R}^{d+d'})$. In order to quantify the goodness of networks, we define the loss:

$$ F_{\text{NNet}}^N(\theta^1, \dots, \theta^N) = \frac{N}{2} \int |y - \Phi^N(\theta^1, \dots, \theta^N; z)|^2 \mu(dz dy). \quad (3.2) $$

It is proposed in [18, 7] that instead of minimizing the original loss (3.2), we consider the mean field output function $\mathbb{E}^{\Theta \sim m}[\Phi(\Theta; \cdot)]$ and minimize the mean field loss

$$ F_{\text{NNet}}^N(m) = \int |y - \mathbb{E}^{\Theta \sim m}[\Phi(\Theta; z)]|^2 \mu(dz dy). \quad (3.3) $$

We also add a quadratic regularizer

$$ F_{\text{Ext}}(m) = \frac{\lambda}{2} \int |\theta|^2 m(d\theta) $$

with regularization parameter $\lambda > 0$. The final optimization problem then reads

$$ \inf_{m \in \mathcal{P}_2(S)} F(m) := F_{\text{NNet}}(m) + F_{\text{Ext}}(m). \quad (3.4) $$

Following the calculations in [7] we can show that if both the truncation and activation function are bounded and has bounded derivatives of up-to-second order, then the conditions (2.1) to (2.3) are verified. Finally, the third-order derivatives $\frac{\delta^3 F}{\delta m^3}$ is a constant thanks to the fact that the loss function is quadratic, and therefore the condition (2.4) is satisfied with $M_{mmm}^F = 0$.
# 3 Application: training neural networks by momentum gradient descent

We have given in Section 3 of our previous work [7] several examples of mean field functionals satisfying conditions (2.1) to (2.3) of our theorems, and the only additional condition that remains to verify is the bound on the higher-order measure derivative (2.4). In the following we will recall the mean field formulation of the loss of two-layer neural networks and its corresponding kinetic dynamics (see [7, Examples 2 and 4]), and verify that it satisfies indeed the additional assumption.

## 3.1 Mean field formulation of neural network

Recall that the structure of a two-layer neural network is determined by its feature map:

$$ \mathbb{R}^d \ni z \mapsto \Phi(\theta; z) := \ell(c)\varphi(a \cdot z + b) \in \mathbb{R}^{d'}, $$

where $\theta := (c, a, b) \in \mathbb{R}^{d'} \times \mathbb{R}^d \times \mathbb{R} := S$ is the parameter of a single neuron, $\varphi: \mathbb{R} \to \mathbb{R}$ is a non-linear activation function satisfying the squashing condition (see [7, (3.4)]), and $\ell: \mathbb{R} \to [-L, L]$ is a truncation function with threshold $L \in (0, +\infty)$. Here the action of the truncation is tensorized: $\ell(c) = \ell(c^1, \dots, c^{d'}) := (\ell(c^1), \dots, \ell(c^{d'}))$ for a $d'$-dimensional vector $c = (c^1, \dots, c^{d'})$. Then given $N$ neurons with respective parameters $\theta^1, \dots, \theta^N$, the associated network's output reads

$$ \mathbb{R}^d \ni z \mapsto \Phi^N(\theta^1, \dots, \theta^N; z) = \frac{1}{N} \sum_{i=1}^{N} \Phi(\theta^i; z) \in \mathbb{R}. \quad (3.1) $$

Here $z$ should be considered as the input of the network, i.e. the feature, and the value $\Phi^N(\theta^1, \dots, \theta^N; z)$ should correspond to the label. We wish to find the optimal neuron parameters $(\theta_i)_{i=1}^N$ for a possibly unknown distribution of feature-label tuples $\mu \in \mathcal{P}(\mathbb{R}^{d+d'})$. In order to quantify the goodness of networks, we define the loss:

$$ F_{\text{NNet}}^N(\theta^1, \dots, \theta^N) = \frac{N}{2} \int |y - \Phi^N(\theta^1, \dots, \theta^N; z)|^2 \mu(dz dy). \quad (3.2) $$

It is proposed in [18, 7] that instead of minimizing the original loss (3.2), we consider the mean field output function $\mathbb{E}^{\Theta \sim m}[\Phi(\Theta; \cdot)]$ and minimize the mean field loss

$$ F_{\text{NNet}}^N(m) = \int |y - \mathbb{E}^{\Theta \sim m}[\Phi(\Theta; z)]|^2 \mu(dz dy). \quad (3.3) $$

We also add a quadratic regularizer

$$ F_{\text{Ext}}(m) = \frac{\lambda}{2} \int |\theta|^2 m(d\theta) $$

with regularization parameter $\lambda > 0$. The final optimization problem then reads

$$ \inf_{m \in \mathcal{P}_2(S)} F(m) := F_{\text{NNet}}(m) + F_{\text{Ext}}(m). \quad (3.4) $$

Following the calculations in [7] we can show that if both the truncation and activation function are bounded and has bounded derivatives of up-to-second order, then the conditions (2.1) to (2.3) are verified. Finally, the third-order derivatives $\frac{\delta^3 F}{\delta m^3}$ is a constant thanks to the fact that the loss function is quadratic, and therefore the condition (2.4) is satisfied with $M_{mmm}^F = 0$.
**Remark 3.1.** Following [7, Remark 3.6], we recognize that the SDE (2.10) describes the continuous version of the gradient descent algorithm with momentum. Among various momentum gradient descent methods commonly used to train neural networks, the most prevalent ones are RMSProp and Adam algorithms (see [17, 22]), where the momentum is accumulated and the step size is adapted along the dynamics. In [26, 37, 31] the authors studied the convergence of these momentum-based algorithms and compared them to algorithms without momentum based on optimization theory. We note that estimates of the discretization error and optimal parameters can also be found in these studies.

## 3.2 Numerical experiments

We present our numerical experiments in this section. Our experiments are based on the discretized version of a particle system dynamics (2.15). We first explain the optimization problem and the numerical algorithm, and then present our two experiments: the first investigates the convergence behavior as the number of particles tends to infinity, and the second compares the kinetic dynamics to the corresponding overdamped dynamics.

**Problem setup and momentum algorithm.** We aim to solve a supervised learning problem: our goal is to classify the handwritten digits "4" and "6" by a two-layer neural network. We randomly choose $K = 10^4$ samples from the MNIST dataset [25] and denote by $(z_k)_{k=1}^K$ the figures in 28 × 28 pixel format, i.e. each $z_k$ belongs to $\mathbb{R}^{28 \times 28} = \mathbb{R}^{784}$, and by $(y_k)_{k=1}^K$ the one-hot vectors for the two classes of digits, i.e. if the k-th figure corresponds to the digit "4", then $y_k = (1, 0)$, otherwise $y_k = (0, 1)$. See Figure 1 for random samples in the dataset. We choose $N$ particles and use the sigmoid function as the activation, i.e. $\varphi(x) = 1/(1 + \exp(-x))$. The truncation function is fixed by

$$ \ell(x) = L \tanh(x/L) = L \frac{\exp(2x/L) - 1}{\exp(2x/L) + 1} $$

and its threshold equals $L$. The quadratic regularization parameter is denoted by $\lambda$. Following the arguments of [7] and the precedent section, all the conditions of our theorems (2.1) to (2.4) are satisfied. In the beginning of training process, the neuron positions $(\Theta_0^i)_{i=1}^N = (C_0^{x,i}, A_0^{x,i}, B_0^{x,i})_{i=1}^N$ and momenta $(\Psi_0^i)_{i=1}^N = (C_0^{v,i}, A_0^{v,i}, B_0^{v,i})_{i=1}^N$ are sampled independently from a given initial distribution $m_0^x, m_0^v \in \mathcal{P}(\mathbb{R}^2 \times \mathbb{R}^{784} \times \mathbb{R})$. We

<figure><img src="image_3.png" /><figcaption>Figure 1: Randomly chosen handwritten digits "4" and "6" from the MNIST dataset.</figcaption></figure>
**Remark 3.1.** Following [7, Remark 3.6], we recognize that the SDE (2.10) describes the continuous version of the gradient descent algorithm with momentum. Among various momentum gradient descent methods commonly used to train neural networks, the most prevalent ones are RMSProp and Adam algorithms (see [17, 22]), where the momentum is accumulated and the step size is adapted along the dynamics. In [26, 37, 31] the authors studied the convergence of these momentum-based algorithms and compared them to algorithms without momentum based on optimization theory. We note that estimates of the discretization error and optimal parameters can also be found in these studies.

## 3.2 Numerical experiments

We present our numerical experiments in this section. Our experiments are based on the discretized version of a particle system dynamics (2.15). We first explain the optimization problem and the numerical algorithm, and then present our two experiments: the first investigates the convergence behavior as the number of particles tends to infinity, and the second compares the kinetic dynamics to the corresponding overdamped dynamics.

**Problem setup and momentum algorithm.** We aim to solve a supervised learning problem: our goal is to classify the handwritten digits "4" and "6" by a two-layer neural network. We randomly choose $K = 10^4$ samples from the MNIST dataset [25] and denote by $(z_k)_{k=1}^K$ the figures in 28 × 28 pixel format, i.e. each $z_k$ belongs to $\mathbb{R}^{28 \times 28} = \mathbb{R}^{784}$, and by $(y_k)_{k=1}^K$ the one-hot vectors for the two classes of digits, i.e. if the k-th figure corresponds to the digit "4", then $y_k = (1, 0)$, otherwise $y_k = (0, 1)$. See Figure 1 for random samples in the dataset. We choose $N$ particles and use the sigmoid function as the activation, i.e. $\varphi(x) = 1/(1 + \exp(-x))$. The truncation function is fixed by

$$ \ell(x) = L \tanh(x/L) = L \frac{\exp(2x/L) - 1}{\exp(2x/L) + 1} $$

and its threshold equals $L$. The quadratic regularization parameter is denoted by $\lambda$. Following the arguments of [7] and the precedent section, all the conditions of our theorems (2.1) to (2.4) are satisfied. In the beginning of training process, the neuron positions $(\Theta_0^i)_{i=1}^N = (C_0^{x,i}, A_0^{x,i}, B_0^{x,i})_{i=1}^N$ and momenta $(\Psi_0^i)_{i=1}^N = (C_0^{v,i}, A_0^{v,i}, B_0^{v,i})_{i=1}^N$ are sampled independently from a given initial distribution $m_0^x, m_0^v \in \mathcal{P}(\mathbb{R}^2 \times \mathbb{R}^{784} \times \mathbb{R})$. We

<figure><img src="image_3.png" /><figcaption>Figure 1: Randomly chosen handwritten digits "4" and "6" from the MNIST dataset.</figcaption></figure>
**Algorithm 1:** Noised momentum gradient descent for training a two-layer neural network

**Input:** number of particles *N*, truncation *L*, data set $(z_k, y_k)_{k=1}^K$, noise $\sigma$, friction $\gamma$, $l_2$ regularization $\lambda$, initial distribution $(m_0^x, m_0^v)$, time step $\Delta t$, time horizon *T*

**Output:** $(\Theta_T^i)_{i=1}^N$
generate i.i.d. $\Theta_0^i = (A_0^{x,i}, B_0^{x,i}, C_0^{x,i}) \sim m_0^x$ for $i = 1, \dots, N$;
generate i.i.d. $\Psi_0^i = (A_0^{v,i}, B_0^{v,i}, C_0^{v,i}) \sim m_0^v$ for $i = 1, \dots, N$;
for $t = 0, \Delta t, 2\Delta t, \dots, T - \Delta t$ do

generate i.i.d. $N_t^i \sim N(0, 1)$ for $i = 1, \dots, N$;
// update particles according to discretized underdamped Langevin
for $i = 1, \dots, N$ do
  $\Psi_{t+\Delta t}^i \leftarrow (1-\gamma\Delta t)\Psi_t^i - D_m F_{NNet}(\frac{1}{N}\sum_{j=1}^N \delta_{\Theta_t^j}, \Theta_t^i)\Delta t - \lambda\Theta_t^i\Delta t + \sigma\sqrt{\Delta t}N_t^i$;
  $\Theta_{t+\Delta t}^i \leftarrow \Theta_t^i + \Psi_{t+\Delta t}^i\Delta t$;
  // where $D_m F_{NNet}(\frac{1}{N}\sum_{j=1}^N \delta_{\Theta_t^j}, \Theta_t^i) = \frac{1}{K}\sum_{k=1}^K (y_k - \Psi^N(\Theta_t^1, \dots, \Theta_t^N; z_k))\frac{\partial \Psi^N}{\partial \theta}(\Theta_t^i; z_k)$

update the parameters $(\Theta_0^i)_{i=1}^N$ and $(\Psi_0^i)_{i=1}^N$ following the discrete-time version of the underdamped Langevin SDE (2.14) with fixed set of parameters $(\alpha, \gamma, \sigma)$, that is, we calculate the neurons' evolution by Algorithm 1.

**Convergence when** $N \to +\infty$. To study the behavior of the momentum training dynamics when $N \to +\infty$ we conduct independent experiments with an increasing number of particles: $N = 2^P$ for $P = 5, 6, \dots, 10$ and repeat the experiment 10 times for each $N$. The hyperparameters for this experiment are listed in the second column of Table 1.

<table><caption>Table 1: Hyperparameters of neural networks' trainings.</caption><thead><tr><th>Hyperparameter</th><th>First Exp.'s Value</th><th>Second Exp.'s Value</th></tr></thead><tbody><tr><td>N</td><td>[128, 256, 512, 1024, 2048]</td><td>256</td></tr><tr><td>Δt</td><td>0.02</td><td>0.01</td></tr><tr><td>T</td><td>300</td><td>500</td></tr><tr><td>m<sub>0</sub><sup>x</sup></td><td>N(0, 0.01)</td><td>N(0, 0.01)</td></tr><tr><td>m<sub>0</sub><sup>v</sup></td><td>N(0, 0.25)</td><td>N(0, 0.01)</td></tr><tr><td>L</td><td>500</td><td>500</td></tr><tr><td>λ</td><td>10<sup>-4</sup></td><td>10<sup>-3</sup></td></tr><tr><td>α</td><td>1</td><td>1</td></tr><tr><td>γ</td><td>0.1</td><td>0.1</td></tr><tr><td>σ</td><td>0.01√2</td><td>0.01√2</td></tr></tbody></table>

To quantity the convergence, we compute $\frac{1}{N}F_{NNet}^N(\Theta_t^1, \dots, \Theta_t^N)$ and $\frac{1}{N}F_{Kinet}^N(\Psi_t^1, \dots, \Psi_t^N)$, where $F_{Kinet}^N(\Psi^1, \dots, \Psi^N) := \frac{1}{2}\sum_{i=1}^N |\Psi^i|^2$. We then compute its average of the respective quantities over the 10 repeated runs. The evolutions of $\frac{1}{N}F_{NNet}^N$ and $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$ are plotted in Figure 2 and Figure 3 respectively, and can be characterized by two distinct phases. In the first phase, both the quantities decrease and the second quantity decreases exponentially, for every $N$. We also find that in this phase the
**Algorithm 1:** Noised momentum gradient descent for training a two-layer neural network

**Input:** number of particles $N$, truncation $L$, data set $(z_k, y_k)_{k=1}^K$, noise $\sigma$, friction $\gamma$, $l_2$ regularization $\lambda$, initial distribution $(m_0^x, m_0^v)$, time step $\Delta t$, time horizon $T$

**Output:** $(\Theta_T^i)_{i=1}^N$
generate i.i.d. $\Theta_0^i = (A_0^{x,i}, B_0^{x,i}, C_0^{x,i}) \sim m_0^x$ for $i = 1, \dots, N$;
generate i.i.d. $\Psi_0^i = (A_0^{v,i}, B_0^{v,i}, C_0^{v,i}) \sim m_0^v$ for $i = 1, \dots, N$;
for $t = 0, \Delta t, 2\Delta t, \dots, T - \Delta t$ do

generate i.i.d. $N_t^i \sim N(0, 1)$ for $i = 1, \dots, N$;
// update particles according to discretized underdamped Langevin
for $i = 1, \dots, N$ do
$$ \begin{aligned} \Psi_{t+\Delta t}^i &\leftarrow (1-\gamma\Delta t)\Psi_t^i - D_m F_{NNet}\left(\frac{1}{N}\sum_{j=1}^N \delta_{\Theta_t^j}, \Theta_t^i\right)\Delta t - \lambda\Theta_t^i\Delta t + \sigma\sqrt{\Delta t}N_t^i; \\ \Theta_{t+\Delta t}^i &\leftarrow \Theta_t^i + \Psi_{t+\Delta t}^i\Delta t; \\ // \text{where } D_m F_{NNet}\left(\frac{1}{N}\sum_{j=1}^N \delta_{\Theta_t^j}, \Theta_t^i\right) &= \frac{1}{K}\sum_{k=1}^K (y_k - \Psi^N(\Theta_t^1, \dots, \Theta_t^N; z_k))\frac{\partial \Psi^N}{\partial \theta}(\Theta_t^i; z_k) \end{aligned} $$

update the parameters $(\Theta_0^i)_{i=1}^N$ and $(\Psi_0^i)_{i=1}^N$ following the discrete-time version of the underdamped Langevin SDE (2.14) with fixed set of parameters $(\alpha, \gamma, \sigma)$, that is, we calculate the neurons' evolution by Algorithm 1.

**Convergence when** $N \to +\infty$. To study the behavior of the momentum training dynamics when $N \to +\infty$ we conduct independent experiments with an increasing number of particles: $N = 2^P$ for $P = 5, 6, \dots, 10$ and repeat the experiment 10 times for each $N$. The hyperparameters for this experiment are listed in the second column of Table 1.

<table><caption>Table 1: Hyperparameters of neural networks' trainings.</caption><thead><tr><th>Hyperparameter</th><th>First Exp.'s Value</th><th>Second Exp.'s Value</th></tr></thead><tbody><tr><td>N</td><td>[128, 256, 512, 1024, 2048]</td><td>256</td></tr><tr><td>Δt</td><td>0.02</td><td>0.01</td></tr><tr><td>T</td><td>300</td><td>500</td></tr><tr><td>m<sub>0</sub><sup>x</sup></td><td>N(0, 0.01)</td><td>N(0, 0.01)</td></tr><tr><td>m<sub>0</sub><sup>v</sup></td><td>N(0, 0.25)</td><td>N(0, 0.01)</td></tr><tr><td>L</td><td>500</td><td>500</td></tr><tr><td>λ</td><td>10<sup>-4</sup></td><td>10<sup>-3</sup></td></tr><tr><td>α</td><td>1</td><td>1</td></tr><tr><td>γ</td><td>0.1</td><td>0.1</td></tr><tr><td>σ</td><td>0.01√2</td><td>0.01√2</td></tr></tbody></table>

To quantity the convergence, we compute $\frac{1}{N}F_{NNet}^N(\Theta_t^1, \dots, \Theta_t^N)$ and $\frac{1}{N}F_{Kinet}^N(\Psi_t^1, \dots, \Psi_t^N)$, where $F_{Kinet}^N(\Psi^1, \dots, \Psi^N) := \frac{1}{2}\sum_{i=1}^N |\Psi^i|^2$. We then compute its average of the respective quantities over the 10 repeated runs. The evolutions of $\frac{1}{N}F_{NNet}^N$ and $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$ are plotted in Figure 2 and Figure 3 respectively, and can be characterized by two distinct phases. In the first phase, both the quantities decrease and the second quantity decreases exponentially, for every $N$. We also find that in this phase the
<figure><img src="image_4.png" /><figcaption>Figure 2: Individual (shadowed) and 10-averaged (bold) losses without kinetic energy versus time.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Individual (shadowed) and 10-averaged (bold) losses with kinetic energy versus time.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Average values of $\frac{1}{N}F_{NNet} + \frac{1}{N}F_{Kinet}$ over the last 500 epochs. The mean (black squares) and standard derivations (error bars) are calculated from the 10 independent runs. Dashed curve fits the data.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Target function $\frac{1}{N}F_{NNet}$ for underdamped Langevin (blue) and overdamped Langevin (red) versus time.</figcaption></figure>

convergence rates are almost the same for different $N$ and this is coherent with the behavior indicated by our theoretical upper bound (2.21). We also observe that $\frac{1}{N}F_{NNet}^N$ fluctuates in a stronger way than $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$. In the second phase, both the values cease to decrease but the remnant values differ for different $N$.

To investigate the relationship between the remnant values in the second phase and the number of particles $N$, we compute the average value of $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$ of the last 500 training epochs for each individual run and plot their values in Figure 4. Motivated by the upper bound (2.21) in Theorem 2.2, we fit the remnant values by $C' + \frac{C}{N}$ and find the values are well fitted by this curve.

**Comparison to algorithm without momentum.** We also investigate the difference between gradient descent algorithms with and without momentum by working on the same set of hyperparameters, listed in the last column of Table 1. It is found that the algorithm with momentum leads to much stronger fluctuations compared the algorithm without momentum (see Figure 5). Both algorithms cease to decrease after certain
<figure><img src="image_4.png" /><figcaption>Figure 2: Individual (shadowed) and 10-averaged (bold) losses without kinetic energy versus time.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Individual (shadowed) and 10-averaged (bold) losses with kinetic energy versus time.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Average values of $\frac{1}{N}F_{NNet} + \frac{1}{N}F_{Kinet}$ over the last 500 epochs. The mean (black squares) and standard derivations (error bars) are calculated from the 10 independent runs. Dashed curve fits the data.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Target function $\frac{1}{N}F_{NNet}$ for underdamped Langevin (blue) and overdamped Langevin (red) versus time.</figcaption></figure>

convergence rates are almost the same for different $N$ and this is coherent with the behavior indicated by our theoretical upper bound (2.21). We also observe that $\frac{1}{N}F_{NNet}^N$ fluctuates in a stronger way than $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$. In the second phase, both the values cease to decrease but the remnant values differ for different $N$.

