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 $T \ge 0$,

$$ W_2^2((P_T^N)^* m^N, (P_T^* m)^{\otimes N}) \le Ce^{CT}W_2^2(m^N, m^{\otimes N}) + C(e^{CT} - 1)(\text{Var } m_0 + d). \quad (5.7) $$

*Proof.* We apply Lemma 5.1 with

$$ \begin{aligned} \beta^i(t, m^N, \mathbf{z}) &= (v^i, b(\mu(\mathbf{x}_t, \mathbf{v}_t), x^i, v^i))^\mathrm{T}, \\ \beta'^i(t, m^N, \mathbf{z}) &= (v^i, b(P_t^* m, x^i, v^i))^\mathrm{T}, \\ \delta_t^2 &:= \sum_{i=1}^N |\delta_t^i|^2 := \sum_{i=1}^N |b(\mu(\mathbf{x}_t, \mathbf{v}_t), X_t^i, V_t^i) - b(P_t^* m, X_t^i, V_t^i)|^2, \end{aligned} $$
and $M_z := \sqrt{2}M_x^b \lor \sqrt{2(M_v^b)^2 + 1}$, $M_m := 0$. We then obtain

$$W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}) \le e^{2M_z t+1} W_2^2(m_0^N, m_0^{\otimes N}) + e^{2t} \int_0^t e^{2M_z(t-s)} \mathbb{E}[\delta_t^2] ds.$$

So it remains to bound $\mathbb{E}[\delta_t^2]$. By enlarging the underlying probability space, we construct the random variable $\tilde{Z}_t = (\tilde{\mathbf{X}}'_t, \tilde{\mathbf{V}}'_t) \sim (P_t^* m)^{\otimes N}$ such that

$$\sum_{i=1}^{N} \mathbb{E} [ |Z_t^i - \tilde{Z}_t^i|^2 ] = W_2^2 ((P_t^N)^* m^N, (P_t^* m)^{\otimes N}).$$

This implies in particular

$$\mathbb{E}[W_2^2(\mu(\mathbf{X}_t, \mathbf{V}_t), \mu(\tilde{\mathbf{X}}'_t, \tilde{\mathbf{V}}'_t))] \le \frac{1}{N} W_2^2(m_t^N, m_t^{\otimes N}). \quad (5.8)$$

For each $i$, we decompose

$$\delta_t^i = (b(P_t^* m, Z_t^i) - b(\mu_{\tilde{Z}_t^{-i}}, Z_t^i)) + (b(\mu_{\tilde{Z}_t^{-i}}, Z_t^i) - b(\mu_{\tilde{Z}_t}, Z_t^i)) \\ + (b(\mu_{\tilde{Z}_t}, Z_t^i) - b(\mu_{\mathbf{Z}_t}, Z_t^i)) =: (\text{I}) + (\text{II}) + (\text{III}).$$

According to the assumption (2.4) we can apply Lemma B.1 to the first term and obtain

$$\mathbb{E}[(\mathbf{I})^2] = \mathbb{E}[\mathbb{E}[(\mathbf{I})^2 | Z_t^i]] \le \frac{(M_m^b)^2 \mathrm{Var} m_t}{N-1} + \frac{M_{mm}^b}{4(N-1)^2}.$$

We then bound the second term by the $M_m^b$-Lipschitz continuity:

$$\mathbb{E}[(\mathbf{II})^2] \le (M_m^b)^2 \mathbb{E}[W_2^2(\mu_{\tilde{Z}_t^{-i}}, \mu_{\tilde{Z}_t})] \\ \le \frac{(M_m^b)^2}{N(N-1)} \sum_{j:j \ne i} \mathbb{E}[|\tilde{Z}_t^j - \tilde{Z}_t^i|^2] = \frac{2(M_m^b)^2}{N} \mathrm{Var} P_t^* m.$$

Finally by (5.8), we have

$$\mathbb{E}[(\mathbf{III})^2] \le (M_m^b)^2 \mathbb{E}[W_2^2(\mu_{\mathbf{Z}_t}, \mu_{\tilde{\mathbf{Z}}'_t})] \le \frac{(M_m^b)^2}{N} W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}).$$

Hence

$$\mathbb{E}[\delta_t^2] = \sum_{i=1}^{N} \mathbb{E}[|\delta_t^i|^2] \le C[1 + \mathrm{Var} m_t + W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N})]$$

for some constant $C = C(M_m^b, M_{mm}^b)$. By Itō's formula the variance $\mathrm{Var} m_t$ satisfies

$$\frac{d}{dt} \mathrm{Var} m_t = 2(\mathbb{E}[\tilde{V}_t^2] - \mathbb{E}[\tilde{V}_t]^2) + 2\mathbb{E}[\tilde{X}_t \cdot b(P_t^* m, \tilde{Z}_t)] \\ - 2\mathbb{E}[\tilde{X}_t] \cdot \mathbb{E}[b(P_t^* m, \tilde{Z}_t)] + 2d \le C'(\mathrm{Var} m_t + d)$$

for some $C' = C'(M_z^b, M_m^b)$. Then Grönwall's lemma yields $\mathrm{Var} m_t \le e^{C't} \mathrm{Var} m_0 + (e^{C't} - 1)d$. Upon redefining the constants, we obtain for every $t \ge 0$,

$$W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}) \le e^{2M_z t+1} W_2^2(m^N, m^{\otimes N}) \\ + Ct \int_0^t e^{2M_z(t-s)} [W_2^2((P_s^N)^* m^N, (P_s^* m)^{\otimes N}) + e^{Cs}(\mathrm{Var} m_0 + d)] ds.$$