To investigate the relationship between the remnant values in the second phase and the number of particles $N$, we compute the average value of $\frac{1}{N}F_{NNet}^N + \frac{1}{N}F_{Kinet}^N$ of the last 500 training epochs for each individual run and plot their values in Figure 4. Motivated by the upper bound (2.21) in Theorem 2.2, we fit the remnant values by $C' + \frac{C}{N}$ and find the values are well fitted by this curve.

**Comparison to algorithm without momentum.** We also investigate the difference between gradient descent algorithms with and without momentum by working on the same set of hyperparameters, listed in the last column of Table 1. It is found that the algorithm with momentum leads to much stronger fluctuations compared the algorithm without momentum (see Figure 5). Both algorithms cease to decrease after certain
training epochs, but the momentum algorithm leads to better loss in the end. This may be explained by the fact that the presence of momentum helps the particles to escape local minima.

# 4 Entropic convergence

## 4.1 Collection of known results

Before moving on to the proofs, we first state some elementary results without proofs. They are either immediate consequences of the corresponding ones in our previous work [7], or easy adaptations thereof.

**Lemma 4.1** (Existence and uniqueness of invariant measures). If $F$ satisfies (2.1) to (2.3), then there exist unique measures $m_\infty$ and $m_\infty^N$ satisfying (2.18) and (2.19) respectively and they have finite exponential moments.

**Lemma 4.2** (Mean field entropy sandwich). Assume $F$ satisfies (2.1) to (2.3). Then for every $m \in \mathcal{P}_2(\mathbb{R}^{2d})$, we have

$$H(m|m_\infty) \le \mathcal{F}(m) - \mathcal{F}(m_\infty) \le H(m|\hat{m}) \le \left(1 + \frac{M_{mm}^F}{\rho} + \frac{(M_{mm}^F)^2}{2\rho^2}\right) H(m|m_\infty), \quad (4.1)$$

where $\rho$ is defined by (2.6). Here, the leftmost inequality holds even without the uniform LSI condition (2.3), once there exists a measure $m_\infty$ satisfying (2.18) and having finite exponential moments.

**Lemma 4.3** (Particle system's entropy inequality). Assume that $F$ satisfies (2.1) and that there exists a measure $m_\infty \in \mathcal{P}_2(\mathbb{R}^{2d})$ verifying (2.18). Then for all $m^N \in \mathcal{P}_2(\mathbb{R}^{dN})$ of finite entropy, we have

$$H(m^N|m_\infty^N) \le \mathcal{F}^N(m^N) - N\mathcal{F}(m_\infty). \quad (4.2)$$

**Lemma 4.4** (Information inequalities). Let $X_1, \dots, X_N$ be measurable spaces, $\mu$ be a probability on the product space $X = X_1 \times \dots \times X_N$ and $\nu = \nu^1 \otimes \dots \otimes \nu^N$ be a $\sigma$-finite measure. Then

$$\sum_{i=1}^{N} H(\mu^i|\nu^i) \le H(\mu|\nu) \le \sum_{i=1}^{N} \int H(\mu^{i|-i}(\cdot|x^{-i})|\nu^i)\mu^{-i}(dx^{-i}). \quad (4.3)$$

Here we set the rightmost term to $+\infty$ if the conditional distribution $\mu^{i|-i}$ does not exist $\mu^{-i}$-a.e.

## 4.2 Mean field system

In this section we study the mean field system described by the Fokker-Planck equation (2.12) and the SDE (2.10). Our aim is to prove Theorem 2.1. To this end, we first show its wellposedness and regularity.

**Lemma 4.5.** Suppose $F$ satisfies (2.2). Then for every initial value $m_0$ of finite second moment, the equation (2.12) admits a unique solution in $C([0, \infty); \mathcal{P}_2(\mathbb{R}^d))$. Moreover, for every $t > 0$, the measure $m_t$ is absolutely continuous with respect to the Lebesgue measure.

*Proof.* Since the drift $D_m F(\cdot, \cdot)$ of the SDE system (2.10) is jointly Lipschitz in measure and in space by our condition (2.2), the existence and uniqueness of the solution is standard.
training epochs, but the momentum algorithm leads to better loss in the end. This may be explained by the fact that the presence of momentum helps the particles to escape local minima.

# 4 Entropic convergence

## 4.1 Collection of known results

Before moving on to the proofs, we first state some elementary results without proofs. They are either immediate consequences of the corresponding ones in our previous work [7], or easy adaptations thereof.

**Lemma 4.1** (Existence and uniqueness of invariant measures). If $F$ satisfies (2.1) to (2.3), then there exist unique measures $m_\infty$ and $m_\infty^N$ satisfying (2.18) and (2.19) respectively and they have finite exponential moments.

**Lemma 4.2** (Mean field entropy sandwich). Assume $F$ satisfies (2.1) to (2.3). Then for every $m \in \mathcal{P}_2(\mathbb{R}^{2d})$, we have

$$H(m|m_\infty) \le \mathcal{F}(m) - \mathcal{F}(m_\infty) \le H(m|\hat{m}) \le \left(1 + \frac{M_{mm}^F}{\rho} + \frac{(M_{mm}^F)^2}{2\rho^2}\right) H(m|m_\infty), \quad (4.1)$$

where $\rho$ is defined by (2.6). Here, the leftmost inequality holds even without the uniform LSI condition (2.3), once there exists a measure $m_\infty$ satisfying (2.18) and having finite exponential moments.

**Lemma 4.3** (Particle system's entropy inequality). Assume that $F$ satisfies (2.1) and that there exists a measure $m_\infty \in \mathcal{P}_2(\mathbb{R}^{2d})$ verifying (2.18). Then for all $m^N \in \mathcal{P}_2(\mathbb{R}^{dN})$ of finite entropy, we have

$$H(m^N|m_\infty^N) \le \mathcal{F}^N(m^N) - N\mathcal{F}(m_\infty). \quad (4.2)$$

**Lemma 4.4** (Information inequalities). Let $X_1, \dots, X_N$ be measurable spaces, $\mu$ be a probability on the product space $X = X_1 \times \dots \times X_N$ and $\nu = \nu^1 \otimes \dots \otimes \nu^N$ be a $\sigma$-finite measure. Then

$$\sum_{i=1}^{N} H(\mu^i|\nu^i) \le H(\mu|\nu) \le \sum_{i=1}^{N} \int H(\mu^{i|-i}(\cdot|x^{-i})|\nu^i)\mu^{-i}(dx^{-i}). \quad (4.3)$$

Here we set the rightmost term to $+\infty$ if the conditional distribution $\mu^{i|-i}$ does not exist $\mu^{-i}$-a.e.

## 4.2 Mean field system

In this section we study the mean field system described by the Fokker-Planck equation (2.12) and the SDE (2.10). Our aim is to prove Theorem 2.1. To this end, we first show its wellposedness and regularity.

**Lemma 4.5.** Suppose $F$ satisfies (2.2). Then for every initial value $m_0$ of finite second moment, the equation (2.12) admits a unique solution in $C([0, \infty); \mathcal{P}_2(\mathbb{R}^d))$. Moreover, for every $t > 0$, the measure $m_t$ is absolutely continuous with respect to the Lebesgue measure.

*Proof.* Since the drift $D_m F(\cdot, \cdot)$ of the SDE system (2.10) is jointly Lipschitz in measure and in space by our condition (2.2), the existence and uniqueness of the solution is standard.
To show the existence of density we recall Kolmogorov's fundamental solution

$$ \rho_t(x, v; x', v') := \left(\frac{\sqrt{3}}{2\pi t^2}\right)^d \exp\left(-\frac{3|x - (x' + tv')|^2}{t^3} + \frac{3(x - (x' + tv')) \cdot (v - v')}{t^2} - \frac{|v - v'|^2}{t}\right) $$

associated to the differential operator $\partial_t - \Delta_v + v \cdot \nabla_x$. Then the Duhamel's formula holds in the sense of distributions:

$$ m_t = \int \rho_t(\cdot; z')m_0(dz') + \int_0^t \iint \rho_s(\cdot; x', v')\nabla_{v'} \cdot \left(m_{t-s}(dx'dv')(v' + D_m F(m_{t-s}^x, x'))\right)ds. \quad (4.4) $$

Since the first moment of $m_t$ is bounded, that is, for every $T > 0$, $\sup_{t \in [0,T]} \int (|v| + |x|)m_t(dxdv) < +\infty$, we can integrate by parts in the second term of (4.4) and obtain

$$ \|m_t\|_{L^1} \le 1 + C \int_0^t \sup_{x',v'} \|\nabla_{v'} \rho_s(\cdot; x', v')\|_{L^1} ds. $$

By explicit computations we have $\sup_{x',v'} \|\nabla_{v'} \rho_s(\cdot; x', v')\|_{L^1} = O(s^{-1/2})$, from which the existence of the density follows. □

We now introduce a technical condition on the mean field functional: the mapping $x \mapsto D_m F(m, x)$ is fourth-order differentiable with derivatives continuous in measure and in space, and satisfying

$$ \sup_{m \in \mathcal{P}_2(\mathbb{R}^d)} \sup_{x \in \mathbb{R}^d} |\nabla^k D_m F(m, x)| < +\infty, \quad k = 2, 3, 4. \quad (4.5) $$

This condition will be used to derive some intermediate results in the following studies of the mean field dynamics.

**Definition 4.6 (Standard algebra).** We define the standard algebra $\mathcal{A}_+$ to be the set of $C^4$ functions $h : \mathbb{R}^{2d} \to (0, \infty)$ for which there exists a constant $C$ such that

$$ |\log h(x, v)| \le C(1 + |x| + |v|) \quad \text{and} \quad \sum_{k=1}^4 |\nabla^k h(x, v)| \le \exp(C(1 + |x| + |v|)) $$

holds for every $(x, v) \in \mathbb{R}^{2d}$. For a collection of functions $(h_\iota)_{\iota \in I}$ we say $h_\iota \in \mathcal{A}_+$ uniformly for $\iota \in I$ or $(h_\iota)_{\iota \in I} \subset \mathcal{A}_+$ uniformly, if there exists a constant $C$ such that the previous bounds holds for every $h_\iota$, $\iota \in I$.

**Proposition 4.7 (Density of $\mathcal{A}_+$).** Assume $F$ satisfies (2.2) and (4.5) and there exists a measure $m_\infty$ satisfying (2.18) and having finite exponential moments. Then for every $m \in \mathcal{P}_2(\mathbb{R}^d)$ with finite entropy and finite Fisher information, there exists a sequence of measures $(m_n)_{n \in \mathbb{N}}$ such that $m_n/m_\infty \in \mathcal{A}_+$ and

$$ W_2(m_n, m) \to 0, \quad H(m_n) \to H(m), \quad I(m_n) \to I(m) $$

when $n \to +\infty$.
To show the existence of density we recall Kolmogorov's fundamental solution

$$ \rho_t(x, v; x', v') := \left(\frac{\sqrt{3}}{2\pi t^2}\right)^d \exp\left(-\frac{3|x - (x' + tv')|^2}{t^3} + \frac{3(x - (x' + tv')) \cdot (v - v')}{t^2} - \frac{|v - v'|^2}{t}\right) $$

associated to the differential operator $\partial_t - \Delta_v + v \cdot \nabla_x$. Then the Duhamel's formula holds in the sense of distributions:

$$ m_t = \int \rho_t(\cdot; z') m_0(dz') + \int_0^t \iint \rho_s(\cdot; x', v') \nabla_{v'} \cdot \left(m_{t-s}(dx'dv')(v' + D_m F(m_{t-s}^x, x'))\right) ds. \quad (4.4) $$

Since the first moment of $m_t$ is bounded, that is, for every $T > 0$, $\sup_{t \in [0,T]} \int (|v| + |x|)m_t(dxdv) < +\infty$, we can integrate by parts in the second term of (4.4) and obtain

$$ \|m_t\|_{L^1} \le 1 + C \int_0^t \sup_{x', v'} \|\nabla_{v'} \rho_s(\cdot; x', v')\|_{L^1} ds. $$

By explicit computations we have $\sup_{x', v'} \|\nabla_{v'} \rho_s(\cdot; x', v')\|_{L^1} = O(s^{-1/2})$, from which the existence of the density follows. □

We now introduce a technical condition on the mean field functional: the mapping $x \mapsto D_m F(m, x)$ is fourth-order differentiable with derivatives continuous in measure and in space, and satisfying

$$ \sup_{m \in \mathcal{P}_2(\mathbb{R}^d)} \sup_{x \in \mathbb{R}^d} |\nabla^k D_m F(m, x)| < +\infty, \quad k = 2, 3, 4. \quad (4.5) $$

This condition will be used to derive some intermediate results in the following studies of the mean field dynamics.

**Definition 4.6 (Standard algebra).** We define the standard algebra $\mathcal{A}_+$ to be the set of $C^4$ functions $h : \mathbb{R}^{2d} \to (0, \infty)$ for which there exists a constant $C$ such that

$$ |\log h(x, v)| \le C(1 + |x| + |v|) \quad \text{and} \quad \sum_{k=1}^4 |\nabla^k h(x, v)| \le \exp(C(1 + |x| + |v|)) $$

holds for every $(x, v) \in \mathbb{R}^{2d}$. For a collection of functions $(h_\iota)_{\iota \in I}$ we say $h_\iota \in \mathcal{A}_+$ uniformly for $\iota \in I$ or $(h_\iota)_{\iota \in I} \subset \mathcal{A}_+$ uniformly, if there exists a constant $C$ such that the previous bounds holds for every $h_\iota$, $\iota \in I$.

**Proposition 4.7 (Density of $\mathcal{A}_+$).** Assume $F$ satisfies (2.2) and (4.5) and there exists a measure $m_\infty$ satisfying (2.18) and having finite exponential moments. Then for every $m \in \mathcal{P}_2(\mathbb{R}^d)$ with finite entropy and finite Fisher information, there exists a sequence of measures $(m_n)_{n \in \mathbb{N}}$ such that $m_n/m_\infty \in \mathcal{A}_+$ and

$$ W_2(m_n, m) \to 0, \quad H(m_n) \to H(m), \quad I(m_n) \to I(m) $$

when $n \to +\infty$.
Proof. Let $\varepsilon$ be arbitrary positive real. Put $h = m/m_\infty$. Define $h'_n = (h \wedge n) \vee \frac{1}{n}$ and the associated probability measure $m'_n = h'_n m_\infty / \int h'_n m_\infty$. Let $N \in \mathbb{N}$ be big enough so that $\int h'_N m_\infty > 0$. Note that

$$ \sup_{n \ge N} |x|^2 m'_n(x) \le \frac{|x|^2 m}{\int h'_N m_\infty}, $$

that is to say, the second moments of $(m'_n)_{n \in \mathbb{N}}$ are uniformly bounded. Together with the fact that the density of $m'_n$ converges to that of $m$ pointwise, we have $m'_n \to m$ in $\mathcal{P}_2$. By the dominated convergence theorem, the sequence of measures $m'_n$ satisfies

$$ H(m'_n) = \frac{\int \log(h'_n m_\infty) h'_n m_\infty}{\int h'_n m_\infty} - \log \int h'_n m_\infty \to \int m \log m, \quad \text{when } n \to +\infty. $$

Moreover, we have the convergence of Fisher information as

$$ \int \frac{|\nabla(h'_n m_\infty)|^2}{h'_n m_\infty} = \int \left[ \frac{|\nabla m_\infty|^2 h'_n}{m_\infty} + \left( 2 \frac{\nabla h \cdot \nabla m_\infty}{hm_\infty} + \frac{|\nabla h|^2 m_\infty}{h} \right) \mathbf{1}_{1/n \le h \le n} \right] $$

converges to $I(m)$ when $n \to +\infty$, where we used the fact that the weak derivatives satisfy $\nabla h'_n = \nabla h \mathbf{1}_{1/n \le h \le n}$. Hence we may choose $n_0 \in \mathbb{N}$ such that

$$ W_2(m_{n_0}, m) + |H(m_{n_0}) - H(m)| + |I(m_{n_0}) - I(m)| \le \frac{\varepsilon}{2}. $$

Now set $m''_n = m'_{n_0} \star \eta_n$, where $(\eta_n)_{n \in \mathbb{N}}$ is a sequence of smooth mollifiers supported in the unit ball. We have $m''_n \to m'_{n_0}$ in $\mathcal{P}_2$. By the convexity of entropy and Fisher information we have $H(m''_n) \le H(m'_{n_0})$ and $I(m''_n) \le I(m'_{n_0})$, and by the lower semicontinuities in Lemma A.1 we have $\liminf_{n \to +\infty} H(m''_n) \ge H(m'_{n_0})$ and $\liminf_{n \to +\infty} I(m''_n) \ge I(m'_{n_0})$. Hence,

$$ W_2(m''_n, m'_{n_0}) + |H(m''_n) - H(m'_{n_0})| + |I(m''_n) - I(m'_{n_0})| \to 0 $$

when $n \to +\infty$. So we pick another $n_1 \in \mathbb{N}$ such that $W_2(m''_{n_1}, m'_{n_0}) + |H(m''_{n_1}) - H(m'_{n_0})| + |I(m''_{n_1}) - I(m'_{n_0})| < \varepsilon/2$.

It remains to verify that $m''_{n_1}/m_\infty$ belongs to $\mathcal{A}_+$. By the definition we have

$$ \frac{m''_{n_1}}{m_\infty} = \frac{(h''m_\infty) \star \rho_{n_1}}{m_\infty} $$

for some $h''$ with $0 < \inf h'' \le \sup h'' < +\infty$. Hence for every $z \in \mathbb{R}^{2d}$,

$$ \inf h'' \frac{\inf_{B(z,1)} m_\infty}{m_\infty(z)} \le \frac{m''_{n_1}(z)}{m_\infty(z)} \le \sup h'' \frac{\sup_{B(z,1)} m_\infty}{m_\infty(z)}. $$

On the other hand, the gradient of $m_\infty$ satisfies $|\nabla \log m_\infty(z)| \le |D_m F(m_\infty, x)| + |v|$ for every $z = (x, v) \in \mathbb{R}^{2d}$. In particular, we have

$$ \exp(-C(1 + |x| + |v|)) \le \frac{\inf_{B(z,1)} m_\infty}{m_\infty(z)} \le \frac{\sup_{B(z,1)} m_\infty}{m_\infty(z)} \le \exp(C(1 + |x| + |v|)), $$

for some constant $C$. Therefore, $m''_{n_1}/m_\infty$ verifies the first condition of $\mathcal{A}_+$.

Now verify the conditions on the derivatives. The derivatives read

$$ \nabla^k \left( \frac{m''_{n_1}}{m_\infty} \right) = \sum_{j=0}^{k} \binom{k}{j} \frac{(h''m_\infty) \star \nabla^j \rho_{n_1}}{m_\infty} \cdot m_\infty \nabla^{k-j} (m_\infty^{-1}). $$

For each term in the sum, we can bound its first part by

$$ \left| \frac{(h''m_\infty) \star \nabla^j \rho_{n_1}(z)}{m_\infty(z)} \right| \le \exp(C(1 + |z|)), $$
Proof. Let $\varepsilon$ be arbitrary positive real. Put $h = m/m_\infty$. Define $h'_n = (h \wedge n) \vee \frac{1}{n}$ and the associated probability measure $m'_n = h'_n m_\infty / \int h'_n m_\infty$. Let $N \in \mathbb{N}$ be big enough so that $\int h'_N m_\infty > 0$. Note that

$$ \sup_{n \ge N} |x|^2 m'_n(x) \le \frac{|x|^2 m}{\int h'_N m_\infty}, $$

that is to say, the second moments of $(m'_n)_{n \in \mathbb{N}}$ are uniformly bounded. Together with the fact that the density of $m'_n$ converges to that of $m$ pointwise, we have $m'_n \to m$ in $\mathcal{P}_2$. By the dominated convergence theorem, the sequence of measures $m'_n$ satisfies

$$ H(m'_n) = \frac{\int \log(h'_n m_\infty) h'_n m_\infty}{\int h'_n m_\infty} - \log \int h'_n m_\infty \to \int m \log m, \quad \text{when } n \to +\infty. $$

Moreover, we have the convergence of Fisher information as

$$ \int \frac{|\nabla(h'_n m_\infty)|^2}{h'_n m_\infty} = \int \left[ \frac{|\nabla m_\infty|^2 h'_n}{m_\infty} + \left( 2 \frac{\nabla h \cdot \nabla m_\infty}{h m_\infty} + \frac{|\nabla h|^2 m_\infty}{h} \right) \mathbf{1}_{1/n \le h \le n} \right] $$

converges to $I(m)$ when $n \to +\infty$, where we used the fact that the weak derivatives satisfy $\nabla h'_n = \nabla h \mathbf{1}_{1/n \le h \le n}$. Hence we may choose $n_0 \in \mathbb{N}$ such that

$$ W_2(m_{n_0}, m) + |H(m_{n_0}) - H(m)| + |I(m_{n_0}) - I(m)| \le \frac{\varepsilon}{2}. $$

Now set $m''_n = m'_{n_0} \star \eta_n$, where $(\eta_n)_{n \in \mathbb{N}}$ is a sequence of smooth mollifiers supported in the unit ball. We have $m''_n \to m'_{n_0}$ in $\mathcal{P}_2$. By the convexity of entropy and Fisher information we have $H(m''_n) \le H(m'_{n_0})$ and $I(m''_n) \le I(m'_{n_0})$, and by the lower semicontinuities in Lemma A.1 we have $\liminf_{n \to +\infty} H(m''_n) \ge H(m'_{n_0})$ and $\liminf_{n \to +\infty} I(m''_n) \ge I(m'_{n_0})$. Hence,

$$ W_2(m''_n, m'_{n_0}) + |H(m''_n) - H(m'_{n_0})| + |I(m''_n) - I(m'_{n_0})| \to 0 $$

when $n \to +\infty$. So we pick another $n_1 \in \mathbb{N}$ such that $W_2(m''_{n_1}, m'_{n_0}) + |H(m''_{n_1}) - H(m'_{n_0})| + |I(m''_{n_1}) - I(m'_{n_0})| < \varepsilon/2$.

It remains to verify that $m''_{n_1}/m_\infty$ belongs to $\mathcal{A}_+$. By the definition we have

$$ \frac{m''_{n_1}}{m_\infty} = \frac{(h'' m_\infty) \star \rho_{n_1}}{m_\infty} $$

for some $h''$ with $0 < \inf h'' \le \sup h'' < +\infty$. Hence for every $z \in \mathbb{R}^{2d}$,

$$ \inf h'' \frac{\inf_{B(z,1)} m_\infty}{m_\infty(z)} \le \frac{m''_{n_1}(z)}{m_\infty(z)} \le \sup h'' \frac{\sup_{B(z,1)} m_\infty}{m_\infty(z)}. $$

On the other hand, the gradient of $m_\infty$ satisfies $|\nabla \log m_\infty(z)| \le |D_m F(m_\infty, x)| + |v|$ for every $z = (x, v) \in \mathbb{R}^{2d}$. In particular, we have

$$ \exp(-C(1 + |x| + |v|)) \le \frac{\inf_{B(z,1)} m_\infty}{m_\infty(z)} \le \frac{\sup_{B(z,1)} m_\infty}{m_\infty(z)} \le \exp(C(1 + |x| + |v|)), $$

for some constant $C$. Therefore, $m''_{n_1}/m_\infty$ verifies the first condition of $\mathcal{A}_+$.

Now verify the conditions on the derivatives. The derivatives read

$$ \nabla^k \left( \frac{m''_{n_1}}{m_\infty} \right) = \sum_{j=0}^{k} \binom{k}{j} \frac{(h'' m_\infty) \star \nabla^j \rho_{n_1}}{m_\infty} \cdot m_\infty \nabla^{k-j} (m_\infty^{-1}). $$

For each term in the sum, we can bound its first part by

$$ \left| \frac{(h'' m_\infty) \star \nabla^j \rho_{n_1}(z)}{m_\infty(z)} \right| \le \exp(C(1 + |z|)), $$
using the same method that we used to verify the first condition of $A_+$. Moreover, since our assumptions (2.2) and (4.5) imply

$$|\nabla \log m_{\infty}(z)| \le C(1 + |z|) \quad \text{and} \quad |\nabla^k \log m_{\infty}(z)| \le C \quad \text{for } k = 2, 3, 4,$$

the second part of each term of the sum, $m_{\infty} \nabla^{k-j}(m_{\infty}^{-1})$, is of polynomial growth. The proof is then complete. □

Then we show the stability of the set $A_+$ under the mean field flow. This property will be used to justify the computations in the proof of Theorem 2.1, as is usual in the analysis of PDE.

**Proposition 4.8** (Stability of $A_+$ under flow). Assume that $F$ satisfies (2.2) and (4.5) and that there exists a measure $m_{\infty}$ satisfying (2.18) and having finite exponential moments. Let $(m_t)_{t \in [0,T]} \in C([0,T]; P_2(\mathbb{R}^d))$ be a solution in the sense of distributions to the mean field Fokker-Planck equation (2.12). If $m_0/m_{\infty} \in A_+$, then $m_t/m_{\infty} \in A_+$ uniformly for $t \in [0,T]$. In particular, $m_t$ is a classical solution to the Fokker-Planck equation.

*Proof.* In the following $C$ will denote a constant depending on $M_{mx}^F, M_{mm}^F$, the initial value $h_0 = h(0, \cdot) := m_0/m_{\infty}$, the time interval $T$ and the bounds on the higher-order derivatives $\max_{k=2,3,4} \sup_{m,x} |\nabla^k D_m F(m, x)|$, and it may change from line to line. For a given quantity $Q$, we denote by $C_Q$ a constant depending additionally on $Q$.

Denote $b_t(x) = -D_m F(m_t, x)$ and $b_{\infty}(x) = -D_m F(m_{\infty}, x)$. We also define $h_t(x) = m_t(x)/m_{\infty}(x)$. The relative density solves

$$\partial_t h = \Delta_v h - v \cdot \nabla_v h - v \cdot \nabla_x h - b_t \cdot \nabla_v h + (b_t - b_{\infty}) \cdot v h. \quad (4.6)$$

Fix $t \in [0, T]$. We construct for every $z = (x, v) \in \mathbb{R}^{2d}$, the stochastic process $Z_s^{t,z} = (X^{t,z}, V^{t,z})$, solving

$$dX_s^{t,z} = -V_s^{t,z} ds, \\ dV_s^{t,z} = -V_s^{t,z} ds - b_{t-s}(X_s^{t,z}) ds + \sqrt{2} dW_s$$

for $s \in [0, t]$ with the initial value $X_0^{t,z} = x, V_0^{t,z} = v$ and the same Brownian motion $(W_s)_{s \in [0, t]}$.

*Regularity of $Z_s^{t,z}$.* Set $M^{t,z} = \sup_{s \in [0,t]} |Z_s^{t,z}|$. By Itō's formula and Doob's maximal inequality, the processes satisfy for every $\alpha \ge 0$,

$$\mathbb{E}[\exp(\alpha M^{t,z})] \le \exp(C_{\alpha}(1 + |z|)). \quad (4.7)$$

Thanks to the assumption on the uniform boundedness of the higher-order derivatives (4.5), the mapping $z \mapsto Z_s^{t,z}$ is $C^4$ and the partial derivatives solve the Cauchy-Lipschitz SDEs for $k = 1, 2, 3, 4$:

$$d\nabla^k X_s^{t,z} = -\nabla^k V_s^{t,z} ds, \\ d\nabla^k V_s^{t,z} = -\nabla^k V_s^{t,z} ds - \sum_{j=1}^k \nabla^j b_{t-s}(X_s^{t,z}) B_{k,j}(\nabla X_s^{t,z}, \dots, \nabla^{k-j+1} X_s^{t,z}) ds,$$

where $B_{k,j}$ is a $k-j+1$-variate polynomial and in particular $B_{k,1}(x_1, \dots, x_k) = x_k$. The initial values of the SDEs read

$$\nabla Z_0^{t,z} = \text{Id} \quad \text{and} \quad \nabla^k Z_0^{t,z} = 0 \quad \text{for } k = 2, 3, 4.$$
using the same method that we used to verify the first condition of $\mathcal{A}_+$. Moreover, since our assumptions (2.2) and (4.5) imply

$$|\nabla \log m_{\infty}(z)| \le C(1 + |z|) \quad \text{and} \quad |\nabla^k \log m_{\infty}(z)| \le C \quad \text{for } k = 2, 3, 4,$$

the second part of each term of the sum, $m_{\infty} \nabla^{k-j}(m_{\infty}^{-1})$, is of polynomial growth. The proof is then complete. $\square$

Then we show the stability of the set $\mathcal{A}_+$ under the mean field flow. This property will be used to justify the computations in the proof of Theorem 2.1, as is usual in the analysis of PDE.

**Proposition 4.8 (Stability of $\mathcal{A}_+$ under flow).** Assume that $F$ satisfies (2.2) and (4.5) and that there exists a measure $m_{\infty}$ satisfying (2.18) and having finite exponential moments. Let $(m_t)_{t \in [0,T]} \in C([0,T]; \mathcal{P}_2(\mathbb{R}^d))$ be a solution in the sense of distributions to the mean field Fokker-Planck equation (2.12). If $m_0/m_{\infty} \in \mathcal{A}_+$, then $m_t/m_{\infty} \in \mathcal{A}_+$ uniformly for $t \in [0,T]$. In particular, $m_t$ is a classical solution to the Fokker-Planck equation.

*Proof.* In the following $C$ will denote a constant depending on $M_{mx}^F$, $M_{mm}^F$, the initial value $h_0 = h(0, \cdot) := m_0/m_{\infty}$, the time interval $T$ and the bounds on the higher-order derivatives $\max_{k=2,3,4} \sup_{m,x} |\nabla^k D_m F(m, x)|$, and it may change from line to line. For a given quantity $Q$, we denote by $C_Q$ a constant depending additionally on $Q$.

Denote $b_t(x) = -D_m F(m_t, x)$ and $b_{\infty}(x) = -D_m F(m_{\infty}, x)$. We also define $h_t(x) = m_t(x)/m_{\infty}(x)$. The relative density solves

$$\partial_t h = \Delta_v h - v \cdot \nabla_v h - v \cdot \nabla_x h - b_t \cdot \nabla_v h + (b_t - b_{\infty}) \cdot v h. \quad (4.6)$$

Fix $t \in [0, T]$. We construct for every $z = (x, v) \in \mathbb{R}^{2d}$, the stochastic process $Z_s^{t,z} = (X^{t,z}, V^{t,z})$, solving

$$dX_s^{t,z} = -V_s^{t,z} ds, \\ dV_s^{t,z} = -V_s^{t,z} ds - b_{t-s}(X_s^{t,z}) ds + \sqrt{2} dW_s$$

for $s \in [0, t]$ with the initial value $X_0^{t,z} = x$, $V_0^{t,z} = v$ and the same Brownian motion $(W_s)_{s \in [0, t]}$.

*Regularity of $Z_s^{t,z}$. Set $M^{t,z} = \sup_{s \in [0,t]} |Z_s^{t,z}|$. By Itō's formula and Doob's maximal inequality, the processes satisfy for every $\alpha \ge 0$,

$$\mathbb{E}[\exp(\alpha M^{t,z})] \le \exp(C_{\alpha}(1 + |z|)). \quad (4.7)$$

Thanks to the assumption on the uniform boundedness of the higher-order derivatives (4.5), the mapping $z \mapsto Z_s^{t,z}$ is $C^4$ and the partial derivatives solve the Cauchy-Lipschitz SDEs for $k = 1, 2, 3, 4$:

$$d\nabla^k X_s^{t,z} = -\nabla^k V_s^{t,z} ds, \\ d\nabla^k V_s^{t,z} = -\nabla^k V_s^{t,z} ds - \sum_{j=1}^k \nabla^j b_{t-s}(X_s^{t,z}) B_{k,j}(\nabla X_s^{t,z}, \dots, \nabla^{k-j+1} X_s^{t,z}) ds,$$

where $B_{k,j}$ is a $k-j+1$-variate polynomial and in particular $B_{k,1}(x_1, \dots, x_k) = x_k$. The initial values of the SDEs read

$$\nabla Z_0^{t,z} = \text{Id} \quad \text{and} \quad \nabla^k Z_0^{t,z} = 0 \quad \text{for } k = 2, 3, 4.$$
By induction we can obtain the almost sure bound

$$ \max_{k=1,2,3,4} \sup_{s \in [0,t]} |\nabla^k Z_s^{t,z}| \le C. \quad (4.8) $$

Regularity of h by Feynman-Kac. Denote $g(t, z) = g(t, x, v) = (b_t(x) - b_\infty(x)) \cdot v$. It satisfies

$$ |g(t, z)| \le M_{mm}^F W_2(m_t, m_\infty) |v| \le M_{mm}^F \sup_{t \in [0,T]} W_2(m_t, m_\infty) |v| = C|v| $$

and also $|\nabla^k g(t, z)| \le C(1 + |z|)$ for $k = 1, 2, 3, 4$. The Feynman-Kac formula for the parabolic equation (4.6) reads

$$ h(t, z) = \mathbb{E} \left[ \exp \left( \int_0^t g(t-s, Z_s^{t,z}) ds \right) h(0, Z_t^{t,z}) \right]. \quad (4.9) $$

Using the method in the proof of [7, Proposition 4.12], we can prove

$$ |\log h(t, z)| \le C(1 + |z|). $$

Moreover, thanks to the estimates (4.8), we can apply the dominated convergence theorem to the Feynman-Kac formula (4.9) and obtain that $z \mapsto h(t, z)$ belongs to $C^4$ with partial derivatives

$$ \nabla^k h(t, z) = \sum_{j=0}^{k} \mathbb{E} \left[ \exp \left( \int_0^t g(t-s, Z_s^{t,z}) ds \right) P_j \left( \int_0^t \nabla_z g(t-s, Z_s^{t,z}) ds, \dots, \int_0^t \nabla_z^j g(t-s, Z_s^{t,z}) ds \right) \nabla_z^{k-j} h(0, Z_t^{t,z}) \right], $$

where $P_j$ is a j-variate polynomial. Note that

$$ \nabla_z^k f(Z_s^{t,z}) = \sum_{\ell=1}^{k} \nabla^\ell f(Z_s^{t,z}) B_{k,\ell} (\nabla Z_s^{t,z}, \dots, \nabla^{k-\ell+1} Z_s^{t,z}) $$

holds for $f = g(t - s, \cdot)$, $s \in [0, t]$ and for $f = h(0, \cdot)$. We apply the bounds on $|\nabla^k g|, |\nabla^k h|$ for $k = 0, 1, 2, 3, 4$ and the exponential moment bound (4.7) to obtain that $|\nabla^k h(t, z)| \le \exp(C(1 + |z|))$ for $k = 1, 2, 3, 4$. Finally, the derivatives $\nabla h, \nabla^2 h$ exist and one can show that they are continuous in time by differentiating (4.6) twice in space. So again by the equation (4.6) we have $\partial_t h$ is continuous and therefore exists classically. Thus $m_t$ is a classical solution to the Fokker-Planck equation (2.12). $\square$

**Remark 4.1.** The polynomials appearing in the previous proof belong to the non-commutative free algebras over $\mathbb{R}$ of respective number of indeterminates instead of the usual polynomial rings, as the tensor product is not commutative.

After the technical preparations we prove Theorem 2.1.

**Proof of Theorem 2.1.** Suppose first that the mean field functional $F$ satisfies additionally (4.5) and the initial value of the dynamics is such that $m_0/m_\infty$ belongs to $A_+$, which is the standard algebra defined in Definition 4.6. According to Proposition 4.8, the measure $m_t$ belongs to $A_+$ uniformly in $t$, for every $T > 0$. Since we have that $z \mapsto \hat{m}_t(z)/m_\infty(z)$ is $C^4$ with

$$ \sup_{z \in \mathbb{R}^{2d}} |\nabla \log \frac{\hat{m}_t}{m_\infty}(z)| \le M_{mm}^F W_2(m_t, m_\infty) $$
By induction we can obtain the almost sure bound

$$ \max_{k=1,2,3,4} \sup_{s \in [0,t]} |\nabla^k Z_s^{t,z}| \le C. \quad (4.8) $$

Regularity of h by Feynman-Kac. Denote $g(t, z) = g(t, x, v) = (b_t(x) - b_\infty(x)) \cdot v$. It satisfies

$$ |g(t, z)| \le M_{mm}^F W_2(m_t, m_\infty) |v| \le M_{mm}^F \sup_{t \in [0,T]} W_2(m_t, m_\infty) |v| = C|v| $$

and also $|\nabla^k g(t, z)| \le C(1 + |z|)$ for $k = 1, 2, 3, 4$. The Feynman-Kac formula for the parabolic equation (4.6) reads

$$ h(t, z) = \mathbb{E} \left[ \exp \left( \int_0^t g(t-s, Z_s^{t,z}) ds \right) h(0, Z_t^{t,z}) \right]. \quad (4.9) $$

Using the method in the proof of [7, Proposition 4.12], we can prove

$$ |\log h(t, z)| \le C(1 + |z|). $$

Moreover, thanks to the estimates (4.8), we can apply the dominated convergence theorem to the Feynman-Kac formula (4.9) and obtain that $z \mapsto h(t, z)$ belongs to $C^4$ with partial derivatives

$$ \nabla^k h(t, z) = \sum_{j=0}^k \mathbb{E} \left[ \exp \left( \int_0^t g(t-s, Z_s^{t,z}) ds \right) P_j \left( \int_0^t \nabla_z g(t-s, Z_s^{t,z}) ds, \dots, \right. \right. \\ \left. \left. \int_0^t \nabla_z^j g(t-s, Z_s^{t,z}) ds \right) \nabla_z^{k-j} h(0, Z_t^{t,z}) \right], $$

where $P_j$ is a j-variate polynomial. Note that

$$ \nabla_z^k f(Z_s^{t,z}) = \sum_{\ell=1}^k \nabla^\ell f(Z_s^{t,z}) B_{k,\ell} (\nabla Z_s^{t,z}, \dots, \nabla^{k-\ell+1} Z_s^{t,z}) $$

holds for $f = g(t - s, \cdot)$, $s \in [0, t]$ and for $f = h(0, \cdot)$. We apply the bounds on $|\nabla^k g|, |\nabla^k h|$ for $k = 0, 1, 2, 3, 4$ and the exponential moment bound (4.7) to obtain that $|\nabla^k h(t, z)| \le \exp(C(1 + |z|))$ for $k = 1, 2, 3, 4$. Finally, the derivatives $\nabla h, \nabla^2 h$ exist and one can show that they are continuous in time by differentiating (4.6) twice in space. So again by the equation (4.6) we have $\partial_t h$ is continuous and therefore exists classically. Thus $m_t$ is a classical solution to the Fokker-Planck equation (2.12). $\square$

**Remark 4.1.** The polynomials appearing in the previous proof belong to the non-commutative free algebras over $\mathbb{R}$ of respective number of indeterminates instead of the usual polynomial rings, as the tensor product is not commutative.

After the technical preparations we prove Theorem 2.1.

**Proof of Theorem 2.1.** Suppose first that the mean field functional $F$ satisfies additionally (4.5) and the initial value of the dynamics is such that $m_0/m_\infty$ belongs to $\mathcal{A}_+$, which is the standard algebra defined in Definition 4.6. According to Proposition 4.8, the measure $m_t$ belongs to $\mathcal{A}_+$ uniformly in $t$, for every $T > 0$. Since we have that $z \mapsto \hat{m}_t(z)/m_\infty(z)$ is $C^4$ with

$$ \sup_{z \in \mathbb{R}^{2d}} |\nabla \log \frac{\hat{m}_t}{m_\infty}(z)| \le M_{mm}^F W_2(m_t, m_\infty) $$
and

$$ \max_{k=2,3,4} \sup_{z \in \mathbb{R}^{2d}} \left| \nabla^k \log \frac{\hat{m}_t}{m_\infty}(z) \right| \le M, $$

for some constant $M$, the alternative relative density $\eta_t(z) := m_t(z)/\hat{m}_t(z)$ is $C^4$ in $z$ and there exists a constant $M_T$ such that

$$ \eta_t(z) + \frac{1}{\eta_t(z)} + \sum_{k=1}^{4} |\nabla^k \eta_t(z)| \le \exp(M_T(1 + |z|)) \quad (4.10) $$

for every $(t, z) \in [0, T] \times \mathbb{R}^{2d}$. The constant $M_T$ may change from line to line in the following.

In the following we will adopt the abstract notations introduced by Villani in his seminal work on the hypocoercivity [40]. Define $\mathcal{H}_t = L^2(\hat{m}_t)$, $A_t = \nabla_v$ and $B_t = v \cdot \nabla_x - D_m F(m_t, x) \cdot \nabla_v$. The adjoint of $A_t$ in $\mathcal{H}_t$ is therefore $A_t^* = -\nabla_v + v$, while $B_t$ is antisymmetric: $B_t^* = -B_t$. Define the commutator $C_t = [A_t, B_t] = A_t B_t - B_t A_t = \nabla_x$. Finally define $L_t = A_t^* A_t + B_t$ and $u_t = \log \eta_t$. The Fokker-Planck equation (2.12) now reads

$$ \frac{\partial_t m_t}{\hat{m}_t} = -L_t \eta_t = -(A_t^* A_t + B_t) \eta_t. \quad (4.11) $$

Adding anisotropic Fisher. Let $a, b, c$ be positive reals to be determined. We define the hypocoercive Lyapunov functional

$$ \mathcal{E}(m) = \mathcal{F}(m) + a \int |\nabla_v \log \frac{m}{\hat{m}}(z)|^2 m(dz) + 2b \int \nabla_v \log \frac{m}{\hat{m}}(z) \cdot \nabla_x \log \frac{m}{\hat{m}}(z) m(dz) + c \int |\nabla_x \log \frac{m}{\hat{m}}(z)|^2 m(dz), \quad (4.12) $$

where $\mathcal{F}(m) = F(m) + \frac{1}{2} \int |v|^2 m + H(m)$ is the free energy. We also denote the sum of the last three terms in (4.12) by $I_{a,b,c}(m_t|\hat{m}_t)$, so that

$$ \mathcal{E}(m) = \mathcal{F}(m) + I_{a,b,c}(m_t|\hat{m}_t). $$

Thanks to Proposition 4.8 and in particular the bound (4.10), we can show that the quantity $\mathcal{E}(m_t)$ is well defined for every $t \ge 0$ and is continuous in $t$. We will show in the following that $t \mapsto \mathcal{E}(m_t)$ is in fact absolutely continuous and calculate its almost everywhere derivative. To this end, for every $t > 0$ and every $h \ge -t$, we define

$$ \begin{aligned} \mathcal{E}(m_{t+h}) - \mathcal{E}(m_t) &= (\mathcal{F}(m_{t+h}) - \mathcal{F}(m_t)) \\ &\quad + (I_{a,b,c}(m_{t+h}|\hat{m}_{t+h}) - I_{a,b,c}(m_t|\hat{m}_{t+h})) \\ &\quad + (I_{a,b,c}(m_t|\hat{m}_{t+h}) - I_{a,b,c}(m_t|\hat{m}_t)) \\ &=: \Delta_1 + \Delta_2 + \Delta_3. \end{aligned} $$

Contributions from $\Delta_1$ and $\Delta_2$. We first calculate the contributions from $\Delta_1$. Using the Fokker-Planck equation (2.12) and the bounds (4.10), one has $|\Delta_1| \le M_T h$ for every $t, h$ such that $t$ and $t+h$ belong to $[0, T]$; moreover, by the dominated convergence theorem one has for almost every $t > 0$,

$$ \lim_{h \to 0} \frac{\Delta_1}{h} = \frac{d\mathcal{F}(m_t)}{dt} = - \int |\nabla_v \log \frac{m_t}{\hat{m}_t}(z)|^2 m_t(z) dz = - \int |A_t u_t|^2 m_t, $$
and

$$ \max_{k=2,3,4} \sup_{z \in \mathbb{R}^{2d}} \left| \nabla^k \log \frac{\hat{m}_t}{m_\infty}(z) \right| \le M, $$

for some constant $M$, the alternative relative density $\eta_t(z) := m_t(z)/\hat{m}_t(z)$ is $C^4$ in $z$ and there exists a constant $M_T$ such that

$$ \eta_t(z) + \frac{1}{\eta_t(z)} + \sum_{k=1}^{4} |\nabla^k \eta_t(z)| \le \exp(M_T(1 + |z|)) \quad (4.10) $$

for every $(t, z) \in [0, T] \times \mathbb{R}^{2d}$. The constant $M_T$ may change from line to line in the following.