We then conclude by applying the integral version of Grönwall's lemma. $\square$

<figure><img src="image_8.png" /></figure>
and $M_z := \sqrt{2}M_x^b \vee \sqrt{2(M_v^b)^2 + 1}$, $M_m := 0$. We then obtain

$$W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}) \le e^{2M_z(t+1)} W_2^2(m_0^N, m_0^{\otimes N}) + e^{2t} \int_0^t e^{2M_z(t-s)} \mathbb{E}[\delta_t^2] ds.$$

So it remains to bound $\mathbb{E}[\delta_t^2]$. By enlarging the underlying probability space, we construct the random variable $\tilde{Z}_t = (\tilde{\mathbf{X}}'_t, \tilde{\mathbf{V}}'_t) \sim (P_t^* m)^{\otimes N}$ such that

$$\sum_{i=1}^{N} \mathbb{E} [ |Z_t^i - \tilde{Z}_t^i|^2 ] = W_2^2 ((P_t^N)^* m^N, (P_t^* m)^{\otimes N}).$$

This implies in particular

$$\mathbb{E}[W_2^2(\mu(\mathbf{X}_t, \mathbf{V}_t), \mu(\tilde{\mathbf{X}}'_t, \tilde{\mathbf{V}}'_t))] \le \frac{1}{N} W_2^2(m_t^N, m_t^{\otimes N}). \quad (5.8)$$

For each $i$, we decompose

$$\delta_t^i = (b(P_t^* m, Z_t^i) - b(\mu_{\tilde{Z}_t^{-i}}, Z_t^i)) + (b(\mu_{\tilde{Z}_t^{-i}}, Z_t^i) - b(\mu_{\tilde{Z}_t}, Z_t^i)) \\ + (b(\mu_{\tilde{Z}_t}, Z_t^i) - b(\mu_{\mathbf{Z}_t}, Z_t^i)) =: (I) + (II) + (III).$$

According to the assumption (2.4) we can apply Lemma B.1 to the first term and obtain

$$\mathbb{E}[(I)^2] = \mathbb{E}[\mathbb{E}[(I)^2|Z_t^i]] \le \frac{(M_m^b)^2 \mathrm{Var} m_t}{N-1} + \frac{M_{mm}^b}{4(N-1)^2}.$$

We then bound the second term by the $M_m^b$-Lipschitz continuity:

$$\mathbb{E}[(II)^2] \le (M_m^b)^2 \mathbb{E}[W_2^2(\mu_{\tilde{Z}_t^{-i}}, \mu_{\tilde{Z}_t})] \\ \le \frac{(M_m^b)^2}{N(N-1)} \sum_{j:j \ne i} \mathbb{E}[|\tilde{Z}_t^j - \tilde{Z}_t^i|^2] = \frac{2(M_m^b)^2}{N} \mathrm{Var} P_t^* m.$$

Finally by (5.8), we have

$$\mathbb{E}[(III)^2] \le (M_m^b)^2 \mathbb{E}[W_2^2(\mu_{\mathbf{Z}_t}, \mu_{\tilde{\mathbf{Z}}_t})] \le \frac{(M_m^b)^2}{N} W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}).$$

Hence

$$\mathbb{E}[\delta_t^2] = \sum_{i=1}^{N} \mathbb{E}[|\delta_t^i|^2] \le C[1 + \mathrm{Var} m_t + W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N})]$$

for some constant $C = C(M_m^b, M_{mm}^b)$. By Itō's formula the variance $\mathrm{Var} m_t$ satisfies

$$\frac{d}{dt} \mathrm{Var} m_t = 2(\mathbb{E}[\tilde{V}_t^2] - \mathbb{E}[\tilde{V}_t]^2) + 2 \mathbb{E}[\tilde{X}_t \cdot b(P_t^* m, \tilde{Z}_t)] \\ - 2 \mathbb{E}[\tilde{X}_t] \cdot \mathbb{E}[b(P_t^* m, \tilde{Z}_t)] + 2d \le C'(\mathrm{Var} m_t + d)$$

for some $C' = C'(M_z^b, M_m^b)$. Then Grönwall's lemma yields $\mathrm{Var} m_t \le e^{C't} \mathrm{Var} m_0 + (e^{C't} - 1)d$. Upon redefining the constants, we obtain for every $t \ge 0$,

$$W_2^2((P_t^N)^* m^N, (P_t^* m)^{\otimes N}) \le e^{2M_z(t+1)} W_2^2(m^N, m^{\otimes N}) \\ + Ct \int_0^t e^{2M_z(t-s)} [W_2^2((P_s^N)^* m^N, (P_s^* m)^{\otimes N}) + e^{Cs}(\mathrm{Var} m_0 + d)] ds.$$

We then conclude by applying the integral version of Grönwall's lemma. $\square$

<figure><img src="image_8.png" /></figure>
## 5.2 From Wasserstein metric to entropy

We study in this section a logarithmic Harnack's inequality for kinetic McKean-Vlasov dynamics and the corresponding particle system. This inequality then implies the regularization from Wasserstein to entropy.