In the following we will adopt the abstract notations introduced by Villani in his seminal work on the hypocoercivity [40]. Define $\mathcal{H}_t = L^2(\hat{m}_t)$, $A_t = \nabla_v$ and $B_t = v \cdot \nabla_x - D_m F(m_t, x) \cdot \nabla_v$. The adjoint of $A_t$ in $\mathcal{H}_t$ is therefore $A_t^* = -\nabla_v + v$, while $B_t$ is antisymmetric: $B_t^* = -B_t$. Define the commutator $C_t = [A_t, B_t] = A_t B_t - B_t A_t = \nabla_x$. Finally define $L_t = A_t^* A_t + B_t$ and $u_t = \log \eta_t$. The Fokker-Planck equation (2.12) now reads

$$ \frac{\partial_t m_t}{\hat{m}_t} = -L_t \eta_t = -(A_t^* A_t + B_t) \eta_t. \quad (4.11) $$

Adding anisotropic Fisher. Let $a, b, c$ be positive reals to be determined. We define the hypocoercive Lyapunov functional

$$ \mathcal{E}(m) = \mathcal{F}(m) + a \int |\nabla_v \log \frac{m}{\hat{m}}(z)|^2 m(dz) + 2b \int \nabla_v \log \frac{m}{\hat{m}}(z) \cdot \nabla_x \log \frac{m}{\hat{m}}(z) m(dz) + c \int |\nabla_x \log \frac{m}{\hat{m}}(z)|^2 m(dz), \quad (4.12) $$

where $\mathcal{F}(m) = F(m) + \frac{1}{2} \int |v|^2 m + H(m)$ is the free energy. We also denote the sum of the last three terms in (4.12) by $I_{a,b,c}(m_t|\hat{m}_t)$, so that

$$ \mathcal{E}(m) = \mathcal{F}(m) + I_{a,b,c}(m_t|\hat{m}_t). $$

Thanks to Proposition 4.8 and in particular the bound (4.10), we can show that the quantity $\mathcal{E}(m_t)$ is well defined for every $t \ge 0$ and is continuous in $t$. We will show in the following that $t \mapsto \mathcal{E}(m_t)$ is in fact absolutely continuous and calculate its almost everywhere derivative. To this end, for every $t > 0$ and every $h \ge -t$, we define

$$ \begin{aligned} \mathcal{E}(m_{t+h}) - \mathcal{E}(m_t) &= (\mathcal{F}(m_{t+h}) - \mathcal{F}(m_t)) \\ &\quad + (I_{a,b,c}(m_{t+h}|\hat{m}_{t+h}) - I_{a,b,c}(m_t|\hat{m}_{t+h})) \\ &\quad + (I_{a,b,c}(m_t|\hat{m}_{t+h}) - I_{a,b,c}(m_t|\hat{m}_t)) \\ &=: \Delta_1 + \Delta_2 + \Delta_3. \end{aligned} $$

Contributions from $\Delta_1$ and $\Delta_2$. We first calculate the contributions from $\Delta_1$. Using the Fokker-Planck equation (2.12) and the bounds (4.10), one has $|\Delta_1| \le M_T h$ for every $t, h$ such that $t$ and $t+h$ belong to $[0, T]$; moreover, by the dominated convergence theorem one has for almost every $t > 0$,

$$ \lim_{h \to 0} \frac{\Delta_1}{h} = \frac{d\mathcal{F}(m_t)}{dt} = - \int |\nabla_v \log \frac{m_t}{\hat{m}_t}(z)|^2 m_t(z) dz = - \int |A_t u_t|^2 m_t, $$
where the right hand side is continuous in $t$. The above inequality then holds for every $t > 0$. Define the $4 \times 4$ matrix

$$K_1 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},$$

and denote the Hilbertian norm by $\| \cdot \| = \| \cdot \|_{L^2(m_t)}$. Introduce the four-dimensional vector

$$Y_t = (\|A_t u_t\|, \|A_t^2 u_t\|, \|C_t u_t\|, \|C_t A_t u_t\|)^\top. \quad (4.13)$$

Then we have for almost every $t > 0$, $\lim_{h \to 0} \Delta_1/h = -Y_t^\top K_1 Y_t$.

Next calculate the contributions from $\Delta_2$. Arguing as we did for $\Delta_1$, again we have $|\Delta_2| \le M_T h$. Applying the dominated convergence theorem and compute as in the proofs of [40, Lemma 32 and Theorem 18], we obtain that for almost every $t > 0$, the limit $\lim_{h \to 0} \Delta_2/h$ exists and is upper bounded by $-Y_t^\top K_2 Y_t$, where

$$K_2 := \begin{pmatrix} 2a - 2M_{mx}^F b & -2b & -2a & 0 \\ 0 & 2a & -2M_{mx}^F c & -4b \\ 0 & 0 & 2b & 0 \\ 0 & 0 & 0 & 2c \end{pmatrix}.$$

*Contributions from* $\Delta_3$. Now we calculate the last term $\Delta_3:=I_{a,b,c}(m_t|\hat{m}_{t+h})-I_{a,b,c}(m_t|\hat{m}_t)$. Note that $\nabla_v \log \hat{m}_t(z) = -v$ and, by the $W_2$-Lipschitz continuity of $m \mapsto D_m F(m, x)$, we have

$$\begin{aligned} |\nabla_x \log \hat{m}_{t+h}(z) - \nabla_x \log \hat{m}_t(z)| &= |D_m F(m_{t+h}^x, x) - D_m F(m_t^x, x)| \\ &\le M_{mm}^F W_2(m_{t+h}^x, m_t^x). \end{aligned}$$

So for each $z \in \mathbb{R}^{2d}$, we know that $\nabla \log \hat{m}_t(z)$ is continuous in $t$, and is absolutely continuous once $t \mapsto m_t^x$ is absolutely continuous with respect to the $W_2$ distance in the sense of [2, Definition 1.1.1]. Let us show the latter. Integrating the speed component in the Fokker-Planck equation (2.12), we obtain

$$\partial_t m_t^x + \nabla_x \cdot (v_t^x m_t^x) = 0, \quad (4.14)$$

where

$$v_t^x(x) := \frac{\int vm_t(x, v)dv}{\int m_t(x, v)dv} = \frac{\int \nabla_v \log \frac{m_t}{\hat{m}_t}(x, v)m_t(x, v)dv}{\int m_t(x, v)dv}$$

is the average speed at the spatial point $x$. The $L^2$ norm of the vector field in the continuity equation (4.14) satisfies

$$\begin{aligned} \|v_t^x\|_{L^2(m_t^x)} &= \left( \int \left| \frac{\nabla_v \log \frac{m_t}{\hat{m}_t}(x, v)m_t(x, v)dv}{\int m_t(x, v)dv} \right|^2 m_t^x(x)dx \right)^{1/2} \\ &\le \left( \int \left| \nabla_v \log \frac{m_t}{\hat{m}_t}(z) \right|^2 m_t(dz) \right)^{1/2} = \|A_t u_t\| \le M_T, \end{aligned}$$

where the first inequality is due to Cauchy-Schwarz. Applying [2, Proposition 8.3.1] to the flow $t \mapsto m_t^x$ and its continuity equation (4.14), and using [2, Theorem 1.1.2], we obtain

$$W_2(m_{t+h}^x, m_t^x) \le \int_t^{t+h} \|A_s u_s\| ds \le M_T h$$
where the right hand side is continuous in $t$. The above inequality then holds for every $t > 0$. Define the $4 \times 4$ matrix

$$K_1 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},$$

and denote the Hilbertian norm by $\| \cdot \| = \| \cdot \|_{L^2(m_t)}$. Introduce the four-dimensional vector

$$Y_t = (\|A_t u_t\|, \|A_t^2 u_t\|, \|C_t u_t\|, \|C_t A_t u_t\|)^\top. \quad (4.13)$$

Then we have for almost every $t > 0$, $\lim_{h \to 0} \Delta_1/h = -Y_t^\top K_1 Y_t$.

Next calculate the contributions from $\Delta_2$. Arguing as we did for $\Delta_1$, again we have $|\Delta_2| \le M_{Th}$. Applying the dominated convergence theorem and compute as in the proofs of [40, Lemma 32 and Theorem 18], we obtain that for almost every $t > 0$, the limit $\lim_{h \to 0} \Delta_2/h$ exists and is upper bounded by $-Y_t^\top K_2 Y_t$, where

$$K_2 := \begin{pmatrix} 2a - 2M_{mx}^F b & -2b & -2a & 0 \\ 0 & 2a & -2M_{mx}^F c & -4b \\ 0 & 0 & 2b & 0 \\ 0 & 0 & 0 & 2c \end{pmatrix}.$$

*Contributions from* $\Delta_3$. Now we calculate the last term $\Delta_3:=I_{a,b,c}(m_t|\hat{m}_{t+h})-I_{a,b,c}(m_t|\hat{m}_t)$. Note that $\nabla_v \log \hat{m}_t(z) = -v$ and, by the $W_2$-Lipschitz continuity of $m \mapsto D_m F(m, x)$, we have

$$\begin{aligned} |\nabla_x \log \hat{m}_{t+h}(z) - \nabla_x \log \hat{m}_t(z)| &= |D_m F(m_{t+h}^x, x) - D_m F(m_t^x, x)| \\ &\le M_{mm}^F W_2(m_{t+h}^x, m_t^x). \end{aligned}$$

So for each $z \in \mathbb{R}^{2d}$, we know that $\nabla \log \hat{m}_t(z)$ is continuous in $t$, and is absolutely continuous once $t \mapsto m_t^x$ is absolutely continuous with respect to the $W_2$ distance in the sense of [2, Definition 1.1.1]. Let us show the latter. Integrating the speed component in the Fokker-Planck equation (2.12), we obtain

$$\partial_t m_t^x + \nabla_x \cdot (v_t^x m_t^x) = 0, \quad (4.14)$$

where

$$v_t^x(x) := \frac{\int vm_t(x, v)dv}{\int m_t(x, v)dv} = \frac{\int \nabla_v \log \frac{m_t}{\hat{m}_t}(x, v)m_t(x, v)dv}{\int m_t(x, v)dv}$$

is the average speed at the spatial point $x$. The $L^2$ norm of the vector field in the continuity equation (4.14) satisfies

$$\begin{aligned} \|v_t^x\|_{L^2(m_t^x)} &= \left( \int \left| \frac{\nabla_v \log \frac{m_t}{\hat{m}_t}(x, v)m_t(x, v)dv}{\int m_t(x, v)dv} \right|^2 m_t^x(x)dx \right)^{1/2} \\ &\le \left( \int \left| \nabla_v \log \frac{m_t}{\hat{m}_t}(z) \right|^2 m_t(dz) \right)^{1/2} = \|A_t u_t\| \le M_T, \end{aligned}$$

where the first inequality is due to Cauchy-Schwarz. Applying [2, Proposition 8.3.1] to the flow $t \mapsto m_t^x$ and its continuity equation (4.14), and using [2, Theorem 1.1.2], we obtain

$$W_2(m_{t+h}^x, m_t^x) \le \int_t^{t+h} \|A_s u_s\| ds \le M_{Th}$$
for every $t, h$ such that $t$ and $t+h$ belong to $[0, T]$. So the mapping $t \mapsto \nabla \log \hat{m}_t(z)$ is absolutely continuous with almost everywhere derivatives satisfying

$$ \begin{gathered} \partial_t \nabla_v \log \hat{m}_t(z) = 0, \\ |\partial_t \nabla_x \log \hat{m}_t(z)| \le M_{mm}^F \|A_t u_t\| \le M_T. \end{gathered} $$

Then we obtain $|\Delta_3| \le M_T h$. Moreover, by the dominated convergence theorem, we have for almost every $t > 0$,

$$ \begin{aligned} \lim_{h \to 0} \frac{|\Delta_3|}{|h|} &\le 2M_{mm}^F \int (|A_t u_t(z)|, |C_t u_t(z)|) \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} 0 \\ \|A_t u_t\| \end{pmatrix} m_t(dz) \\ &\le 2M_{mm}^F (b \|A_t u_t\| \|A_t u_t\| + c \|A_t u_t\| \|C_t u_t\|) = Y_t^\top K_3 Y_t \end{aligned} $$

by applying Cauchy-Schwarz again, where

$$ K_3 := 2M_{mm}^F \begin{pmatrix} b & 0 & c & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}. $$

**Hypocoercivity.** Our previous bounds on $\Delta_k$, $k = 1, 2, 3$ establish that $t \mapsto \mathcal{E}(m_t)$ is absolutely continuous (locally Lipschitz, in fact) with its almost everywhere derivative satisfying $\frac{d}{dt}\mathcal{E}(m_t) \le -Y_t^\top K Y_t$, where $K$ is defined by $K_1 + K_2 - K_3$ and is equal to

$$ \begin{pmatrix} 1 + 2M_{mm}^F a - 2(M_{mx}^F + M_{mm}^F)b & -2b & -2a - 2M_{mm}^F c & 0 \\ 0 & 2a & -2M_{mx}^F c & -4b \\ 0 & 0 & 2b & 0 \\ 0 & 0 & 0 & 2c \end{pmatrix}. $$

As in the end of the proof of [40, Theorem 18], we can pick constants $a, b, c > 0$, depending only on $M_{mx}^F$ and $M_{mm}^F$, such that $ac > b^2$ and the matrix $K$ is a positive-definite. Let $\alpha$ be the smallest eigenvalue of $K$. Then we have

$$ \begin{aligned} \frac{d\mathcal{E}(m_t)}{dt} &\le -\alpha (\|A_t u_t\|^2 + \|C_t u_t\|^2 + \|A_t^2 u_t\|^2 + \|C_t A_t u_t\|^2) \\ &\le -\alpha (\|A_t u_t\|^2 + \|C_t u_t\|^2) = -\alpha I(m_t |\hat{m}_t|). \end{aligned} $$

Hence for every $t, s$ such that $t \ge s \ge 0$,

$$ \mathcal{E}(m_t) \le \mathcal{E}(m_s) - \alpha \int_s^t I(m_u |\hat{m}_u|) du. \quad (4.15) $$

**Approximation.** We now show that the inequality (4.15) holds without additional assumptions on the mean field functional $F$ and the initial value $m_0$.

First, suppose still that $F$ satisfies (4.5) but no longer suppose $m_0$ is such that $m_0/m_\infty \in A_+$. The initial value $m_0$ belongs to $P_2(\mathbb{R}^d)$ and both $H(m_0)$ and $I(m_0)$ are finite, so thanks to Proposition 4.7, we can pick a sequence of measures $(m'_{n,0})_{n \in \mathbb{N}}$, each of which belongs to $A_+$, such that

$$ \lim_{n \to \infty} W_2(m'_{n,0}, m_0) + |H(m'_{n,0}) - H(m_0)| + |I(m'_{n,0}) - I(m_0)| = 0. $$

As proved above, the inequality (4.15) holds for the flow $(m'_{n,t})_{t \ge 0}$, that is,

$$ \mathcal{E}(m'_{n,t}) \le \mathcal{E}(m'_{n,0}) - \alpha \int_0^t I(m'_{n,s} |\hat{m}'_{n,s}) ds. $$
for every $t, h$ such that $t$ and $t+h$ belong to $[0, T]$. So the mapping $t \mapsto \nabla \log \hat{m}_t(z)$ is absolutely continuous with almost everywhere derivatives satisfying

$$ \begin{gathered} \partial_t \nabla_v \log \hat{m}_t(z) = 0, \\ |\partial_t \nabla_x \log \hat{m}_t(z)| \le M_{mm}^F \|A_t u_t\| \le M_T. \end{gathered} $$

Then we obtain $|\Delta_3| \le M_T h$. Moreover, by the dominated convergence theorem, we have for almost every $t > 0$,

$$ \begin{aligned} \lim_{h \to 0} \frac{|\Delta_3|}{|h|} &\le 2M_{mm}^F \int (|A_t u_t(z)|, |C_t u_t(z)|) \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} 0 \\ \|A_t u_t\| \end{pmatrix} m_t(dz) \\ &\le 2M_{mm}^F (b \|A_t u_t\| \|A_t u_t\| + c \|A_t u_t\| \|C_t u_t\|) = Y_t^T K_3 Y_t \end{aligned} $$

by applying Cauchy-Schwarz again, where

$$ K_3 := 2M_{mm}^F \begin{pmatrix} b & 0 & c & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}. $$

**Hypocoercivity.** Our previous bounds on $\Delta_k$, $k = 1, 2, 3$ establish that $t \mapsto \mathcal{E}(m_t)$ is absolutely continuous (locally Lipschitz, in fact) with its almost everywhere derivative satisfying $\frac{d}{dt}\mathcal{E}(m_t) \le -Y_t^T K Y_t$, where $K$ is defined by $K_1 + K_2 - K_3$ and is equal to

$$ \begin{pmatrix} 1 + 2M_{mm}^F a - 2(M_{mx}^F + M_{mm}^F)b & -2b & -2a - 2M_{mm}^F c & 0 \\ 0 & 2a & -2M_{mx}^F c & -4b \\ 0 & 0 & 2b & 0 \\ 0 & 0 & 0 & 2c \end{pmatrix}. $$

As in the end of the proof of [40, Theorem 18], we can pick constants $a, b, c > 0$, depending only on $M_{mx}^F$ and $M_{mm}^F$, such that $ac > b^2$ and the matrix $K$ is a positive-definite. Let $\alpha$ be the smallest eigenvalue of $K$. Then we have

$$ \begin{aligned} \frac{d\mathcal{E}(m_t)}{dt} &\le -\alpha (\|A_t u_t\|^2 + \|C_t u_t\|^2 + \|A_t^2 u_t\|^2 + \|C_t A_t u_t\|^2) \\ &\le -\alpha (\|A_t u_t\|^2 + \|C_t u_t\|^2) = -\alpha I(m_t |\hat{m}_t|). \end{aligned} $$

Hence for every $t, s$ such that $t \ge s \ge 0$,

$$ \mathcal{E}(m_t) \le \mathcal{E}(m_s) - \alpha \int_s^t I(m_u |\hat{m}_u|) du. \quad (4.15) $$

**Approximation.** We now show that the inequality (4.15) holds without additional assumptions on the mean field functional $F$ and the initial value $m_0$.

First, suppose still that $F$ satisfies (4.5) but no longer suppose $m_0$ is such that $m_0/m_\infty \in A_+$. The initial value $m_0$ belongs to $P_2(\mathbb{R}^d)$ and both $H(m_0)$ and $I(m_0)$ are finite, so thanks to Proposition 4.7, we can pick a sequence of measures $(m'_{n,0})_{n \in \mathbb{N}}$, each of which belongs to $A_+$, such that

$$ \lim_{n \to \infty} W_2(m'_{n,0}, m_0) + |H(m'_{n,0}) - H(m_0)| + |I(m'_{n,0}) - I(m_0)| = 0. $$

As proved above, the inequality (4.15) holds for the flow $(m'_{n,t})_{t \ge 0}$, that is,

$$ \mathcal{E}(m'_{n,t}) \le \mathcal{E}(m'_{n,0}) - \alpha \int_0^t I(m'_{n,s} |\hat{m}'_{n,s}) ds. $$
By the continuity with respect to the initial value of the SDE system (2.10), we have also $m'_{n,t} \to m_t$ in the weak topology of $P_2$. We recall in Lemma A.1 that both the entropy and the Fisher information are lower semicontinuous with respect to the weak topology of $P_2$. Taking the lower limit on both sides of the inequality above, we obtain (4.15) with $s=0$ for the original flow $(m_t)_{t \ge 0}$.

Second, we no longer require $F$ to satisfy (4.5) and set $F_k(m) = F(m \star \rho_k)$ for a sequence of smooth and symmetric mollifiers $(\rho_k)_{k \in \mathbb{N}}$ in $\mathbb{R}^d$ with $\text{supp } \rho_k \subset B(0, 1/k)$. The linear derivative of the regularized mean field functional reads $\frac{\delta F_k}{\delta m}(m, \cdot) = \frac{\delta F}{\delta m}(m \star \rho_k, \cdot) \star \rho_k$, and its intrinsic derivative reads $D_m F_k(m, \cdot) = D_m F(m \star \rho_k, \cdot) \star \rho_k$. Consequently,

$$|D_m F_k(m', x') - D_m F(m, x)| \le M_{mm}^F W_2(m', m) + M_{mx}^F |x' - x| + \frac{M_{mx}^F + M_{mm}^F}{k}. \quad (4.16)$$

Moreover, $\nabla D_m F_k(m, \cdot) = \nabla D_m F(m \star \rho_k, \cdot) \star \rho_k$ and

$$\nabla^k D_m F_k(m, \cdot) = D_m F(m \star \rho_k, \cdot) \star \nabla^k \rho_k = \nabla D_m F(m \star \rho_k, \cdot) \star \nabla^{k-1} \rho_k$$

is continuous for $k \ge 0$ and bounded for $k \ge 1$. In particular $F_k$ satisfies (4.5). Define $\mathcal{E}_k(m) = F_k(m) + \frac{1}{2} \int |v|^2 m + H(m) + I_{a,b,c}(m|\hat{m}|)$ and here $\hat{m}$ should be understood as the Gibbs-type measure defined with $F_k$ instead of $F$. Let $(m''_{k,t})_{t \ge 0}$ be the flow of measures driven by $F_k$ with the initial value $m''_{k,0} = m_0$. Our previous result yields for every $t \ge 0$,

$$\mathcal{E}_k(m''_{k,t}) \le \mathcal{E}_k(m_0) - \alpha \int_0^t I(m''_{k,s}|\hat{m}'_{k,s})ds,$$

where $\hat{m}''_{k,s}$ is the probability measure proportional to

$$\exp\left(-\frac{\delta F_k}{\delta m}(m''_{k,s}, x) - \frac{1}{2}|v|^2\right)dx dv.$$

From the bound (4.16) we deduce that $m''_{k,t} \to m_t$ in $P_2$ for every $t \ge 0$ by the synchronous coupling result in Lemma 5.1. So taking the lower limit on both sides of the previous inequality, we obtain the inequality (4.15) with $s=0$ holds for general initial values and general mean field functionals. In particular, for every $t \ge 0$, the measure $m_t$ has finite entropy and finite Fisher information. Then we apply the same argument to the flow with the initial value $m_s$ and obtain the inequality (4.15) for general $s \ge 0$.