**Lemma 5.3** (Log-Harnack inequality for 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 $m^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, every measurable function $f^N: \mathbb{R}^{2dN} \to (0, +\infty)$ that is lower bounded away from 0 and upper bounded, and every $T > 0$,

$$ (P_T^N \log f^N)(m^N) \le \log(P_T^{\otimes N} f^N)(m) + C \left( \frac{1}{(T \wedge 1)^3} + e^{CT} \right) W_2^2(m^N, m^{\otimes N}) \\ + C(e^{CT} - 1)(\mathrm{Var} m + d). \quad (5.9) $$

Consequently,

$$ H((P_T^N)^* m^N | (P_T^* m)^{\otimes N}) \le C \left( \frac{1}{(T \wedge 1)^3} + e^{CT} \right) W_2^2(m^N, m^{\otimes N}) \\ + C(e^{CT} - 1)(\mathrm{Var} m + d). \quad (5.10) $$

**Proof.** Let us first prove the log-Harnack inequality (5.9) for compactly supported $m$ and $m^N$.

**Constructing a bridge.** Fix $T > 0$ and let $(\tilde{X}_t^i, \tilde{V}_t^i)_{i=1}^N$ be $N$ independent duplicates of the solution to (5.5) with the initial condition $\mathrm{Law}(\tilde{X}_0^i, \tilde{V}_0^i) = m$ for $i = 1, \dots, N$. We denote the $N$-independent Brownians by $\tilde{W}_t^i$. By enlarging the underlying probability space, we construct random variables $X_0, V_0$ such that

$$ \sum_{i=1}^{N} \mathbb{E} \left[ |X_0^i - \tilde{X}_0^i|^2 + |V_0^i - \tilde{V}_0^i|^2 \right] = W_2^2(m^N, m^{\otimes N}). $$

Define for $i = 1, \dots, N$ the stochastic processes

$$ dX_t^i = V_t^i dt, \quad (5.11) $$

$$ dV_t^i = \left( b(P_t^* m, \tilde{X}_t^i, \tilde{V}_t^i) - \frac{V_0^i - \tilde{V}_0^i}{T} + \frac{d}{dt}(t(T-t))v_t^i \right) dt + \sqrt{2d} \tilde{W}_t^i, \quad (5.12) $$

where

$$ v^i := \frac{6}{T^3} \left( -(X_0^i - \tilde{X}_0^i) + \frac{T}{2} (V_0^i - \tilde{V}_0^i) \right). \quad (5.13) $$

The difference processes $(X_t^i - \tilde{X}_t^i, V_t^i - \tilde{V}_t^i)$ satisfy

$$ d(V_t^i - \tilde{V}_t^i) = -\frac{V_0^i - \tilde{V}_0^i}{T} dt + d(t(T-t))v_t^i, $$

$$ d(X_t^i - \tilde{X}_t^i) = \frac{t-T}{T} (V_0^i - \tilde{V}_0^i) dt + t(T-t)v_t^i dt, $$

so that

$$ V_t^i - \tilde{V}_t^i = \frac{T-t}{T}(V_0^i - \tilde{V}_0^i) + \frac{6t(T-t)}{T^3} \left( -(X_0^i - \tilde{X}_0^i) + \frac{T}{2}(V_0^i - \tilde{V}_0^i) \right), \quad (5.14) $$

$$ X_t^i - \tilde{X}_t^i = -\frac{t(t-T)^2}{T}(V_0^i - \tilde{V}_0^i) + \frac{T^3 - 3Tt^2 + 2t^3}{T^3}(X_0^i - \tilde{X}_0^i). \quad (5.15) $$

In particular $X_T^i = \tilde{X}_T^i$ and $V_T^i = \tilde{V}_T^i$.
## 5.2 From Wasserstein metric to entropy

We study in this section a logarithmic Harnack's inequality for kinetic McKean-Vlasov dynamics and the corresponding particle system. This inequality then implies the regularization from Wasserstein to entropy.

**Lemma 5.3** (Log-Harnack inequality for 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 $m^N \in \mathcal{P}_2(\mathbb{R}^{2dN})$, every measurable function $f^N: \mathbb{R}^{2dN} \to (0, +\infty)$ that is lower bounded away from 0 and upper bounded, and every $T > 0$,

$$ (P_T^N \log f^N)(m^N) \le \log(P_T^{\otimes N} f^N)(m) + C \left( \frac{1}{(T \wedge 1)^3} + e^{CT} \right) W_2^2(m^N, m^{\otimes N}) \\ + C(e^{CT} - 1)(\mathrm{Var} m + d). \quad (5.9) $$

Consequently,

$$ H\left(\left(P_T^N\right)^* m^N \middle| (P_T^* m)^{\otimes N}\right) \le C \left( \frac{1}{(T \wedge 1)^3} + e^{CT} \right) W_2^2(m^N, m^{\otimes N}) \\ + C(e^{CT} - 1)(\mathrm{Var} m + d). \quad (5.10) $$

**Proof.** Let us first prove the log-Harnack inequality (5.9) for compactly supported $m$ and $m^N$.

**Constructing a bridge.** Fix $T > 0$ and let $(\tilde{X}_t^i, \tilde{V}_t^i)_{i=1}^N$ be $N$ independent duplicates of the solution to (5.5) with the initial condition $\mathrm{Law}(\tilde{X}_0^i, \tilde{V}_0^i) = m$ for $i = 1, \dots, N$. We denote the $N$-independent Brownians by $\tilde{W}_t^i$. By enlarging the underlying probability space, we construct random variables $X_0, V_0$ such that

$$ \sum_{i=1}^{N} \mathbb{E} \left[ |X_0^i - \tilde{X}_0^i|^2 + |V_0^i - \tilde{V}_0^i|^2 \right] = W_2^2(m^N, m^{\otimes N}). $$

Define for $i = 1, \dots, N$ the stochastic processes

$$ dX_t^i = V_t^i dt, \quad (5.11) $$

$$ dV_t^i = \left( b(P_t^* m, \tilde{X}_t^i, \tilde{V}_t^i) - \frac{V_0^i - \tilde{V}_0^i}{T} + \frac{d}{dt}(t(T-t))v_t^i \right) dt + \sqrt{2d} \tilde{W}_t^i, \quad (5.12) $$

where

$$ v^i := \frac{6}{T^3} \left( -(X_0^i - \tilde{X}_0^i) + \frac{T}{2} (V_0^i - \tilde{V}_0^i) \right). \quad (5.13) $$

The difference processes $(X_t^i - \tilde{X}_t^i, V_t^i - \tilde{V}_t^i)$ satisfy

$$ d(V_t^i - \tilde{V}_t^i) = - \frac{V_0^i - \tilde{V}_0^i}{T} dt + d(t(T-t))v_t^i, $$

$$ d(X_t^i - \tilde{X}_t^i) = \frac{t-T}{T} (V_0^i - \tilde{V}_0^i) dt + t(T-t) v_t^i dt, $$

so that

$$ V_t^i - \tilde{V}_t^i = \frac{T-t}{T} (V_0^i - \tilde{V}_0^i) + \frac{6t(T-t)}{T^3} \left( -(X_0^i - \tilde{X}_0^i) + \frac{T}{2} (V_0^i - \tilde{V}_0^i) \right), \quad (5.14) $$

$$ X_t^i - \tilde{X}_t^i = - \frac{t(t-T)^2}{T} (V_0^i - \tilde{V}_0^i) + \frac{T^3 - 3Tt^2 + 2t^3}{T^3} (X_0^i - \tilde{X}_0^i). \quad (5.15) $$

In particular $X_T^i = \tilde{X}_T^i$ and $V_T^i = \tilde{V}_T^i$.
*Change of measure. Define*

$$ \xi_t^i := \frac{1}{\sqrt{2}} \left( b(P_t^* m, \tilde{X}_t^i, \tilde{V}_t^i) - b(\mu(\mathbf{X}_t, \mathbf{V}_t), X_t^i, V_t^i) + \frac{\tilde{V}_0^i - V_0^i}{T} + \frac{d}{dt} (t(T-t)) v_t^i \right) $$

and $\delta b_t^i := b(P_t^* m, Z_t^i) - b(\mu Z_t, Z_t^i)$. It satisfies

$$ |\xi_t^i| \le C |\delta b_t^i| + C \left( M_x^b + \frac{M_v^b}{T} + \frac{1}{T^2} \right) \left( |X_0^i - \tilde{X}_0^i| + T |V_0^i - \tilde{V}_0^i| \right) \quad (5.16) $$

for some universal constant C. In the following C may change from line to line and depend on the constants $M_x^b$, $M_v^b$, $M_m^b$ and $M_{mm}^b$. Set $W_t^i := \tilde{W}_t^i + \int_0^t \xi_t^i dt$ and

$$ R. := \exp \left[ - \sum_{i=1}^{N} \left( \int_{0}^{T} \xi_{t}^{i} d\tilde{W}_{t}^{i} + \frac{1}{2} \int_{0}^{T} |\xi_{s}^{i}|^{2} dt \right) \right] $$

which is a local martingale. Then $(X^i, V^i, W^i)$ solves (5.6). Since $m, m^N$ are both compactly supported, $|X_0^i - \tilde{X}_0^i|, |V_0^i - \tilde{V}_0^i|$ are bounded almost surely. The difference in drift $\delta b_t^i$ has uniform linear growth in $\mathbf{X}_t, \mathbf{V}_t$, and therefore uniform linear growth in $\tilde{\mathbf{X}}_t, \tilde{\mathbf{V}}_t$. We then apply Lemma C.1 in the appendix to obtain that $R$ is really a martingale. By Girsanov's theorem $W_t^i$ are independent Brownians under the new probability $\mathbb{Q} = \mathbb{RP}$. Since $\mathbf{X}_0, \mathbf{V}_0, \tilde{\mathbf{X}}_0, \tilde{\mathbf{V}}_0$ are independent from the Brownian motions we have

$$ \sum_{i=1}^{N} \mathbb{E}^{\mathbb{Q}} \left[ |X_0^i - \tilde{X}_0^i|^2 + |V_0^i - \tilde{V}_0^i|^2 \right] = W_2^2(m^N, m^{\otimes N}). $$

Hence for measurable functions $f^N : \mathbb{R}^{2dN} \to \mathbb{R}$ that are lower bounded away from 0 and upper bounded, we have

$$ \begin{aligned} (P_T^N \log f^N)(m^N) &= \mathbb{E}[R_T \log f^N(\mathbf{X}_T, \mathbf{V}_T)] \\ &\le \mathbb{E}[R_T \log R_T] + \log \mathbb{E}[f^N(\mathbf{X}_T, \mathbf{V}_T)] \\ &= \mathbb{E}[R_T \log R_T] + \log \mathbb{E}[f^N(\tilde{\mathbf{X}}_T, \tilde{\mathbf{V}}_T)] \\ &= \mathbb{E}[R_T \log R_T] + \log(P_T^{\otimes N} f^N)(m). \end{aligned} $$