**Conclusion.** Define the matrix

$$S = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$$

and denote by $|S|$ its largest eigenvalue. The Fisher information satisfies for every $t \ge 0$,

$$\begin{aligned} I(m_t|\hat{m}_t) &= \frac{1}{2}I(m_t|\hat{m}_t) + \frac{1}{2}I(m_t|\hat{m}_t) \\ &\ge 2\rho H(m_t|\hat{m}_t) + \frac{1}{2}I(m_t|\hat{m}_t) \\ &\ge 2\rho(\mathcal{F}(m_t) - \mathcal{F}(m_\infty)) + \frac{1}{2|S|}I_{a,b,c}(m_t|\hat{m}_t) \\ &\ge \left(2\rho \wedge \frac{1}{2|S|}\right)(\mathcal{E}(m_t) - \mathcal{E}(m_\infty)), \end{aligned}$$

where on the second line we applied the uniform LSI (2.7), with $\rho$ defined by (2.6), and on the third line we used Lemma 4.2, $m_\infty = \hat{m}_\infty$ and $S \subseteq \lambda_2$. Applying Grönwall's lemma<fnref n="2" /> to (4.15), we obtain the desired contractivity (2.20) with $\kappa := \alpha(2\rho \wedge (2|S|)^{-1})$. □

<footnote n="2">²The mapping $t \mapsto \mathcal{E}(m_t)$ is lower semicontinuous by Lemma A.1 and non-increasing by the inequality (4.15). So it is càdlàg. It then suffices to convolute the mapping $t \mapsto \mathcal{E}(m_t)$ by a sequence of mollifiers compactly supported in $(0, 1)$, apply the classical Grönwall's lemma and take the limit.</footnote>
By the continuity with respect to the initial value of the SDE system (2.10), we have also $m'_{n,t} \to m_t$ in the weak topology of $P_2$. We recall in Lemma A.1 that both the entropy and the Fisher information are lower semicontinuous with respect to the weak topology of $P_2$. Taking the lower limit on both sides of the inequality above, we obtain (4.15) with $s=0$ for the original flow $(m_t)_{t \ge 0}$.

Second, we no longer require $F$ to satisfy (4.5) and set $F_k(m) = F(m \star \rho_k)$ for a sequence of smooth and symmetric mollifiers $(\rho_k)_{k \in \mathbb{N}}$ in $\mathbb{R}^d$ with $\text{supp } \rho_k \subset B(0, 1/k)$. The linear derivative of the regularized mean field functional reads $\frac{\delta F_k}{\delta m}(m, \cdot) = \frac{\delta F}{\delta m}(m \star \rho_k, \cdot) \star \rho_k$, and its intrinsic derivative reads $D_m F_k(m, \cdot) = D_m F(m \star \rho_k, \cdot) \star \rho_k$. Consequently,

$$ |D_m F_k(m', x') - D_m F(m, x)| \le M_{mm}^F W_2(m', m) + M_{mx}^F |x' - x| + \frac{M_{mx}^F + M_{mm}^F}{k}. \quad (4.16) $$

Moreover, $\nabla D_m F_k(m, \cdot) = \nabla D_m F(m \star \rho_k, \cdot) \star \rho_k$ and

$$ \nabla^k D_m F_k(m, \cdot) = D_m F(m \star \rho_k, \cdot) \star \nabla^k \rho_k = \nabla D_m F(m \star \rho_k, \cdot) \star \nabla^{k-1} \rho_k $$

is continuous for $k \ge 0$ and bounded for $k \ge 1$. In particular $F_k$ satisfies (4.5). Define $\mathcal{E}_k(m) = F_k(m) + \frac{1}{2} \int |v|^2 m + H(m) + I_{a,b,c}(m|\hat{m}|)$ and here $\hat{m}$ should be understood as the Gibbs-type measure defined with $F_k$ instead of $F$. Let $(m''_{k,t})_{t \ge 0}$ be the flow of measures driven by $F_k$ with the initial value $m''_{k,0} = m_0$. Our previous result yields for every $t \ge 0$,

$$ \mathcal{E}_k(m''_{k,t}) \le \mathcal{E}_k(m_0) - \alpha \int_0^t I(m''_{k,s}|\hat{m}'_{k,s})ds, $$

where $\hat{m}''_{k,s}$ is the probability measure proportional to

$$ \exp\left(-\frac{\delta F_k}{\delta m}(m''_{k,s}, x) - \frac{1}{2}|v|^2\right)dx dv. $$

From the bound (4.16) we deduce that $m''_{k,t} \to m_t$ in $P_2$ for every $t \ge 0$ by the synchronous coupling result in Lemma 5.1. So taking the lower limit on both sides of the previous inequality, we obtain the inequality (4.15) with $s=0$ holds for general initial values and general mean field functionals. In particular, for every $t \ge 0$, the measure $m_t$ has finite entropy and finite Fisher information. Then we apply the same argument to the flow with the initial value $m_s$ and obtain the inequality (4.15) for general $s \ge 0$.

**Conclusion.** Define the matrix

$$ S = \begin{pmatrix} a & b \\ b & c \end{pmatrix} $$

and denote by $|S|$ its largest eigenvalue. The Fisher information satisfies for every $t \ge 0$,

$$ \begin{aligned} I(m_t|\hat{m}_t) &= \frac{1}{2}I(m_t|\hat{m}_t) + \frac{1}{2}I(m_t|\hat{m}_t) \\ &\ge 2\rho H(m_t|\hat{m}_t) + \frac{1}{2}I(m_t|\hat{m}_t) \\ &\ge 2\rho(\mathcal{F}(m_t) - \mathcal{F}(m_\infty)) + \frac{1}{2|S|}I_{a,b,c}(m_t|\hat{m}_t) \\ &\ge \left(2\rho \wedge \frac{1}{2|S|}\right)(\mathcal{E}(m_t) - \mathcal{E}(m_\infty)), \end{aligned} $$

where on the second line we applied the uniform LSI (2.7), with $\rho$ defined by (2.6), and on the third line we used Lemma 4.2, $m_\infty = \hat{m}_\infty$ and $S \subseteq \lambda_2$. Applying Grönwall's lemma<fnref n="2" /> to (4.15), we obtain the desired contractivity (2.20) with $\kappa := \alpha(2\rho \wedge (2|S|)^{-1})$. □

<footnote n="2">²The mapping $t \mapsto \mathcal{E}(m_t)$ is lower semicontinuous by Lemma A.1 and non-increasing by the inequality (4.15). So it is càdlàg. It then suffices to convolute the mapping $t \mapsto \mathcal{E}(m_t)$ by a sequence of mollifiers compactly supported in $(0, 1)$, apply the classical Grönwall's lemma and take the limit.</footnote>
**Remark 4.2.** Our Theorem 2.1 can be compared to [40, Theorem 56], where kinetic mean field Langevin dynamics with two-body interaction are studied and $O(t^{-\infty})$ entropic convergence to equilibrium is shown, under the assumption that the mean field dependence is small. This restriction is lifted by our method which leverages the functional convexity.

**Remark 4.3.** The regularized energy functional $F_k$ is such that $x \mapsto D_m F_k(m, x)$ has bounded derivatives of every order. However $m \mapsto D_m F_k(m, x)$ remains only Lipschitz continuous and we are not aware of any approximation argument that allows us to obtain differentiability in the measure argument. Consequently we use still the result from [2] to treat this low regularity.

## 4.3 Particle system

In this section we study the system of particles described by the linear Fokker-Planck equation (2.13) and the SDE (2.11). Note that since the dynamics is linear, its wellposedness is classical and we omit its proof.

We first show that for our model we can construct hypocoercive functionals whose constants are independent of the number of particles.

**Lemma 4.9 (Uniform-in-N hypocoercivity).** Assume $F$ satisfies (2.2) and there exists a measure $m_{\infty}^N$ satisfying (2.19) and having finite exponential moments. Let $t \mapsto m_t^N$ be a solution to the $N$-particle Fokker-Planck equation (2.13) in $C([0, T]; \mathcal{P}_2(\mathbb{R}^{2dN}))$ whose initial value $m_0^N$ has finite entropy and finite Fisher information. Then there exist constants $a, b, c, \alpha > 0$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and the functional

$$ 
\begin{aligned}
\mathcal{E}^N(m^N) &:= \mathcal{F}^N(m^N) + I_{a,b,c}(m^N|m_{\infty}^N) \\
&:= \mathcal{F}^N(m^N) + \sum_{i=1}^N \left( a \int |\nabla_{v^i} \log h^N(\mathbf{z})|^2 m^N(d\mathbf{z}) \right. \\
& \qquad \left. + 2b \int \nabla_{v^i} \log h^N(\mathbf{z}) \cdot \nabla_{x^i} \log h^N(\mathbf{z}) m^N(d\mathbf{z}) \right. \\
& \qquad \left. + c \int |\nabla_{x^i} \log h^N(\mathbf{z})|^2 m^N(d\mathbf{z}) \right), \quad (4.17)
\end{aligned}
$$

where $h^N := m^N/m_{\infty}^N$, is finite on $m_t^N$ for $t > 0$; moreover, the mapping $t \mapsto \mathcal{E}^N(m_t^N)$ satisfies

$$ \mathcal{E}^N(m_t^N) \le \mathcal{E}^N(m_s^N) - \alpha \int_s^t I(m_u^N|m_{\infty}^N)du \quad (4.18) $$

for every $t, s$ such that $t \ge s \ge 0$.

**Remark 4.4.** The constants $a, b, c$ are possibly different from those appearing in the proof of Theorem 2.1.

**Proof.** We first show that the condition (2.2) implies a bound on the second-order derivatives of $x \mapsto U^N(x) := NF(\mu_x)$. The first-order derivatives satisfy

$$ 
\begin{aligned}
|\nabla_i U^N(\mathbf{x}) - \nabla_i U^N(\mathbf{x}')| &= |D_m F(\mu_{\mathbf{x}}, x^i) - D_m F(\mu_{\mathbf{x}'}, x'^i)| \\
&\le M_{mm}^F W_2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}'}) + M_{mx}^F |x^i - x'^i|.
\end{aligned}
$$

Summing over $i$, we obtain for every $\varepsilon > 0$,

$$ 
\begin{aligned}
|\nabla U^N(\mathbf{x}) - \nabla U^N(\mathbf{x}')|^2 &\le (1+\varepsilon)(M_{mm}^F)^2 N W_2^2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}'}) + (1+\varepsilon^{-1})(M_{mx}^F)^2 |\mathbf{x} - \mathbf{x}'|^2 \\
&\le ((1+\varepsilon)(M_{mm}^F)^2 + (1+\varepsilon^{-1})(M_{mx}^F)^2) |\mathbf{x} - \mathbf{x}'|^2.
\end{aligned}
$$
**Remark 4.2.** Our Theorem 2.1 can be compared to [40, Theorem 56], where kinetic mean field Langevin dynamics with two-body interaction are studied and $O(t^{-\infty})$ entropic convergence to equilibrium is shown, under the assumption that the mean field dependence is small. This restriction is lifted by our method which leverages the functional convexity.

**Remark 4.3.** The regularized energy functional $F_k$ is such that $x \mapsto D_m F_k(m, x)$ has bounded derivatives of every order. However $m \mapsto D_m F_k(m, x)$ remains only Lipschitz continuous and we are not aware of any approximation argument that allows us to obtain differentiability in the measure argument. Consequently we use still the result from [2] to treat this low regularity.

## 4.3 Particle system

In this section we study the system of particles described by the linear Fokker-Planck equation (2.13) and the SDE (2.11). Note that since the dynamics is linear, its wellposedness is classical and we omit its proof.

We first show that for our model we can construct hypocoercive functionals whose constants are independent of the number of particles.

**Lemma 4.9 (Uniform-in-N hypocoercivity).** Assume $F$ satisfies (2.2) and there exists a measure $m_{\infty}^N$ satisfying (2.19) and having finite exponential moments. Let $t \mapsto m_t^N$ be a solution to the $N$-particle Fokker-Planck equation (2.13) in $C([0, T]; \mathcal{P}_2(\mathbb{R}^{2dN}))$ whose initial value $m_0^N$ has finite entropy and finite Fisher information. Then there exist constants $a, b, c, \alpha > 0$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and the functional

$$ 
\begin{aligned}
\mathcal{E}^N(m^N) &:= \mathcal{F}^N(m^N) + I_{a,b,c}(m^N|m_{\infty}^N) \\
&:= \mathcal{F}^N(m^N) + \sum_{i=1}^N \left( a \int |\nabla_{v^i} \log h^N(\mathbf{z})|^2 m^N(d\mathbf{z}) \right. \\
&\qquad \left. + 2b \int \nabla_{v^i} \log h^N(\mathbf{z}) \cdot \nabla_{x^i} \log h^N(\mathbf{z}) m^N(d\mathbf{z}) \right. \\
&\qquad \left. + c \int |\nabla_{x^i} \log h^N(\mathbf{z})|^2 m^N(d\mathbf{z}) \right), \quad (4.17)
\end{aligned}
$$

where $h^N := m^N/m_{\infty}^N$, is finite on $m_t^N$ for $t > 0$; moreover, the mapping $t \mapsto \mathcal{E}^N(m_t^N)$ satisfies

$$ \mathcal{E}^N(m_t^N) \le \mathcal{E}^N(m_s^N) - \alpha \int_s^t I(m_u^N|m_{\infty}^N) du \quad (4.18) $$

for every $t, s$ such that $t \ge s \ge 0$.

**Remark 4.4.** The constants $a, b, c$ are possibly different from those appearing in the proof of Theorem 2.1.

**Proof.** We first show that the condition (2.2) implies a bound on the second-order derivatives of $x \mapsto U^N(x) := NF(\mu_x)$. The first-order derivatives satisfy

$$ 
\begin{aligned}
|\nabla_i U^N(\mathbf{x}) - \nabla_i U^N(\mathbf{x}')| &= |D_m F(\mu_{\mathbf{x}}, x^i) - D_m F(\mu_{\mathbf{x}'}, x'^i)| \\
&\le M_{mm}^F W_2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}'}) + M_{mx}^F |x^i - x'^i|.
\end{aligned}
$$

Summing over $i$, we obtain for every $\varepsilon > 0$,

$$ 
\begin{aligned}
|\nabla U^N(\mathbf{x}) - \nabla U^N(\mathbf{x}')|^2 &\le (1 + \varepsilon)(M_{mm}^F)^2 N W_2^2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}'}) + (1 + \varepsilon^{-1})(M_{mx}^F)^2 |\mathbf{x} - \mathbf{x}'|^2 \\
&\le ((1 + \varepsilon)(M_{mm}^F)^2 + (1 + \varepsilon^{-1})(M_{mx}^F)^2) |\mathbf{x} - \mathbf{x}'|^2.
\end{aligned}
$$
Uniform propagation of chaos for kinetic MFL

Optimizing $\epsilon$ yields $|\nabla U^N(\mathbf{x}) - \nabla U^N(\mathbf{x}')| \le (M_{mm}^F + M_{mx}^F)|\mathbf{x} - \mathbf{x}'|$. Define

$$ \|\nabla^2 U^N\|_{\infty} = \|\nabla^2 U^N\|_{\text{op},\infty} = \underset{\mathbf{x} \in \mathbb{R}^{2dN}}{\operatorname{ess\,sup}} \sup_{\mathbf{x}' \in \mathbb{R}^{2dN}: |\mathbf{x}'|_2 = 1} |\nabla^2 U^N(\mathbf{x})\mathbf{x}'|_2. $$

From the Lipschitz bound we obtain

$$ \|\nabla^2 U^N\|_{\text{op},\infty} \le M_{mx}^F + M_{mm}^F. \quad (4.19) $$

Now suppose there exist a constant $M$ such that $U^N$ satisfies

$$ \mathbf{x} \mapsto U^N(\mathbf{x}) \text{ is } C^4 \quad \text{and} \quad \sum_{k=3}^{4} \|\nabla^k U^N\|_{\infty} \le M, \quad (4.20) $$

and that $h_0^N = m_0^N/m_\infty^N$ satisfies

$$ h_0^N(\mathbf{z}) + \frac{1}{h_0^N(\mathbf{z})} + \sum_{k=1}^{4} |\nabla^k h_t^N(\mathbf{z})| \le M \quad (4.21) $$

for every $\mathbf{z} \in \mathbb{R}^{2dN}$. We apply Proposition 4.8 to show that under our assumptions, there exists a constant $M_T$ such that

$$ h_t^N(\mathbf{z}) + \frac{1}{h_t^N(\mathbf{z})} + \sum_{k=1}^{4} |\nabla^k h_t^N(\mathbf{z})| \le \exp(M_T(1 + |\mathbf{z}|)) \quad (4.22) $$

for every $(t, \mathbf{z}) \in [0, T] \times \mathbb{R}^{2dN}$ (in fact, $\mathbf{z} \mapsto h_t^N(\mathbf{z})$ remains lower and upper bounded and its up-to-fourth-order derivatives grow at most polynomially).

We denote $u_t^N = \log h_t^N$. In view of the regularity bound (4.22), we have

$$ \begin{aligned} -\frac{d\mathcal{F}^N(m_t^N)}{dt} &= \sum_{i=1}^N \int |\nabla_{v_i} u_t^N|^2 m_t^N, \\ -\frac{d}{dt} \int |\nabla_{v_i} u_t^N|^2 m_t^N &= 2 \int \left( \nabla_{x^i} u_t^N \cdot \nabla_{v_i} u_t^N + |\nabla_{v_i}^2 u_t^N|^2 \right. \\ & \qquad \left. + \nabla_{v_i} u_t^N \cdot \nabla_{v_i} u_t^N \right) m_t^N, \\ -\frac{d}{dt} \int \nabla_{v_i} u_t^N \cdot \nabla_{x^i} u_t^N m_t^N &= \int \left( -\sum_{j=1}^N \nabla_{v_i} u_t^N \nabla_{ij}^2 U^N \nabla_{v_j} u_t^N + |\nabla_{x^i} u_t^N|^2 \right. \\ & \qquad \left. + 2 \nabla_{v_i}^2 u_t^N \cdot \nabla_{v_i} \nabla_{x^i} u_t^N + \nabla_{v_i} u_t^N \cdot \nabla_{x^i} u_t^N \right) m_t^N, \\ -\frac{d}{dt} \int \nabla_{x^i} u_t^N \cdot \nabla_{x^i} u_t^N m_t^N &= \int \left( -2 \sum_{j=1}^N \nabla_{x^i} u_t^N \nabla_{ij}^2 U^N \nabla_{v_j} \nabla_{v_i} u_t^N \right. \\ & \qquad \left. + 2 |\nabla_{x^i} \nabla_{v_i} u_t^N|^2 \right) m_t^N, \end{aligned} $$

as is computed in [40]. Denote the Hilbertian norm by $\|\cdot\| = \|\cdot\|_{L^2(m_t^N)}$ and define the four-dimensional vector

$$ Y_t^N = (\|\nabla_{\mathbf{v}} u_t^N\|, \|\nabla_{\mathbf{v}}^2 u_t^N\|, \|\nabla_{\mathbf{x}} u_t^N\|, \|\nabla_{\mathbf{x}} \nabla_{\mathbf{v}} u_t^N\|)^\mathrm{T}. \quad (4.23) $$

By Cauchy-Schwarz we have $-\frac{d}{dt}\mathcal{E}^N(m_t^N) \ge (Y_t^N)^\mathrm{T} K Y_t^N$ where

$$ K := \begin{pmatrix} 1 + 2a - 2\|\nabla^2 U^N\|_{\text{op},\infty} & b & -2b & -2a & 0 \\ & 2a & -2\|\nabla^2 U^N\|_{\text{op},\infty} & c & -4b \\ & & 2b & 0 & 2c \end{pmatrix}, $$
Uniform propagation of chaos for kinetic MFL

Optimizing $\epsilon$ yields $|\nabla U^N(\mathbf{x}) - \nabla U^N(\mathbf{x}')| \le (M_{mm}^F + M_{mx}^F)|\mathbf{x} - \mathbf{x}'|$. Define

$$ \|\nabla^2 U^N\|_{\infty} = \|\nabla^2 U^N\|_{\text{op},\infty} = \underset{\mathbf{x} \in \mathbb{R}^{2dN}}{\operatorname{ess\,sup}} \sup_{\mathbf{x}' \in \mathbb{R}^{2dN}: |\mathbf{x}'|_2 = 1} |\nabla^2 U^N(\mathbf{x})\mathbf{x}'|_2. $$

From the Lipschitz bound we obtain

$$ \|\nabla^2 U^N\|_{\text{op},\infty} \le M_{mx}^F + M_{mm}^F. \quad (4.19) $$

Now suppose there exist a constant $M$ such that $U^N$ satisfies

$$ \mathbf{x} \mapsto U^N(\mathbf{x}) \text{ is } C^4 \quad \text{and} \quad \sum_{k=3}^{4} \|\nabla^k U^N\|_{\infty} \le M, \quad (4.20) $$

and that $h_0^N = m_0^N/m_\infty^N$ satisfies

$$ h_0^N(\mathbf{z}) + \frac{1}{h_0^N(\mathbf{z})} + \sum_{k=1}^{4} |\nabla^k h_t^N(\mathbf{z})| \le M \quad (4.21) $$

for every $\mathbf{z} \in \mathbb{R}^{2dN}$. We apply Proposition 4.8 to show that under our assumptions, there exists a constant $M_T$ such that

$$ h_t^N(\mathbf{z}) + \frac{1}{h_t^N(\mathbf{z})} + \sum_{k=1}^{4} |\nabla^k h_t^N(\mathbf{z})| \le \exp(M_T(1 + |\mathbf{z}|)) \quad (4.22) $$

for every $(t, \mathbf{z}) \in [0, T] \times \mathbb{R}^{2dN}$ (in fact, $\mathbf{z} \mapsto h_t^N(\mathbf{z})$ remains lower and upper bounded and its up-to-fourth-order derivatives grow at most polynomially).