So it remains to bound $\mathbb{E}[R_T \log R_T]$. We observe

$$ \begin{aligned} \mathbb{E}[R_T \log R_T] &= \mathbb{E}^{\mathbb{Q}}[\log R_T] = \frac{1}{2} \mathbb{E}^{\mathbb{Q}} \left[ \sum_{i=1}^{N} \int_{0}^{T} |\xi_{t}^{i}|^{2} dt \right] \\ &\le C T \left( M_{x}^{b} + \frac{M_{v}^{b}}{T} + \frac{1}{T^2} \right)^{2} \mathbb{E}^{\mathbb{Q}} \left[ |X_{0}^{i} - \tilde{X}_{0}^{i}|^{2} + T^{2} |V_{0}^{i} - \tilde{V}_{0}^{i}|^{2} \right] + C \mathbb{E}^{\mathbb{Q}} \left[ \int_{0}^{T} \sum_{i=1}^{N} |\delta b_{t}^{i}|^{2} dt \right] \\ &\le \frac{C(T \vee 1)^{3}}{(T \wedge 1)^{3}} W_{2}^{2}(m^{N}, m^{\otimes N}) + C \mathbb{E}^{\mathbb{Q}} \left[ \int_{0}^{T} \sum_{i=1}^{N} |\delta b_{t}^{i}|^{2} dt \right]. \end{aligned} $$

Arguing as in the proof of Proposition 5.2, we have

$$ \sum_{i=1}^{N} \mathbb{E}^{\mathbb{Q}} \left[ |\delta b_t^i|^2 \right] \le C e^{Ct} \left( W_2^2(m^N, m^{\otimes N}) + \mathrm{Var} m + d \right), $$

So the log-Harnack inequality (5.9) is proved for compactly supported $m^N$ and $m$.
*Change of measure. Define*

$$ \xi_t^i := \frac{1}{\sqrt{2}} \left( b(P_t^* m, \tilde{X}_t^i, \tilde{V}_t^i) - b(\mu(\mathbf{X}_t, \mathbf{V}_t), X_t^i, V_t^i) + \frac{\tilde{V}_0^i - V_0^i}{T} + \frac{d}{dt} (t(T-t)) v_t^i \right) $$

and $\delta b_t^i := b(P_t^* m, Z_t^i) - b(\mu_{\mathbf{Z}_t}, Z_t^i)$. It satisfies

$$ |\xi_t^i| \le C |\delta b_t^i| + C \left( M_x^b + \frac{M_v^b}{T} + \frac{1}{T^2} \right) \left( |X_0^i - \tilde{X}_0^i| + T |V_0^i - \tilde{V}_0^i| \right) \quad (5.16) $$

for some universal constant C. In the following C may change from line to line and depend on the constants $M_x^b$, $M_v^b$, $M_m^b$ and $M_{mm}^b$. Set $W_t^i := \tilde{W}_t^i + \int_0^t \xi_t^i dt$ and

$$ R. := \exp \left[ - \sum_{i=1}^{N} \left( \int_{0}^{T} \xi_{t}^{i} d\tilde{W}_{t}^{i} + \frac{1}{2} \int_{0}^{T} |\xi_{s}^{i}|^{2} dt \right) \right] $$

which is a local martingale. Then $(X^i, V^i, W^i)$ solves (5.6). Since $m, m^N$ are both compactly supported, $|X_0^i - \tilde{X}_0^i|, |V_0^i - \tilde{V}_0^i|$ are bounded almost surely. The difference in drift $\delta b_t^i$ has uniform linear growth in $\mathbf{X}_t, \mathbf{V}_t$, and therefore uniform linear growth in $\tilde{\mathbf{X}}_t, \tilde{\mathbf{V}}_t$. We then apply Lemma C.1 in the appendix to obtain that $R$ is really a martingale. By Girsanov's theorem $W_t^i$ are independent Brownians under the new probability $Q = RP$. Since $\mathbf{X}_0, \mathbf{V}_0, \tilde{\mathbf{X}}_0, \tilde{\mathbf{V}}_0$ are independent from the Brownian motions we have

$$ \sum_{i=1}^{N} \mathrm{E}^{Q} \left[ |X_{0}^{i} - \tilde{X}_{0}^{i}|^{2} + |V_{0}^{i} - \tilde{V}_{0}^{i}|^{2} \right] = W_{2}^{2}(m^{N}, m^{\otimes N}). $$

Hence for measurable functions $f^N: \mathbb{R}^{2dN} \to \mathbb{R}$ that are lower bounded away from 0 and upper bounded, we have

$$ \begin{aligned} (P_T^N \log f^N)(m^N) &= \mathrm{E}[R_T \log f^N(\mathbf{X}_T, \mathbf{V}_T)] \\ &\le \mathrm{E}[R_T \log R_T] + \log \mathrm{E}[f^N(\mathbf{X}_T, \mathbf{V}_T)] \\ &= \mathrm{E}[R_T \log R_T] + \log \mathrm{E}[f^N(\tilde{\mathbf{X}}_T, \tilde{\mathbf{V}}_T)] \\ &= \mathrm{E}[R_T \log R_T] + \log(P_T^{\otimes N} f^N)(m). \end{aligned} $$

So it remains to bound $\mathrm{E}[R_T \log R_T]$. We observe

$$ \begin{aligned} \mathrm{E}[R_T \log R_T] &= \mathrm{E}^Q[\log R_T] = \frac{1}{2} \mathrm{E}^Q \left[ \sum_{i=1}^{N} \int_{0}^{T} |\xi_t^i|^2 dt \right] \\ &\le C T \left( M_x^b + \frac{M_v^b}{T} + \frac{1}{T^2} \right)^2 \mathrm{E}^Q \left[ |X_0^i - \tilde{X}_0^i|^2 + T^2 |V_0^i - \tilde{V}_0^i|^2 \right] + C \mathrm{E}^Q \left[ \int_0^T \sum_{i=1}^N |\delta b_t^i|^2 dt \right] \\ &\le \frac{C(T \vee 1)^3}{(T \wedge 1)^3} W_2^2(m^N, m^{\otimes N}) + C \mathrm{E}^Q \left[ \int_0^T \sum_{i=1}^N |\delta b_t^i|^2 dt \right]. \end{aligned} $$

Arguing as in the proof of Proposition 5.2, we have

$$ \sum_{i=1}^{N} \mathrm{E}^{Q} \left[ |\delta b_{t}^{i}|^{2} \right] \le C e^{Ct} \left( W_{2}^{2}(m^{N}, m^{\otimes N}) + \mathrm{Var} m + d \right), $$

So the log-Harnack inequality (5.9) is proved for compactly supported $m^N$ and $m$.
**Approximation.** Now treat general $m^N$, $m$ of finite second moment, but not necessarily compact supported. Take two sequences $(m_k^N)_{k \in \mathbb{N}}$, $(m_k)_{k \in \mathbb{N}}$ of compactly supported measures such that $m_k^N \to m^N$ and $m_k \to m$ in respective topologies of $P_2$. For continuous $f^N$ such that $\log f^N$ is bounded, we have

$$ (P_t^N \log f^N)(m_k^N) \to (P_t^N \log f^N)(m^N), \quad (P_t^{\otimes N} f^N)(m_k) \to (P_t^{\otimes N} f^N)(m) $$

by the $P_2$-continuities of $(P_t^N)^*$ and $P_t^*$. So the log-Harnack inequality (5.9) is shown for every continuous $f^N$ which is both lower and upper bounded, and for general $m^N$ and $m$ of finite second moment. For a doubly bounded but not necessarily continuous $f^N$ we take a sequence of continuous and uniformly bounded $(f_k^N)_{k \in \mathbb{N}}$ that converges to $f^N$ in the $\sigma(L^\infty, L^1)$ topology. We have

$$ (P_t^N \log f_k^N)(m^N) \to (P_t^N \log f^N)(m^N), \quad (P_t^{\otimes N} f_k^N)(m) \to (P_t^{\otimes N} f^N)(m) $$

since both $(P_t^N)^* m^N$ and $P_t^* m$ are absolutely continuous with respect to the Lebesgue measure according to Lemma 4.5. So the desired inequality (5.9) is shown in full generality. Finally, to obtain (5.10) we define another sequence

$$ g_k^N := \left( \frac{(P_T^N)^* m^N}{(P_T^* m)^{\otimes N}} \wedge k \right) \vee \frac{1}{k} $$

for $k \in \mathbb{N}$. We apply the Harnack's inequality (5.9) to $g_k^N$ and take the limit $k \to +\infty$. $\square$

Using the known results on log-Harnack inequalities we can also obtain the regularization in the beginning of the dynamics.

**Proposition 5.4.** Assume $F$ satisfies (2.2) and there exist probabilities $m_\infty$, $m_\infty^N$ satisfying (2.18) and (2.19) respectively and having finite exponential moments. Let $m_0$ (resp. $m_0^N$) be the initial value of the mean field dynamics (2.12) (resp. the particle system dynamics (2.13)) of finite second moment. Then there exist a positive constant $C$ depending on $M_{mm}^F$ and $M_{mx}^F$ such that for every $t \in (0, 1]$,

$$ H(m_t|m_\infty) \le \frac{C}{t^3} W_2^2(m_0, m_\infty) \quad (\text{resp. } H(m_t^N|m_\infty^N) \le \frac{C}{t^3} W_2^2(m_0^N, m_\infty^N)). \quad (5.17) $$

**Proof.** Note that $m_t = P_t^* m_0$ and $m_t^N = (P_t^N)^* m_0^N$ where $P_t$ and $P_t^N$ are the McKean-Vlasov and the linear semigroup corresponding to the SDEs (2.10) and (2.11), respectively. We then apply the log-Harnack inequality for McKean-Vlasov diffusions [32, Proposition 5.1] and obtain

$$ H(m_t|m_\infty) \le \frac{C}{t^3} W_2^2(m_t, m_\infty) $$

for $t \in (0, 1]$. For the particle system we apply the classical log-Harnack inequality (which corresponds to the case where $M_m^b$ and $M_{mm}^b$ are both equal to 0 in our Lemma 5.3, i.e. no mean field dependence) and obtain

$$ H(m_t^N|m_\infty^N) \le \frac{C}{t^3} W_2^2(m_t^N, m_\infty^N) $$

for $t \in (0, 1]$ and it is clear from the computations in Lemma 5.3 that the constant $C$ can be chosen to depend only on $M_{mx}^F$ and $M_{mm}^F$. $\square$
**Approximation.** Now treat general $m^N$, $m$ of finite second moment, but not necessarily compact supported. Take two sequences $(m_k^N)_{k \in \mathbb{N}}$, $(m_k)_{k \in \mathbb{N}}$ of compactly supported measures such that $m_k^N \to m^N$ and $m_k \to m$ in respective topologies of $P_2$. For continuous $f^N$ such that $\log f^N$ is bounded, we have

$$ (P_t^N \log f^N)(m_k^N) \to (P_t^N \log f^N)(m^N), \quad (P_t^{\otimes N} f^N)(m_k) \to (P_t^{\otimes N} f^N)(m) $$

by the $P_2$-continuities of $(P_t^N)^*$ and $P_t^*$. So the log-Harnack inequality (5.9) is shown for every continuous $f^N$ which is both lower and upper bounded, and for general $m^N$ and $m$ of finite second moment. For a doubly bounded but not necessarily continuous $f^N$ we take a sequence of continuous and uniformly bounded $(f_k^N)_{k \in \mathbb{N}}$ that converges to $f^N$ in the $\sigma(L^\infty, L^1)$ topology. We have