We denote $u_t^N = \log h_t^N$. In view of the regularity bound (4.22), we have

$$ \begin{aligned} -\frac{d\mathcal{F}^N(m_t^N)}{dt} &= \sum_{i=1}^N \int |\nabla_{v_i} u_t^N|^2 m_t^N, \\ -\frac{d}{dt} \int |\nabla_{v_i} u_t^N|^2 m_t^N &= 2 \int \left( \nabla_{x^i} u_t^N \cdot \nabla_{v_i} u_t^N + |\nabla_{v_i}^2 u_t^N|^2 \right. \\ & \qquad \left. + \nabla_{v_i} u_t^N \cdot \nabla_{v_i} u_t^N \right) m_t^N, \\ -\frac{d}{dt} \int \nabla_{v_i} u_t^N \cdot \nabla_{x^i} u_t^N m_t^N &= \int \left( -\sum_{j=1}^N \nabla_{v_i} u_t^N \nabla_{ij}^2 U^N \nabla_{v_j} u_t^N + |\nabla_{x^i} u_t^N|^2 \right. \\ & \qquad \left. + 2 \nabla_{v_i}^2 u_t^N \cdot \nabla_{v_i} \nabla_{x^i} u_t^N + \nabla_{v_i} u_t^N \cdot \nabla_{x^i} u_t^N \right) m_t^N, \\ -\frac{d}{dt} \int \nabla_{x^i} u_t^N \cdot \nabla_{x^i} u_t^N m_t^N &= \int \left( -2 \sum_{j=1}^N \nabla_{x^i} u_t^N \nabla_{ij}^2 U^N \nabla_{v_j} \nabla_{v_i} u_t^N \right. \\ & \qquad \left. + 2 |\nabla_{x^i} \nabla_{v_i} u_t^N|^2 \right) m_t^N, \end{aligned} $$

as is computed in [40]. Denote the Hilbertian norm by $\|\cdot\| = \|\cdot\|_{L^2(m_t^N)}$ and define the four-dimensional vector

$$ Y_t^N = (\|\nabla_{\mathbf{v}} u_t^N\|, \|\nabla_{\mathbf{v}}^2 u_t^N\|, \|\nabla_{\mathbf{x}} u_t^N\|, \|\nabla_{\mathbf{x}} \nabla_{\mathbf{v}} u_t^N\|)^\mathrm{T}. \quad (4.23) $$

By Cauchy-Schwarz we have $-\frac{d}{dt}\mathcal{E}^N(m_t^N) \ge (Y_t^N)^\mathrm{T} K Y_t^N$ where

$$ K := \begin{pmatrix} 1 + 2a - 2\|\nabla^2 U^N\|_{\text{op},\infty} & b & -2b & -2a & 0 \\ & 2a & -2\|\nabla^2 U^N\|_{\text{op},\infty} & c & -4b \\ & & 2b & 0 & 2c \end{pmatrix}, $$
where $||∇²U²||_{op,∞}$ is bounded by (4.19). We then apply the same argument as in the proof of Theorem 2.1 to pick a, b, c such that $ac > b²$ and K is positive-definite with its smallest eigenvalue $α > 0$. Then,

$$-\frac{d\mathcal{E}^N(m_t^N)}{dt} \geq (Y_t^N)^T K Y_t^N \geq \alpha I(m_t^N|m_\infty^N),$$

from which the desired inequality (4.18) follows.

We then show the inequality (4.18) holds for general mean field functional F and initial value $m_0^N$. First, suppose still that $U^N$ satisfies additionally the bound (4.20) but no longer suppose $m_0^N$ satisfies additionally (4.21). As $m_0^N$ has finite second moment, finite entropy and finite Fisher information, we can find a sequence of measures $(m_{n,0}^N)_{n \in \mathbb{N}}$, each of which satisfies the bound (4.21), such that

$$\lim_{n \to +\infty} W_2(m_{n,0}^N, m_0^N) + |H(m_{n,0}^N) - H(m_0^N)| + |I(m_{n,0}^N) - I(m_0^N)| = 0,$$

by the procedure in the proof of Proposition 4.7. We have the convergence $m_{n,t}^N \to m_t^N$ in $P_2$. So taking the lower limit on both sides of

$$\mathcal{E}^N(m_{n,t}^N) - \mathcal{E}^N(m_{n,0}^N) + \alpha \int_0^t I(m_{n,s}^N|m_\infty^N)ds \leq 0$$

yields (4.18) for $s = 0$, thanks to the $P_2$-continuity of F and the $P_2$-lower-semicontinuity of entropy and Fisher information, proved in Lemma A.1.

Second, we no longer suppose $U^N$ satisfies the bound (4.20) and set

$$U_k^N = U^N \star \rho_k$$

for a sequence of smooth mollifiers $(\rho_k)_{k \in \mathbb{N}}$ in $\mathbb{R}^{dN}$. Then $U_k^N$ is $C^4$ and satisfies its second and fourth-order derivatives $\nabla^\nu U_k^N = \nabla^2 U^N \star \nabla^{\nu-2} \rho_k$ with $\nu = 3, 4$ are bounded as $||\nabla^2 U^N||_\infty \leq M_{mx}^F + M_{mm}^F$. Moreover, from the bound (4.19) on $\nabla^2 U^N$ we deduce

$$\|\nabla(U_k^N - U^N)\|_{\infty} \to 0 \qquad (4.24)$$

and $||\nabla^2 U_k^N||_\infty \leq ||\nabla^2 U^N||_\infty \leq M_{mx}^F + M_{mm}^F$. Let $(m_{k,t}^{\prime\prime N})_{t \geq 0}$ be the flow of measures driven by the regularized potential $U_k^N$ with the initial value $m_{k,0}^{\prime\prime N} = m_0^N$ and denote its invariant measure by $m_{k,\infty}^{\prime\prime N}$. That is to say, $m_{k,\infty}^{\prime\prime N}$ is the probability measure proportional to $\exp(-U_k^N(\mathbf{x}) - \frac{1}{2}|\mathbf{v}|^2)d\mathbf{x}d\mathbf{v}$. Thanks to the bound (4.24), we can apply the synchronous coupling result in Lemma 5.1 and obtain $m_{k,t}^{\prime\prime N} \to m_t^N$ in $P_2$ for every $t \geq 0$. The result obtained in the previous paragraph writes

$$H(m_{k,t}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N}) + I_{a,b,c}(m_{k,t}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N}) - H(m_0^N|m_{k,\infty}^{\prime\prime N}) - I_{a,b,c}(m_0^N|m_{k,\infty}^{\prime\prime N}) \\ + \alpha \int_0^t I(m_{k,s}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N})ds \leq 0$$

for every $t \geq 0$ and we take the lower limit on both sides to obtain (4.18) with $s = 0$ for general initial values and general mean field functional. In particular, this implies for every $t \geq 0$, $m_t^N$ has finite entropy and finite Fisher information. Then we apply the same argument to the flow with $m_s^N$ as the initial value and obtain (4.18) for general $s \geq 0$. □

**Remark 4.5.** If we additionally assume a uniform-in-N LSI for $m_\infty^N$, then we can directly establish

$$\frac{d\mathcal{E}_N(m_t^N)}{dt} \leq -\kappa \mathcal{E}_N(m_t^N),$$
where $||∇²U²||_{op,∞}$ is bounded by (4.19). We then apply the same argument as in the proof of Theorem 2.1 to pick a, b, c such that $ac > b²$ and K is positive-definite with its smallest eigenvalue $α > 0$. Then,

$$-\frac{d\mathcal{E}^N(m_t^N)}{dt} \geq (Y_t^N)^\top K Y_t^N \geq \alpha I(m_t^N|m_\infty^N),$$

from which the desired inequality (4.18) follows.

We then show the inequality (4.18) holds for general mean field functional F and initial value $m_0^N$. First, suppose still that $U^N$ satisfies additionally the bound (4.20) but no longer suppose $m_0^N$ satisfies additionally (4.21). As $m_0^N$ has finite second moment, finite entropy and finite Fisher information, we can find a sequence of measures $(m_{n,0}^N)_{n \in \mathbb{N}}$, each of which satisfies the bound (4.21), such that

$$\lim_{n \to +\infty} W_2(m_{n,0}^N, m_0^N) + |H(m_{n,0}^N) - H(m_0^N)| + |I(m_{n,0}^N) - I(m_0^N)| = 0,$$

by the procedure in the proof of Proposition 4.7. We have the convergence $m_{n,t}^N \to m_t^N$ in $P_2$. So taking the lower limit on both sides of

$$\mathcal{E}^N(m_{n,t}^N) - \mathcal{E}^N(m_{n,0}^N) + \alpha \int_0^t I(m_{n,s}^N|m_\infty^N)ds \leq 0$$

yields (4.18) for $s = 0$, thanks to the $P_2$-continuity of F and the $P_2$-lower-semicontinuity of entropy and Fisher information, proved in Lemma A.1.

Second, we no longer suppose $U^N$ satisfies the bound (4.20) and set

$$U_k^N = U^N \star \rho_k$$

for a sequence of smooth mollifiers $(\rho_k)_{k \in \mathbb{N}}$ in $\mathbb{R}^{dN}$. Then $U_k^N$ is $C^4$ and satisfies its second and fourth-order derivatives $\nabla^\nu U_k^N = \nabla^2 U^N \star \nabla^{\nu-2} \rho_k$ with $\nu = 3, 4$ are bounded as $||\nabla^2 U^N||_\infty \leq M_{mx}^F + M_{mm}^F$. Moreover, from the bound (4.19) on $\nabla^2 U^N$ we deduce

$$\|\nabla(U_k^N - U^N)\|_\infty \to 0 \qquad (4.24)$$

and $||\nabla^2 U_k^N||_\infty \leq ||\nabla^2 U^N||_\infty \leq M_{mx}^F + M_{mm}^F$. Let $(m_{k,t}^{\prime\prime N})_{t \geq 0}$ be the flow of measures driven by the regularized potential $U_k^N$ with the initial value $m_{k,0}^{\prime\prime N} = m_0^N$ and denote its invariant measure by $m_{k,\infty}^{\prime\prime N}$. That is to say, $m_{k,\infty}^{\prime\prime N}$ is the probability measure proportional to $\exp(-U_k^N(\mathbf{x}) - \frac{1}{2}|\mathbf{v}|^2)d\mathbf{x}d\mathbf{v}$. Thanks to the bound (4.24), we can apply the synchronous coupling result in Lemma 5.1 and obtain $m_{k,t}^{\prime\prime N} \to m_t^N$ in $P_2$ for every $t \geq 0$. The result obtained in the previous paragraph writes

$$H(m_{k,t}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N}) + I_{a,b,c}(m_{k,t}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N}) - H(m_0^N|m_{k,\infty}^{\prime\prime N}) - I_{a,b,c}(m_0^N|m_{k,\infty}^{\prime\prime N}) \\ + \alpha \int_0^t I(m_{k,s}^{\prime\prime N}|m_{k,\infty}^{\prime\prime N})ds \leq 0$$

for every $t \geq 0$ and we take the lower limit on both sides to obtain (4.18) with $s = 0$ for general initial values and general mean field functional. In particular, this implies for every $t \geq 0$, $m_t^N$ has finite entropy and finite Fisher information. Then we apply the same argument to the flow with $m_s^N$ as the initial value and obtain (4.18) for general $s \geq 0$. □

**Remark 4.5.** If we additionally assume a uniform-in-$N$ LSI for $m_\infty^N$, then we can directly establish

$$\frac{d\mathcal{E}_N(m_t^N)}{dt} \leq -\kappa \mathcal{E}_N(m_t^N),$$
for a constant $\kappa > 0$ independent of $N$. This approach has been explored in a number of previous works. We do not impose such an assumption or sufficient conditions for it, as they often requires the mean field interaction to be small enough or (semi-)convex enough, excluding the application to neural networks in Section 3.

We then give the proof of Theorem 2.2. The method of proof is similar to [7, Theorem 2.3] and we only need to take into account of the additional kinetic terms. We give a complete proof only for the sake of self-containedness.

*Proof of Theorem 2.2.* We pick the positive constants $a, b, c, \alpha$ depending only on $M_{max}^F$ and $M_{mm}^F$ such that $ac > b^2$ and (4.18) holds for every $t \ge 0$, according to Lemma 4.9. Then, as in the proof of [7, Theorem 2.3], we will establish a lower bound of the relative Fisher information $I_t := I(m_t^N | m_\infty^N)$ in order to obtain the desired result.

*Regularity of conditional distribution.* By local hypoelliptic positivity (see e.g. [40, Theorem A.19 and Corollary A.21]), we know that for every $t > 0$ and every $\mathbf{z} \in \mathbb{R}^{2dN}$, $m_t^N(\mathbf{z}) > 0$. Let $i \in \{1, \dots, N\}$. Define the marginal density $m_t^{N,-i}(\mathbf{z}^{-i}) = \int m_t^N(\mathbf{z})d\mathbf{z}^i$, which is strictly positive by the local positivity of $m_t^N$ and is lower semicontinuous by Fatou's lemma. By the Fubini theorem, we have $\int m_t^{N,-i}(\mathbf{z}^{-i})d\mathbf{z}^{-i} = 1$. Together with the lower semicontinuity, we obtain that $m_t^{N,-i}(\mathbf{z}^{-i})$ is finite everywhere. We are therefore able to define the conditional probability density

$$m_t^{N,i|-i}(z^i|\mathbf{z}^{-i}) = \frac{m_t^N(\mathbf{z})}{m_t^{N,-i}(\mathbf{z}^{-i})} = \frac{m_t^N(\mathbf{z})}{\int m_t^N(\mathbf{z})d\mathbf{z}^i},$$

which is weakly differentiable in $z^i$ and strictly positive everywhere. We can also define the conditional density $m_\infty^{N,i|-i}$ for the invariant measure $m_\infty^N$, and the regularity follows directly from its explicit expression.

*Decomposing Fisher componentwise.* Using the conditional distributions, we can decompose the relative Fisher information as

$$ 
\begin{aligned}
I_t &= \int \left| \nabla \log \frac{m_t^N(\mathbf{z})}{m_\infty^N(\mathbf{z})} \right|^2 m_t^N(d\mathbf{z}) = \mathbb{E} \left[ \left| \nabla \log \frac{m_t^N(\mathbf{Z}_t)}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log \frac{m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) m_t^{N,-i} (\mathbf{Z}_t^{-i})}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log \frac{m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i})}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t}, X_t^i) + V_t^i \right|^2 \right].
\end{aligned}
$$

*Change of empirical measure and componentwise LSI.* We replace the empirical measure $\mu_{\mathbf{X}}$ in $D_m F$ by $\mu_{\mathbf{X}^{-i}}$. Define the difference $\delta_1^i(\mathbf{x}; y) = D_m F(\mu_{\mathbf{X}}, y) - D_m F(\mu_{\mathbf{X}^{-i}}, y)$ and denote by $\hat{\mu}_{\mathbf{X}^{-i}}$ the probability on $\mathbb{R}^{2d}$ such that

$$\hat{\mu}_{\mathbf{X}^{-i}}(dxdv) \propto \exp\left(-\frac{\delta F}{\delta m}(\mu_{\mathbf{X}^{-i}}, x) - \frac{1}{2}|v|^2\right) dxdv.$$
for a constant $\kappa > 0$ independent of $N$. This approach has been explored in a number of previous works. We do not impose such an assumption or sufficient conditions for it, as they often requires the mean field interaction to be small enough or (semi-)convex enough, excluding the application to neural networks in Section 3.

We then give the proof of Theorem 2.2. The method of proof is similar to [7, Theorem 2.3] and we only need to take into account of the additional kinetic terms. We give a complete proof only for the sake of self-containedness.

**Proof of Theorem 2.2.** We pick the positive constants $a, b, c, \alpha$ depending only on $M_{max}^F$ and $M_{mm}^F$ such that $ac > b^2$ and (4.18) holds for every $t \ge 0$, according to Lemma 4.9. Then, as in the proof of [7, Theorem 2.3], we will establish a lower bound of the relative Fisher information $I_t := I(m_t^N | m_\infty^N)$ in order to obtain the desired result.

**Regularity of conditional distribution.** By local hypoelliptic positivity (see e.g. [40, Theorem A.19 and Corollary A.21]), we know that for every $t > 0$ and every $\mathbf{z} \in \mathbb{R}^{2dN}$, $m_t^N(\mathbf{z}) > 0$. Let $i \in \{1, \dots, N\}$. Define the marginal density $m_t^{N,-i}(\mathbf{z}^{-i}) = \int m_t^N(\mathbf{z})dz^i$, which is strictly positive by the local positivity of $m_t^N$ and is lower semicontinuous by Fatou's lemma. By the Fubini theorem, we have $\int m_t^{N,-i}(\mathbf{z}^{-i})dz^{-i} = 1$. Together with the lower semicontinuity, we obtain that $m_t^{N,-i}(\mathbf{z}^{-i})$ is finite everywhere. We are therefore able to define the conditional probability density

$$m_t^{N,i|-i}(z^i|\mathbf{z}^{-i}) = \frac{m_t^N(\mathbf{z})}{m_t^{N,-i}(\mathbf{z}^{-i})} = \frac{m_t^N(\mathbf{z})}{\int m_t^N(\mathbf{z})dz^i},$$

which is weakly differentiable in $z^i$ and strictly positive everywhere. We can also define the conditional density $m_\infty^{N,i|-i}$ for the invariant measure $m_\infty^N$, and the regularity follows directly from its explicit expression.

**Decomposing Fisher componentwise.** Using the conditional distributions, we can decompose the relative Fisher information as

$$ 
\begin{aligned}
I_t &= \int \left| \nabla \log \frac{m_t^N(\mathbf{z})}{m_\infty^N(\mathbf{z})} \right|^2 m_t^N(d\mathbf{z}) = \mathbb{E} \left[ \left| \nabla \log \frac{m_t^N(\mathbf{Z}_t)}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log \frac{m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) m_t^{N,-i} (\mathbf{Z}_t^{-i})}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log \frac{m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i})}{m_\infty^N(\mathbf{Z}_t)} \right|^2 \right] \\
&= \sum_{i=1}^N \mathbb{E} \left[ \left| \nabla_{Z^i} \log m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t}, X_t^i) + V_t^i \right|^2 \right].
\end{aligned}
$$

**Change of empirical measure and componentwise LSI.** We replace the empirical measure $\mu_{\mathbf{X}}$ in $D_m F$ by $\mu_{\mathbf{X}^{-i}}$. Define the difference $\delta_1^i(\mathbf{x}; y) = D_m F(\mu_{\mathbf{X}}, y) - D_m F(\mu_{\mathbf{X}^{-i}}, y)$ and denote by $\hat{\mu}_{\mathbf{X}^{-i}}$ the probability on $\mathbb{R}^{2d}$ such that

$$\hat{\mu}_{\mathbf{X}^{-i}}(dxdv) \propto \exp\left(-\frac{\delta F}{\delta m}(\mu_{\mathbf{X}^{-i}}, x) - \frac{1}{2}|v|^2\right) dxdv.$$
For every $\epsilon \in (0, 1)$, the Fisher information satisfies

$$ 
\begin{aligned}
I_t &= \sum_{i=1}^{N} \mathrm{E} \left[ \left| \nabla_{x^i} \log m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) + V_t^i + \delta_1^i (\mathbf{X}_t; X_t^i) \right|^2 \right] \\
&\geq \sum_{i=1}^{N} \mathrm{E} \left[ (1-\epsilon) \left| \nabla_{x^i} \log m_t^{N,i|-i} (X_t^i | \mathbf{X}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) + V_t^i \right|^2 \right. \\
&\qquad \left. - (\epsilon^{-1}-1) |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right] \\
&= (1-\epsilon) \sum_{i=1}^{N} \mathrm{E} \left[ I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) \right] - (\epsilon^{-1}-1) \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right],
\end{aligned}
$$

where we used the elementary inequality $(a+b)^2 \geq (1-\epsilon)|a|^2 - (\epsilon^{-1}-1)|b|^2$. Define the first error

$$ \Delta_1 := \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right] := \sum_{i=1}^{N} \mathrm{E} \left[ \left| D_m F(\mu_{\mathbf{X}_t}, X_t^i) - D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) \right|^2 \right]. \quad (4.25) $$

The previous inequality writes

$$ I_t \geq (1-\epsilon) \sum_{i=1}^{N} \mathrm{E} \left[ I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) \right] - (\epsilon^{-1}-1) \Delta_1. \quad (4.26) $$

We apply the uniform $\rho$-log-Sobolev inequality (2.7) for $\hat{\mu}_{\mathbf{X}_t^i}$ with $\rho$ defined by (2.6) and obtain

$$ 
\begin{aligned}
\frac{1}{4\rho} I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) &\geq H(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) \\
&= \int \left( \log m_t^{N,i|-i} (x^i | \mathbf{Z}_t^{-i}) + \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) m_t^{N,i|-i} (dz^i | \mathbf{Z}_t^{-i}) \\
&\qquad + \log Z(\hat{\mu}_{\mathbf{X}_t^{-i}}),
\end{aligned}
$$

where the last quantity is the normalization factor

$$ Z(\hat{\mu}_{\mathbf{X}_t^{-i}}) := \int \exp\left(-\frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x) - \frac{1}{2}|v|^2\right) dx dv. $$