$$ (P_t^N \log f_k^N)(m^N) \to (P_t^N \log f^N)(m^N), \quad (P_t^{\otimes N} f_k^N)(m) \to (P_t^{\otimes N} f^N)(m) $$

since both $(P_t^N)^* m^N$ and $P_t^* m$ are absolutely continuous with respect to the Lebesgue measure according to Lemma 4.5. So the desired inequality (5.9) is shown in full generality. Finally, to obtain (5.10) we define another sequence

$$ g_k^N := \left( \frac{(P_T^N)^* m^N}{(P_T^* m)^{\otimes N}} \wedge k \right) \vee \frac{1}{k} $$

for $k \in \mathbb{N}$. We apply the Harnack's inequality (5.9) to $g_k^N$ and take the limit $k \to +\infty$. $\square$

Using the known results on log-Harnack inequalities we can also obtain the regularization in the beginning of the dynamics.

**Proposition 5.4.** Assume $F$ satisfies (2.2) and there exist probabilities $m_\infty$, $m_\infty^N$ satisfying (2.18) and (2.19) respectively and having finite exponential moments. Let $m_0$ (resp. $m_0^N$) be the initial value of the mean field dynamics (2.12) (resp. the particle system dynamics (2.13)) of finite second moment. Then there exist a positive constant $C$ depending on $M_{mm}^F$ and $M_{mx}^F$ such that for every $t \in (0, 1]$,

$$ H(m_t|m_\infty) \le \frac{C}{t^3} W_2^2(m_0, m_\infty) \quad (\text{resp. } H(m_t^N|m_\infty^N) \le \frac{C}{t^3} W_2^2(m_0^N, m_\infty^N)). \quad (5.17) $$

**Proof.** Note that $m_t = P_t^* m_0$ and $m_t^N = (P_t^N)^* m_0^N$ where $P_t$ and $P_t^N$ are the McKean-Vlasov and the linear semigroup corresponding to the SDEs (2.10) and (2.11), respectively. We then apply the log-Harnack inequality for McKean-Vlasov diffusions [32, Proposition 5.1] and obtain

$$ H(m_t|m_\infty) \le \frac{C}{t^3} W_2^2(m_t, m_\infty) $$

for $t \in (0, 1]$. For the particle system we apply the classical log-Harnack inequality (which corresponds to the case where $M_m^b$ and $M_{mm}^b$ are both equal to 0 in our Lemma 5.3, i.e. no mean field dependence) and obtain

$$ H(m_t^N|m_\infty^N) \le \frac{C}{t^3} W_2^2(m_t^N, m_\infty^N) $$

for $t \in (0, 1]$ and it is clear from the computations in Lemma 5.3 that the constant $C$ can be chosen to depend only on $M_{mx}^F$ and $M_{mm}^F$. $\square$
## 5.3 From entropy to Fisher information

We then adapt Hérau's functional to our setting to obtain the regularization from entropy to Fisher information.

**Proposition 5.5.** Assume that $F$ satisfies (2.1) and (2.2), and that there exist probabilities $m_\infty, m_\infty^N$ satisfying (2.18) and (2.19) respectively and having finite exponential moments. Let $m_0$ (resp. $m_0^N$) be the initial value of the mean field dynamics (2.12) (resp. the particle system dynamics (2.13)) of finite second moment and finite entropy. Then there exist a positive constant $C$ depending on $M_{mm}^F$ and $M_{mx}^F$ such that for every $t \in (0, 1]$,

$$I(m_t|\hat{m}_t) \le \frac{C}{t^3}(\mathcal{F}(m_0) - \mathcal{F}(m_\infty)) \quad (\text{resp. } I(m_t^N|m_\infty^N) \le \frac{C}{t^3}H(m_0^N|m_\infty^N)). \quad (5.18)$$

*Proof.* First derive the bound for the mean field system. We suppose additionally $F$ satisfies (4.5) and $m_0/m_\infty \in A_+$ without loss of generality, as they can be removed by the approximation argument in the end of the proof of Theorem 2.1. Let $a, b, c$ be positive constants to be determined. Motivated by [40, Theorem A.18], we define Hérau's Lyapunov functional for mean field measures:

$$\mathcal{E}(t, m) = F(m) + \frac{1}{2} \int |v^2|m + H(m) + at \int |\nabla_v \log \eta|^2 m + 2bt^2 \int \nabla_x \log \eta \cdot \nabla_v \log \eta m + ct^3 \int |\nabla_x \log \eta|^2 m$$

where $\eta := m/\hat{m}$. From the argument of Theorem 2.1, we know that $\mathcal{E}(t, m_t)$ is well defined and $t \mapsto \mathcal{E}(t, m_t)$ admits derivative satisfying $\frac{d}{dt}\mathcal{E}(t, m_t) \le -Y_t^T K'_t Y_t$, where $K'_t$ is equal to

$$\begin{pmatrix} 1 - a + 2at - 2(M_{mx}^F + M_{mm}^F)bt^2 & -2bt^2 & -2at - 4bt - 2M_{mm}^F ct^3 & 0 \\ 0 & 2at & -2M_{mm}^F ct^3 & -4bt^2 \\ 0 & 0 & 2bt^2 - 3ct^2 & 0 \\ 0 & 0 & 0 & 2ct^3 \end{pmatrix}$$