Then we apply Jensen's inequality to $\log Z(\hat{\mu}_{\mathbf{X}_t^{-i}})$ to obtain

$$ \log Z(\hat{\mu}_{\mathbf{X}_t^{-i}}) \geq - \int \left( \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) m_\infty(dz^i) - \int m_\infty(z^i) \log m_\infty(z^i) dz^i. $$

Chaining the previous two inequalities and summing over $i$, we have

$$ 
\begin{aligned}
\frac{1}{4\rho} \sum_{i=1}^{N} I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) &\geq \sum_{i=1}^{N} \left[ \int \left( \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) \right. \\
&\qquad \left. (m_t^{N,i|-i} (dz^i | \mathbf{Z}_t^{-i}) - m_\infty(dz^i)) + H(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) - H(m_\infty) \right].
\end{aligned}
\quad (4.27) $$

Another change of empirical measure. We are going to replace $\mu_{\mathbf{X}_t^{-i}}$ by $\mu_{\mathbf{X}}$ in (4.27). Define $\delta_2^i(\mathbf{x}; y) := \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, y) - \frac{\delta F}{\delta m}(\mu_{\mathbf{X}}, y)$ and the second error

$$ \Delta_2 := \sum_{i=1}^{N} \int \delta_2^i(\mathbf{x}; x^i) m_t^N(d\mathbf{z}) - \sum_{i=1}^{N} \iint \delta_2^i(\mathbf{x}; x') m_\infty(d\mathbf{z}') m_t^N(d\mathbf{z}). \quad (4.28) $$
For every $\epsilon \in (0, 1)$, the Fisher information satisfies

$$ 
\begin{aligned}
I_t &= \sum_{i=1}^{N} \mathrm{E} \left[ \left| \nabla_{x^i} \log m_t^{N,i|-i} (Z_t^i | \mathbf{Z}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) + V_t^i + \delta_1^i (\mathbf{X}_t; X_t^i) \right|^2 \right] \\
&\geq \sum_{i=1}^{N} \mathrm{E} \left[ (1-\epsilon) \left| \nabla_{x^i} \log m_t^{N,i|-i} (X_t^i | \mathbf{X}_t^{-i}) + D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) + V_t^i \right|^2 \right. \\
&\qquad \left. - (\epsilon^{-1}-1) |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right] \\
&= (1-\epsilon) \sum_{i=1}^{N} \mathrm{E} \left[ I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) \right] - (\epsilon^{-1}-1) \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right],
\end{aligned}
$$

where we used the elementary inequality $(a+b)^2 \geq (1-\epsilon)|a|^2 - (\epsilon^{-1}-1)|b|^2$. Define the first error

$$ \Delta_1 := \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i (\mathbf{X}_t; X_t^i)|^2 \right] := \sum_{i=1}^{N} \mathrm{E} \left[ \left| D_m F(\mu_{\mathbf{X}_t}, X_t^i) - D_m F(\mu_{\mathbf{X}_t^{-i}}, X_t^i) \right|^2 \right]. \quad (4.25) $$

The previous inequality writes

$$ I_t \geq (1 - \epsilon) \sum_{i=1}^{N} \mathrm{E} \left[ I \left( m_t^{N,i|-i} \left( \cdot \middle| \mathbf{Z}_t^{-i} \right) \middle| \hat{\mu}_{\mathbf{X}_t^{-i}} \right) \right] - (\epsilon^{-1} - 1) \Delta_1. \quad (4.26) $$

We apply the uniform $\rho$-log-Sobolev inequality (2.7) for $\hat{\mu}_{\mathbf{X}_t^i}$ with $\rho$ defined by (2.6) and obtain

$$ 
\begin{aligned}
\frac{1}{4\rho} I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) &\geq H(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) \\
&= \int \left( \log m_t^{N,i|-i} (x^i | \mathbf{Z}_t^{-i}) + \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) m_t^{N,i|-i} (dz^i | \mathbf{Z}_t^{-i}) \\
&\qquad + \log Z(\hat{\mu}_{\mathbf{X}_t^{-i}}),
\end{aligned}
$$

where the last quantity is the normalization factor

$$ Z(\hat{\mu}_{\mathbf{X}_t^{-i}}) := \int \exp\left(-\frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x) - \frac{1}{2}|v|^2\right) dx dv. $$

Then we apply Jensen's inequality to $\log Z(\hat{\mu}_{\mathbf{X}_t^{-i}})$ to obtain

$$ \log Z(\hat{\mu}_{\mathbf{X}_t^{-i}}) \geq - \int \left( \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) m_\infty(dz^i) - \int m_\infty(z^i) \log m_\infty(z^i) dz^i. $$

Chaining the previous two inequalities and summing over $i$, we have

$$ 
\begin{aligned}
\frac{1}{4\rho} \sum_{i=1}^{N} I(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) | \hat{\mu}_{\mathbf{X}_t^{-i}}) &\geq \sum_{i=1}^{N} \left[ \int \left( \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, x^i) + \frac{1}{2} |v^i|^2 \right) (m_t^{N,i|-i} (dz^i | \mathbf{Z}_t^{-i}) - m_\infty(dz^i)) + H(m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) - H(m_\infty) \right]. \quad (4.27)
\end{aligned}
$$

Another change of empirical measure. We are going to replace $\mu_{\mathbf{X}_t^{-i}}$ by $\mu_{\mathbf{X}}$ in (4.27). Define $\delta_2^i(\mathbf{x}; y) := \frac{\delta F}{\delta m}(\mu_{\mathbf{X}_t^{-i}}, y) - \frac{\delta F}{\delta m}(\mu_{\mathbf{X}}, y)$ and the second error

$$ \Delta_2 := \sum_{i=1}^{N} \int \delta_2^i(\mathbf{x}; x^i) m_t^N(d\mathbf{z}) - \sum_{i=1}^{N} \iint \delta_2^i(\mathbf{x}; x') m_\infty(d\mathbf{z}') m_t^N(d\mathbf{z}). \quad (4.28) $$
Taking expectations on both sides of (4.27), we obtain

$$
\begin{aligned}
& \frac{1}{4\rho} \sum_{i=1}^{N} \mathrm{E} \left[ I \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \middle| \hat{\mu}_{\mathbf{X}_t^{-i}} \right) \right] \\
& \geq N \mathrm{E} \left[ \int \left( \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t}, x) + \frac{1}{2} |v|^2 \right) (\mu_{\mathbf{Z}_t} - m_{\infty}) (dz) \right] + \sum_{i=1}^{N} \mathrm{E} \left[ H \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \right) \right] \\
& \qquad - N H(m_{\infty}) + \Delta_2.
\end{aligned}
\quad (4.29)
$$

Thanks to the convexity of $F$, the first term satisfies the tangent inequality

$$
\begin{aligned}
& N \mathrm{E} \left[ \int \left( \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t}, x) + \frac{1}{2} |v|^2 \right) (\mu_{\mathbf{Z}_t} - m_{\infty}) (dz) \right] \\
& \geq N \mathrm{E} \left[ F(\mu_{\mathbf{X}_t}) - F(m_{\infty}^x) \right] + \frac{1}{2} \int |\mathbf{v}|^2 m_t^N (d\mathbf{z}) - \frac{N}{2} \int |v|^2 |m_{\infty}(dz) \\
& = F^N(m_t^N) - N F(m_{\infty}) + \frac{1}{2} \int |\mathbf{v}|^2 m_t^N (d\mathbf{z}) - \frac{N}{2} \int |v|^2 |m_{\infty}(dz). \quad (4.30)
\end{aligned}
$$

For the second term we apply the information inequality (4.3) to obtain

$$
\sum_{i=1}^{N} \mathrm{E}^{-i} \left[ H \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \right) \right] \geq H(m_t^N).
$$

Hence,

$$
\sum_{i=1}^{N} \mathrm{E} \left[ I \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \middle| \hat{\mu}_{\mathbf{X}_t^{-i}} \right) \right] \geq 4\rho \left( \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) + \Delta_2 \right)
$$

by the definition of free energies $\mathcal{F}(m) = F(m) + H(m) + \frac{1}{2} \int |v|^2 m, \mathcal{F}^N(m^N) = F^N(m^N) + H(m^N) + \frac{1}{2} \int |v|^2 m^N$. Using (4.26), we obtain

$$
I_t = I(m_t^N | m_\infty^N) \geq 4\rho(1 - \varepsilon)\left(\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) + \Delta_2\right) - (\varepsilon^{-1} - 1)\Delta_1. \quad (4.31)
$$

Bounding the errors $\Delta_1, \Delta_2$. The transport plan between $\mu_x$ and $\mu_{x^{-i}}$

$$
\pi^i = \frac{1}{N} \sum_{j \neq i} \delta_{(x^j, x^i)} + \frac{1}{N(N-1)} \sum_{j \neq i} \delta_{(x^j, x^i)} \quad (4.32)
$$

gives the bound $W_1(\mu_x, \mu_{x^{-i}}) \le \frac{1}{N(N-1)} \sum_{j \ne i} |x^j - x^i|$. We use this transport plan to bound the errors $\Delta_1, \Delta_2$.

Let us treat the first error $\Delta_1$. Since $m \mapsto D_m F(m, x)$ is $M_{mm}^F$-Lipschitz continuous in $W_2$ metric, we have

$$
|\delta_1^i(\mathbf{x}; y)| \leq M_{mm}^F W_2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}).
$$
Taking expectations on both sides of (4.27), we obtain

$$
\begin{aligned}
& \frac{1}{4\rho} \sum_{i=1}^{N} \mathrm{E} \left[ I \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \middle| \hat{\mu}_{\mathbf{X}_t^{-i}} \right) \right] \\
& \geq N \mathrm{E} \left[ \int \left( \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t}, x) + \frac{1}{2} |v|^2 \right) (\mu_{\mathbf{Z}_t} - m_{\infty}) (dz) \right] + \sum_{i=1}^{N} \mathrm{E} \left[ H \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \right) \right] \\
& \qquad - N H(m_{\infty}) + \Delta_2.
\end{aligned}
\quad (4.29)
$$

Thanks to the convexity of $F$, the first term satisfies the tangent inequality

$$
\begin{aligned}
& N \mathrm{E} \left[ \int \left( \frac{\delta F}{\delta m} (\mu_{\mathbf{X}_t}, x) + \frac{1}{2} |v|^2 \right) (\mu_{\mathbf{Z}_t} - m_{\infty}) (dz) \right] \\
& \geq N \mathrm{E} \left[ F(\mu_{\mathbf{X}_t}) - F(m_{\infty}^x) \right] + \frac{1}{2} \int |\mathbf{v}|^2 m_t^N (d\mathbf{z}) - \frac{N}{2} \int |v|^2 |m_{\infty}(dz) \\
& = F^N(m_t^N) - N F(m_{\infty}) + \frac{1}{2} \int |\mathbf{v}|^2 m_t^N (d\mathbf{z}) - \frac{N}{2} \int |v|^2 |m_{\infty}(dz). \quad (4.30)
\end{aligned}
$$

For the second term we apply the information inequality (4.3) to obtain

$$
\sum_{i=1}^{N} \mathrm{E}^{-i} \left[ H \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \right) \right] \geq H(m_t^N).
$$

Hence,

$$
\sum_{i=1}^{N} \mathrm{E} \left[ I \left( m_t^{N,i|-i} (\cdot | \mathbf{Z}_t^{-i}) \middle| \hat{\mu}_{\mathbf{X}_t^{-i}} \right) \right] \geq 4\rho \left( \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) + \Delta_2 \right)
$$

by the definition of free energies $\mathcal{F}(m) = F(m) + H(m) + \frac{1}{2} \int |v|^2 m, \mathcal{F}^N(m^N) = F^N(m^N) + H(m^N) + \frac{1}{2} \int |v|^2 m^N$. Using (4.26), we obtain

$$
I_t = I(m_t^N | m_\infty^N) \geq 4\rho(1 - \varepsilon)\left(\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty) + \Delta_2\right) - (\varepsilon^{-1} - 1)\Delta_1. \quad (4.31)
$$

Bounding the errors $\Delta_1, \Delta_2$. The transport plan between $\mu_x$ and $\mu_{x^{-i}}$

$$
\pi^i = \frac{1}{N} \sum_{j \neq i} \delta_{(x^j, x^i)} + \frac{1}{N(N-1)} \sum_{j \neq i} \delta_{(x^j, x^i)} \quad (4.32)
$$

gives the bound $W_1(\mu_x, \mu_{x^{-i}}) \le \frac{1}{N(N-1)} \sum_{j \ne i} |x^j - x^i|$. We use this transport plan to bound the errors $\Delta_1, \Delta_2$.

Let us treat the first error $\Delta_1$. Since $m \mapsto D_m F(m, x)$ is $M_{mm}^F$-Lipschitz continuous in $W_2$ metric, we have

$$
|\delta_1^i(\mathbf{x}; y)| \leq M_{mm}^F W_2(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}).
$$
Under the L²-optimal transport plan Law((Z_t^i)_{i=1}^N, (\tilde{Z}_\infty^i)_{i=1}^N) \in \Pi(m_t^N, m_\infty^{\otimes N}) we have

$$
\begin{aligned}
\Delta_1 &= \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i(\mathbf{X}_t; X_t^i)|^2 \right] \le (M_{mm}^F)^2 \sum_{i=1}^{N} \mathrm{E} \left[ W_1^2 (\mu_{\mathbf{X}_t}, \mu_{\mathbf{X}_t^{-i}}) \right] \\
&\le \frac{(M_{mm}^F)^2}{N(N-1)} \mathrm{E} \left[ \sum_{1 \le i,j \le N \atop i \ne j} |X_t^j - X_t^i|^2 \right] \\
&\le \frac{3(M_{mm}^F)^2}{N(N-1)} \mathrm{E} \left[ \sum_{1 \le i,j \le N \atop i \ne j} \left( |X_t^i - \tilde{X}_\infty^i|^2 + |\tilde{X}_\infty^i - X_t^j|^2 + |X_t^j - \tilde{X}_\infty^j|^2 \right) \right] \\
&\le \frac{3(M_{mm}^F)^2}{N(N-1)} \left( 2(N-1) \mathrm{E} \left[ \sum_{i=1}^{N} |X_t^i - \tilde{X}_\infty^i|^2 \right] + N(N-1) \mathrm{E} \left[ |\tilde{X}_\infty^1 - \tilde{X}_\infty^2|^2 \right] \right).
\end{aligned}
$$

The first term $\mathrm{E}[\sum_{i=1}^N |X_t^i - \tilde{X}_\infty^i|^2]$ is bounded by the Wasserstein distance $W_2^2(m_t^N, m_\infty^{\otimes N})$, while the second $\mathrm{E}[|\tilde{X}_\infty^1 - \tilde{X}_\infty^2|^2]$ equals $2 \mathrm{Var} m_\infty^x$. Hence the first error satisfies the bound

$$
\Delta_1 \le 6(M_{mm}^F)^2 \left( \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + \mathrm{Var} m_\infty \right). \quad (4.33)
$$

Now treat the second error $\Delta_2$. The Lipschitz constant of $y \mapsto \delta_2^i(\mathbf{x}; y) = \frac{\delta F}{\delta m}(\mu_{\mathbf{x}^{-i}}, y) - \frac{\delta F}{\delta m}(\mu_{\mathbf{x}}, y)$ is controlled by

$$
|\nabla_y \delta_2^i(\mathbf{x}; y)| = |D_m F(\mu_{\mathbf{x}}, y) - D_m F(\mu_{\mathbf{x}^{-i}}, y)| \le M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}).
$$

Hence $|\delta_2^i(\mathbf{x}; y) - \delta_2^i(\mathbf{x}; y')| \le M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}})|y - y'|$. Use Fubini's theorem to first integrate $z'$ in the definition of the second error (4.28) and let $\tilde{Z}_\infty'$ be independent from $Z_t$. We obtain

$$
\begin{aligned}
|\Delta_2| &\le \sum_{i=1}^{N} \iint \left( \int |\delta_2^i(\mathbf{x}; x^i) - \delta_2^i(\mathbf{x}; x')| m_\infty(dz') \right) m_t^N(d\mathbf{z}) \\
&\le \sum_{i=1}^{N} \iint M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}) |x' - x^i| m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \sum_{i=1}^{N} \iint \frac{M_{mm}^F}{N(N-1)} \sum_{j=1, j \ne i}^{N} |x^i - x^j| |x' - x^i| m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \frac{M_{mm}^F}{2N(N-1)} \sum_{i=1}^{N} \iint \sum_{j=1, j \ne i}^{N} (|x^i - x^j|^2 + |x' - x^i|^2) m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \frac{M_{mm}^F}{2N(N-1)} \left( \sum_{\substack{i,j=1 \\ i \ne j}}^{N} \mathrm{E}[|X_t^i - X_t^j|^2] + (N-1) \sum_{i=1}^{N} \mathrm{E}[|X_t^i - \tilde{X}_\infty^i|^2] \right).
\end{aligned}
$$

Using the same method we used for $\Delta_1$, we control the first term by

$$
\sum_{\substack{i,j=1 \\ i \neq j}}^{N} \mathrm{E} \left[ |X_t^i - X_t^j|^2 \right] \le 6N(N-1) \left( \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + \mathrm{Var} m_\infty^x \right).
$$

For the second term we work again under the L²-optimal plan Law((Z_t^i)_{i=1}^N, (\tilde{Z}_\infty^i)_{i=1}^N) ∈
Under the $L^2$-optimal transport plan Law((Z_t^i)_{i=1}^N, (\tilde{Z}_\infty^i)_{i=1}^N) \in \Pi(m_t^N, m_\infty^{\otimes N})$ we have

$$
\begin{aligned}
\Delta_1 &= \sum_{i=1}^{N} \mathrm{E} \left[ |\delta_1^i(\mathbf{X}_t; X_t^i)|^2 \right] \le (M_{mm}^F)^2 \sum_{i=1}^{N} \mathrm{E} \left[ W_1^2 (\mu_{\mathbf{X}_t}, \mu_{\mathbf{X}_t^{-i}}) \right] \\
&\le \frac{(M_{mm}^F)^2}{N(N-1)} \mathrm{E} \left[ \sum_{1 \le i,j \le N \atop i \ne j} |X_t^j - X_t^i|^2 \right] \\
&\le \frac{3(M_{mm}^F)^2}{N(N-1)} \mathrm{E} \left[ \sum_{1 \le i,j \le N \atop i \ne j} \left( |X_t^i - \tilde{X}_\infty^i|^2 + |\tilde{X}_\infty^i - X_t^j|^2 + |X_t^j - \tilde{X}_\infty^j|^2 \right) \right] \\
&\le \frac{3(M_{mm}^F)^2}{N(N-1)} \left( 2(N-1) \mathrm{E} \left[ \sum_{i=1}^{N} |X_t^i - \tilde{X}_\infty^i|^2 \right] + N(N-1) \mathrm{E} \left[ |\tilde{X}_\infty^1 - \tilde{X}_\infty^2|^2 \right] \right).
\end{aligned}
$$

The first term $\mathrm{E}[\sum_{i=1}^N |X_t^i - \tilde{X}_\infty^i|^2]$ is bounded by the Wasserstein distance $W_2^2(m_t^N, m_\infty^{\otimes N})$, while the second $\mathrm{E}[|\tilde{X}_\infty^1 - \tilde{X}_\infty^2|^2]$ equals $2 \mathrm{Var} m_\infty^x$. Hence the first error satisfies the bound

$$
\Delta_1 \le 6(M_{mm}^F)^2 \left( \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + \mathrm{Var} m_\infty \right). \quad (4.33)
$$

Now treat the second error $\Delta_2$. The Lipschitz constant of $y \mapsto \delta_2^i(\mathbf{x}; y) = \frac{\delta F}{\delta m}(\mu_{\mathbf{x}^{-i}}, y) - \frac{\delta F}{\delta m}(\mu_{\mathbf{x}}, y)$ is controlled by

$$
|\nabla_y \delta_2^i(\mathbf{x}; y)| = |D_m F(\mu_{\mathbf{x}}, y) - D_m F(\mu_{\mathbf{x}^{-i}}, y)| \le M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}).
$$

Hence $|\delta_2^i(\mathbf{x}; y) - \delta_2^i(\mathbf{x}; y')| \le M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}})|y - y'|$. Use Fubini's theorem to first integrate $z'$ in the definition of the second error (4.28) and let $\tilde{Z}_\infty'$ be independent from $Z_t$. We obtain

$$
\begin{aligned}
|\Delta_2| &\le \sum_{i=1}^N \iint \left( \int |\delta_2^i(\mathbf{x}; x^i) - \delta_2^i(\mathbf{x}; x')| m_\infty(dz') \right) m_t^N(d\mathbf{z}) \\
&\le \sum_{i=1}^N \iint M_{mm}^F W_1(\mu_{\mathbf{x}}, \mu_{\mathbf{x}^{-i}}) |x' - x^i| m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \sum_{i=1}^N \iint \frac{M_{mm}^F}{N(N-1)} \sum_{j=1, j \ne i}^N |x^i - x^j| |x' - x^i| m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \frac{M_{mm}^F}{2N(N-1)} \sum_{i=1}^N \iint \sum_{j=1, j \ne i}^N (|x^i - x^j|^2 + |x' - x^i|^2) m_\infty(dz') m_t^N(d\mathbf{z}) \\
&\le \frac{M_{mm}^F}{2N(N-1)} \left( \sum_{\substack{i,j=1 \\ i \ne j}}^N \mathrm{E}[|X_t^i - X_t^j|^2] + (N-1) \sum_{i=1}^N \mathrm{E}[|X_t^i - \tilde{X}_\infty^i|^2] \right).
\end{aligned}
$$

Using the same method we used for $\Delta_1$, we control the first term by

$$
\sum_{\substack{i,j=1 \\ i \neq j}}^{N} \mathrm{E} \left[ |X_t^i - X_t^j|^2 \right] \le 6N(N-1) \left( \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + \mathrm{Var} m_\infty^x \right).
$$

For the second term we work again under the $L^2$-optimal plan Law((Z_t^i)_{i=1}^N, (\tilde{Z}_\infty^i)_{i=1}^N) \in
Π($m_t^N$, $m_\infty^N$) and let $\tilde{Z}'_\infty$ remain independent from the other variables. We have

$$ \sum_{i=1}^{N} \mathbb{E} [ |X_t^i - \tilde{X}'_\infty|^2 ] \le 2 \sum_{i=1}^{N} \left( \mathbb{E} [ |X_t^i - \tilde{X}_\infty^i|^2 ] + \mathbb{E} [ |\tilde{X}_\infty^i - \tilde{X}'_\infty|^2 ] \right) \\ \le 2N \left( \frac{1}{N} W_2^2 (m_t^N, m_\infty^{\otimes N}) + 2 \text{Var} m_\infty^x \right). $$

As a result,

$$ |\Delta_2| \le M_{mm}^F \left( \frac{4}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + 5 \text{Var} m_\infty^x \right). \quad (4.34) $$

**Conclusion.** Inserting the bounds on the errors (4.33) and (4.34) to the lower bound of Fisher information (4.31), we obtain

$$ I(m_t^N | m_\infty^N) \ge 4\rho(1-\varepsilon)(\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty)) \\ - (16\rho M_{mm}^F + 6(\varepsilon^{-1}-1)(M_{mm}^F)^2)\frac{1}{N}W_2^2(m_t^N, m_\infty^{\otimes N}) \\ - (20\rho M_{mm}^F + 6(\varepsilon^{-1}-1)(M_{mm}^F)^2)\text{Var} m_\infty^x. $$

Thanks to the Poincaré inequality (2.8) for $m_\infty = \hat{m}_\infty$, its spatial variance satisfies

$$ 2\rho \text{Var}_{m_\infty}(x^i) \le \mathbb{E}_{m_\infty} [ |\nabla x^i|^2 ] = 1. \quad (4.35) $$

So $\text{Var}_{m_\infty^x} = \sum_{i=1}^d \text{Var}_{m_\infty}(x^i) \le d/2\rho$. Using the $T_2$-transport inequality (2.9) for $m_\infty^{\otimes N}$ and the entropy sandwich Lemma 4.3 we bound the transport cost by

$$ W_2^2(m_t^N, m_\infty^{\otimes N}) \le \frac{1}{\rho} H(m_t^N | m_\infty^{\otimes N}) \le \frac{1}{\rho} (\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_t)). $$

In the end we obtain $\mathcal{E}^N(m_T^N) \le \mathcal{E}^N(m_s^N) - \alpha \int_s^T I_t dt$ where

$$ I_t = \frac{1}{2} I(m_t^N | m_\infty^N) + \frac{1}{2} I(m_t^N | m_\infty^N) \\ \ge \frac{1}{2} \left[ 4(1-\varepsilon)\rho - \frac{M_{mm}^F}{N} \left( 16 + 6(\varepsilon^{-1}-1)\frac{M_{mm}^F}{\rho} \right) \right] (\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty)) \\ + \frac{1}{2} I(m_t^N | m_\infty^N) - \frac{dM_{mm}^F}{2\rho} (10\rho + 3(\varepsilon^{-1}-1)M_{mm}^F). $$

We conclude by applying Grönwall's lemma, as in the end of the proof of Theorem 2.1. □

# 5 Short-time behaviors and propagation of chaos

Our proof of the main theorem on the uniform-in-time propagation of chaos (Theorem 2.3) relies on the exponential convergence in Theorems 2.1 and 2.2, where the initial conditions are required to have finite entropy and finite Fisher information. We aim to demonstrate in this section that the non-linear kinetic Langevin dynamics exhibits the same regularization effects in short time as the linear ones, where the contributions from the non-linearity can be controlled. We will first show the short-time Wasserstein propagation of chaos using synchronous coupling. Then we adapt the regularization results for the linear dynamics to our setting and show that for measure initial values of finite second moment, the entropy and the Fisher information are finite for the flow at every positive time, where the short-time Wasserstein propagation of chaos also plays a role. Finally we combine all the estimates obtained to derive Theorem 2.3.
Π($m_t^N, m_\infty^N$) and let $\tilde{Z}'_\infty$ remain independent from the other variables. We have

$$ \sum_{i=1}^{N} \mathrm{E} [|X_t^i - \tilde{X}'_\infty|^2] \le 2 \sum_{i=1}^{N} \left( \mathrm{E} [|X_t^i - \tilde{X}_\infty^i|^2] + \mathrm{E} [|\tilde{X}_\infty^i - \tilde{X}'_\infty|^2] \right) \\ \le 2N \left( \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + 2 \mathrm{Var} m_\infty^x \right). $$

As a result,

$$ |\Delta_2| \le M_{mm}^F \left( \frac{4}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) + 5 \mathrm{Var} m_\infty^x \right). \quad (4.34) $$

**Conclusion.** Inserting the bounds on the errors (4.33) and (4.34) to the lower bound of Fisher information (4.31), we obtain

$$ I(m_t^N | m_\infty^N) \ge 4\rho(1-\varepsilon)(\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty)) \\ - (16\rho M_{mm}^F + 6(\varepsilon^{-1}-1)(M_{mm}^F)^2) \frac{1}{N} W_2^2(m_t^N, m_\infty^{\otimes N}) \\ - (20\rho M_{mm}^F + 6(\varepsilon^{-1}-1)(M_{mm}^F)^2) \mathrm{Var} m_\infty^x. $$

Thanks to the Poincaré inequality (2.8) for $m_\infty = \hat{m}_\infty$, its spatial variance satisfies

$$ 2\rho \mathrm{Var}_{m_\infty}(x^i) \le E_{m_\infty} [|\nabla x^i|^2] = 1. \quad (4.35) $$

So $\mathrm{Var}_{m_\infty^x} = \sum_{i=1}^d \mathrm{Var}_{m_\infty}(x^i) \le d/2\rho$. Using the $T_2$-transport inequality (2.9) for $m_\infty^{\otimes N}$ and the entropy sandwich Lemma 4.3 we bound the transport cost by

$$ W_2^2(m_t^N, m_\infty^{\otimes N}) \le \frac{1}{\rho} H(m_t^N | m_\infty^{\otimes N}) \le \frac{1}{\rho} (\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_t)). $$

In the end we obtain $\mathcal{E}^N(m_T^N) \le \mathcal{E}^N(m_s^N) - \alpha \int_s^T I_t dt$ where

$$ I_t = \frac{1}{2} I(m_t^N | m_\infty^N) + \frac{1}{2} I(m_t^N | m_\infty^N) \\ \ge \frac{1}{2} \left[ 4(1-\varepsilon)\rho - \frac{M_{mm}^F}{N} \left( 16 + 6(\varepsilon^{-1}-1)\frac{M_{mm}^F}{\rho} \right) \right] (\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_\infty)) \\ + \frac{1}{2} I(m_t^N | m_\infty^N) - \frac{dM_{mm}^F}{2\rho} (10\rho + 3(\varepsilon^{-1}-1)M_{mm}^F). $$

We conclude by applying Grönwall's lemma, as in the end of the proof of Theorem 2.1. □

# 5 Short-time behaviors and propagation of chaos

Our proof of the main theorem on the uniform-in-time propagation of chaos (Theorem 2.3) relies on the exponential convergence in Theorems 2.1 and 2.2, where the initial conditions are required to have finite entropy and finite Fisher information. We aim to demonstrate in this section that the non-linear kinetic Langevin dynamics exhibits the same regularization effects in short time as the linear ones, where the contributions from the non-linearity can be controlled. We will first show the short-time Wasserstein propagation of chaos using synchronous coupling. Then we adapt the regularization results for the linear dynamics to our setting and show that for measure initial values of finite second moment, the entropy and the Fisher information are finite for the flow at every positive time, where the short-time Wasserstein propagation of chaos also plays a role. Finally we combine all the estimates obtained to derive Theorem 2.3.
## 5.1 Synchronous coupling

We first show a lemma where synchronous coupling is applied to general McKean-Vlasov diffusions. This lemma is also used to justify the approximation arguments in the proof of Theorems 2.1 and 2.2.

**Lemma 5.1.** Let $T > 0$ and $\beta, \beta' : [0, T] \times \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ be measurable and uniformly Lipschitz continuous in the last two variables and $\sigma$ be a $d \times d$ real matrix. Suppose the integral $\int_0^T (|\beta(t, \delta_0, 0)| + |\beta'(t, \delta_0, 0)|)dt$ is finite. Let $(Z_t)_{t \in [0, T]}$, $(Z'_t)_{t \in [0, T]}$ be respective solutions to

$$dZ_t = \beta(t, \text{Law}(Z_t), Z_t)dt + \sigma dW_t, \\ dZ'_t = \beta'(t, \text{Law}(Z'_t), Z'_t)dt + \sigma dW'_t,$$

where $W, W'$ are $d$-dimensional Brownians. If there exist constants $M_m, M_z$ and a progressively measurable $\delta : \Omega \times [0, T] \to \mathbb{R}$ such that for every $t \in [0, T]$, every $m, m' \in \mathcal{P}_2(\mathbb{R}^d)$ and every $x, x' \in \mathbb{R}^d$,

$$|\beta(t, m, Z_t) - \beta'(t, m', Z'_t)| \leq M_m W_2(m, m') + M_z |Z_t - Z'_t| + \delta_t \quad (5.1)$$

almost surely, then for every $t \in [0, T]$,

$$W_2^2(\text{Law}(Z_t), \text{Law}(Z'_t)) \leq e^{(2M_m+2M_z)t+1} W_2^2(\text{Law}(Z_0), \text{Law}(Z'_0)) \\ + e^2t \int_0^t e^{(2M_m+2M_z)(t-s)} \mathbb{E}[\delta_s^2] ds.$$

**Proof.** From the uniformly Lipschitz continuity of $b$ and $b'$ we have the uniqueness in law and the existence of strong solution for both diffusions. So we can construct $(Z_t, Z'_t)_{t \in [0, T]}$ such that they share the same Brownian motion and satisfy

$$\mathbb{E}[|Z_0 - Z'_0|^2] = W_2^2(\text{Law}(Z_0), \text{Law}(Z'_0)).$$

Consequently,

$$d(Z_s - Z'_s) = [b(s, \text{Law}(Z_t), Z_s) - b'(s, \text{Law}(Z'_s), Z'_s)]dt$$

and by Itô's formula,

$$d|Z_s - Z'_s|^2 = 2(Z_s - Z'_s) \cdot [\beta(s, \text{Law}(Z_s), Z_s) - \beta'(s, \text{Law}(Z'_s), Z'_s)] ds.$$

By (5.1) we have

$$|\beta(s, \text{Law}(Z_s), Z_s) - \beta'(s, \text{Law}(Z_s), Z_s)| \leq M_m W_2(\text{Law}(Z_s), \text{Law}(Z'_s)) \\ + M_z |Z_s - Z'_s| + \delta_s \leq M_m \mathbb{E}[|Z_s - Z'_s|^2]^{1/2} + M_z |Z_s - Z'_s| + \delta_s.$$

Hence

$$\frac{1}{2} d|Z_s - Z'_s|^2 \leq M_z |Z_s - Z'_s|^2 + M_m \mathbb{E}[|Z_s - Z'_s|^2]^{1/2} |Z_s - Z'_s| + |Z_s - Z'_s| \delta_s.$$

By Cauchy-Schwarz,

$$d|Z_s - Z'_s|^2 \leq (2M_z + M_m + t^{-1})|Z_s - Z'_s|^2 + M_m \mathbb{E}[|Z_s - Z'_s|^2] + t\delta_s^2.$$

Taking expectations on both sides and applying Grönwall's lemma, we obtain

$$\mathbb{E}[|Z_t - Z'_t|^2] = e^{2(M_z+M_m)t+1} \mathbb{E}[|Z_0 - Z'_0|^2] + t \int_0^t e^{2(M_z+M_m+t^{-1})(t-s)} \mathbb{E}[\delta_s^2] ds \\ \leq e^{2(M_z+M_m)t+1} \mathbb{E}[|Z_0 - Z'_0|^2] + e^2t \int_0^t e^{2(M_z+M_m)(t-s)} \mathbb{E}[\delta_s^2] ds,$$

from which the desired inequality follows. □
## 5.1 Synchronous coupling

We first show a lemma where synchronous coupling is applied to general McKean-Vlasov diffusions. This lemma is also used to justify the approximation arguments in the proof of Theorems 2.1 and 2.2.

**Lemma 5.1.** Let $T > 0$ and $\beta, \beta' : [0, T] \times \mathcal{P}_2(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d$ be measurable and uniformly Lipschitz continuous in the last two variables and $\sigma$ be a $d \times d$ real matrix. Suppose the integral $\int_0^T (|\beta(t, \delta_0, 0)| + |\beta'(t, \delta_0, 0)|)dt$ is finite. Let $(Z_t)_{t \in [0, T]}$, $(Z'_t)_{t \in [0, T]}$ be respective solutions to

$$dZ_t = \beta(t, \text{Law}(Z_t), Z_t)dt + \sigma dW_t, \\ dZ'_t = \beta'(t, \text{Law}(Z'_t), Z'_t)dt + \sigma dW'_t,$$

where $W, W'$ are $d$-dimensional Brownians. If there exist constants $M_m, M_z$ and a progressively measurable $\delta : \Omega \times [0, T] \to \mathbb{R}$ such that for every $t \in [0, T]$, every $m, m' \in \mathcal{P}_2(\mathbb{R}^d)$ and every $x, x' \in \mathbb{R}^d$,

$$|\beta(t, m, Z_t) - \beta'(t, m', Z'_t)| \le M_m W_2(m, m') + M_z |Z_t - Z'_t| + \delta_t \quad (5.1)$$

almost surely, then for every $t \in [0, T]$,

$$W_2^2(\text{Law}(Z_t), \text{Law}(Z'_t)) \le e^{(2M_m+2M_z)t+1} W_2^2(\text{Law}(Z_0), \text{Law}(Z'_0)) \\ + e^2t \int_0^t e^{(2M_m+2M_z)(t-s)} \mathbb{E}[\delta_s^2] ds.$$

**Proof.** From the uniformly Lipschitz continuity of $b$ and $b'$ we have the uniqueness in law and the existence of strong solution for both diffusions. So we can construct $(Z_t, Z'_t)_{t \in [0, T]}$ such that they share the same Brownian motion and satisfy

$$\mathbb{E}[|Z_0 - Z'_0|^2] = W_2^2(\text{Law}(Z_0), \text{Law}(Z'_0)).$$

Consequently,

$$d(Z_s - Z'_s) = [b(s, \text{Law}(Z_t), Z_s) - b'(s, \text{Law}(Z'_s), Z'_s)]dt$$

and by Itō's formula,

$$d|Z_s - Z'_s|^2 = 2(Z_s - Z'_s) \cdot [\beta(s, \text{Law}(Z_s), Z_s) - \beta'(s, \text{Law}(Z'_s), Z'_s)] ds.$$

By (5.1) we have

$$|\beta(s, \text{Law}(Z_s), Z_s) - \beta'(s, \text{Law}(Z_s), Z_s)| \le M_m W_2(\text{Law}(Z_s), \text{Law}(Z'_s)) \\ + M_z |Z_s - Z'_s| + \delta_s \le M_m \mathbb{E}[|Z_s - Z'_s|^2]^{1/2} + M_z |Z_s - Z'_s| + \delta_s.$$

Hence

$$\frac{1}{2} d|Z_s - Z'_s|^2 \le M_z |Z_s - Z'_s|^2 + M_m \mathbb{E}[|Z_s - Z'_s|^2]^{1/2} |Z_s - Z'_s| + |Z_s - Z'_s| \delta_s.$$

By Cauchy-Schwarz,

$$d|Z_s - Z'_s|^2 \le (2M_z + M_m + t^{-1})|Z_s - Z'_s|^2 + M_m \mathbb{E}[|Z_s - Z'_s|^2] + t\delta_s^2.$$

Taking expectations on both sides and applying Grönwall's lemma, we obtain

$$\mathbb{E}[|Z_t - Z'_t|^2] = e^{2(M_z+M_m)t+1} \mathbb{E}[|Z_0 - Z'_0|^2] + t \int_0^t e^{2(M_z+M_m+t^{-1})(t-s)} \mathbb{E}[\delta_s^2] ds \\ \le e^{2(M_z+M_m)t+1} \mathbb{E}[|Z_0 - Z'_0|^2] + e^2t \int_0^t e^{2(M_z+M_m)(t-s)} \mathbb{E}[\delta_s^2] ds,$$

from which the desired inequality follows. □
Since the finite-time propagation of chaos does not depend on the gradient structure of the diffusions, we introduce a more general setting. Let $b: \mathcal{P}_2(\mathbb{R}^{2d}) \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ be a mapping that is Lipschitz in space and velocity: there exist positive constants $M_x^b, M_v^b$ such that

$$ \forall x, x', v, v' \in \mathbb{R}^d, \forall m \in \mathcal{P}_2(\mathbb{R}^{2d}), \\ |b(m, x, v) - b(m, x', v')| \le M_x^b|x - x'| + M_v^b|v - v'|. \quad (5.2) $$

We suppose also that the functional derivatives $\frac{\delta b}{\delta m}$, $\frac{\delta^2 b}{\delta m^2}$ exist with the following bounds: there exist positive constants $M_m^b, M_m^b$ such that

$$ \forall m \in \mathcal{P}_2(\mathbb{R}^{2d}), \forall z, z' \in \mathbb{R}^{2d}, \quad |D_m b(m, z, z')|_{\text{op}} \le M_m^b, \quad (5.3) $$

and

$$ \forall m, m' \in \mathcal{P}_2(\mathbb{R}^{2d}), \forall z \in \mathbb{R}^{2d}, \\ \left|\iint \left[ \frac{\delta^2 b}{\delta m^2}(m', z, z', z') - \frac{\delta^2 b}{\delta m^2}(m', z, z', z'') \right] m(dz')m(dz'')\right| \le M_{mm}^b. \quad (5.4) $$

We consider the following mean field dynamics:

$$ \begin{aligned} dX_t &= V_t dt, \\ dV_t &= b(\text{Law}(X_t, V_t), X_t, V_t) dt + \sqrt{2} dW_t, \end{aligned} \quad (5.5) $$

and the corresponding particle system:

$$ \begin{aligned} dX_t^i &= V_t^i dt, \\ dV_t^i &= b(\mu(\mathbf{x}_t, \mathbf{v}_t), X_t^i, V_t^i) dt + \sqrt{2} dW_t^i, \quad \text{where } \mu(\mathbf{x}_t, \mathbf{v}_t) = \frac{1}{N} \sum_{i=1}^{N} \delta_{(X_t^i, V_t^i)}, \end{aligned} \quad (5.6) $$

and $i = 1, \dots, N$. In both equations $W_t, W_t^i$ are standard Brownians and $(W_t^i)_{i=1}^N$ are independent from each other. The dynamics (5.5) and (5.6) are well defined globally in time thanks to the Lipschitz continuity (5.2) and we denote by $P$ and $P^N$ the respective associated semigroups. That is to say, if $(X_t, V_t)$ solves (5.5) and $\text{Law}(X_0, V_0) = \mu$, then $P_t^*\mu := \text{Law}(X_t, V_t)$ and $(P_t f)(\mu) := \langle f, (P_t)^*\mu \rangle$ for bounded measurable $f: \mathbb{R}^{2d} \to \mathbb{R}$; if $(X_t^i, V_t^i)_{i=1}^N$ solves (5.6) and $\text{Law}(\mathbf{X}_0, \mathbf{V}_0) = \mu^N$, then $(P_t^N)^*\mu^N := \text{Law}(\mathbf{X}_t, \mathbf{V}_t)$ and $(P_t^N f^N)(\mu^N) := \langle f^N, (P_t^N)^*\mu^N \rangle$ for bounded measurable $f^N: \mathbb{R}^{2dN} \to \mathbb{R}$. We also define the tensor product of the mean field semigroup: $(P_t^{\otimes N} f^N)(\mu) := \langle f^N, (P_t^*\mu)^{\otimes N} \rangle$.

Using the previous Lemma 5.1 as a building block, we now show the finite-time propagation of chaos result.

**Proposition 5.2 (Finite-time propagation of chaos).** Assume $b$ satisfies (5.2) to (5.4) and let $N \ge 2$. Then there exist a positive constant $C$ depending on $M_x^b, M_v^b, M_m^b$ and $M_{mm}^b$ such that for every $m \in \mathcal{P}_2(\mathbb{R}^{2d})$ and $m^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, and every $T \ge 0$,

$$ W_2^2 \left( (P_T^N)^* m^N, (P_T^* m)^{\otimes N} \right) \le C e^{CT} W_2^2(m^N, m^{\otimes N}) + C(e^{CT} - 1)(\text{Var } m_0 + d). \quad (5.7) $$

*Proof.* We apply Lemma 5.1 with

$$ \begin{aligned} \beta^i(t, m^N, \mathbf{z}) &= (v^i, b(\mu(\mathbf{x}_t, \mathbf{v}_t), x^i, v^i))^\top, \\ \beta'^i(t, m^N, \mathbf{z}) &= (v^i, b(P_t^* m, x^i, v^i))^\top, \\ \delta_t^2 &:= \sum_{i=1}^N |\delta_t^i|^2 := \sum_{i=1}^N |b(\mu(\mathbf{x}_t, \mathbf{v}_t), X_t^i, V_t^i) - b(P_t^* m, X_t^i, V_t^i)|^2, \end{aligned} $$