and $Y_t$ is defined by (4.13). We then choose the constants $a, b, c$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and $K'_t \ge 0$ for $t \in [0, 1]$. Hence $t \mapsto \mathcal{E}(t, m_t)$ is non-increasing on $[0, 1]$ and the Fisher bound follows: for every $t \in (0, 1]$,

$$\begin{align*} I(m_t|\hat{m}_t) &\le \frac{C}{t^3}(\mathcal{E}(t, m_t) - \mathcal{F}(m_t)) \\ &\le \frac{C}{t^3}(\mathcal{E}(t, m_t) - \mathcal{F}(m_\infty)) \\ &\le \frac{C}{t^3}(\mathcal{E}(0, m_0) - \mathcal{F}(m_\infty)) \\ &= \frac{C}{t^3}(\mathcal{F}(m_0) - \mathcal{F}(m_\infty)). \end{align*}$$

Here, in the second inequality, we use $\mathcal{F}(m_t) \ge \mathcal{F}(m_\infty)$ which is a consequence of Lemma 4.2. Note that this inequality relies on the convexity of $F$.

For the particle system we suppose additionally $U^N$ satisfies (4.20) and $m_0^N/m_\infty^N$ satisfies (4.21) without loss of generality, as they can be removed by the argument in the end of the proof of Lemma 4.9. We define

$$\mathcal{E}^N(t, m^N) = F^N(m^N) + \frac{1}{2} \int |\mathbf{v}^2|m^N + H(m^N) + at \int |\nabla_{\mathbf{v}} \log h^N|^2 m^N + 2bt^2 \int \nabla_{\mathbf{x}} \log h^N \cdot \nabla_{\mathbf{v}} \log h^N m^N + ct^3 \int |\nabla_{\mathbf{x}} \log h^N|^2 m^N$$
## 5.3 From entropy to Fisher information

We then adapt Hérau's functional to our setting to obtain the regularization from entropy to Fisher information.

**Proposition 5.5.** Assume that $F$ satisfies (2.1) and (2.2), and that there exist probabilities $m_\infty, m_\infty^N$ satisfying (2.18) and (2.19) respectively and having finite exponential moments. Let $m_0$ (resp. $m_0^N$) be the initial value of the mean field dynamics (2.12) (resp. the particle system dynamics (2.13)) of finite second moment and finite entropy. Then there exist a positive constant $C$ depending on $M_{mm}^F$ and $M_{mx}^F$ such that for every $t \in (0, 1]$,

$$I(m_t|\hat{m}_t) \le \frac{C}{t^3}(\mathcal{F}(m_0) - \mathcal{F}(m_\infty)) \quad (\text{resp. } I(m_t^N|m_\infty^N) \le \frac{C}{t^3}H(m_0^N|m_\infty^N)). \quad (5.18)$$

*Proof.* First derive the bound for the mean field system. We suppose additionally $F$ satisfies (4.5) and $m_0/m_\infty \in A_+$ without loss of generality, as they can be removed by the approximation argument in the end of the proof of Theorem 2.1. Let $a, b, c$ be positive constants to be determined. Motivated by [40, Theorem A.18], we define Hérau's Lyapunov functional for mean field measures:

$$\mathcal{E}(t, m) = F(m) + \frac{1}{2} \int |v^2|m + H(m) + at \int |\nabla_v \log \eta|^2 m + 2bt^2 \int \nabla_x \log \eta \cdot \nabla_v \log \eta m + ct^3 \int |\nabla_x \log \eta|^2 m$$

where $\eta := m/\hat{m}$. From the argument of Theorem 2.1, we know that $\mathcal{E}(t, m_t)$ is well defined and $t \mapsto \mathcal{E}(t, m_t)$ admits derivative satisfying $\frac{d}{dt}\mathcal{E}(t, m_t) \le -Y_t^T K'_t Y_t$, where $K'_t$ is equal to

$$\begin{pmatrix} 1 - a + 2at - 2(M_{mx}^F + M_{mm}^F)bt^2 & -2bt^2 & -2at - 4bt - 2M_{mm}^F ct^3 & 0 \\ 0 & 2at & -2M_{mm}^F ct^3 & -4bt^2 \\ 0 & 0 & 2bt^2 - 3ct^2 & 0 \\ 0 & 0 & 0 & 2ct^3 \end{pmatrix}$$

and $Y_t$ is defined by (4.13). We then choose the constants $a, b, c$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and $K'_t \ge 0$ for $t \in [0, 1]$. Hence $t \mapsto \mathcal{E}(t, m_t)$ is non-increasing on $[0, 1]$ and the Fisher bound follows: for every $t \in (0, 1]$,

$$\begin{align*} I(m_t|\hat{m}_t) &\le \frac{C}{t^3}(\mathcal{E}(t, m_t) - \mathcal{F}(m_t)) \\ &\le \frac{C}{t^3}(\mathcal{E}(t, m_t) - \mathcal{F}(m_\infty)) \\ &\le \frac{C}{t^3}(\mathcal{E}(0, m_0) - \mathcal{F}(m_\infty)) \\ &= \frac{C}{t^3}(\mathcal{F}(m_0) - \mathcal{F}(m_\infty)). \end{align*}$$

Here, in the second inequality, we use $\mathcal{F}(m_t) \ge \mathcal{F}(m_\infty)$ which is a consequence of Lemma 4.2. Note that this inequality relies on the convexity of $F$.

For the particle system we suppose additionally $U^N$ satisfies (4.20) and $m_0^N/m_\infty^N$ satisfies (4.21) without loss of generality, as they can be removed by the argument in the end of the proof of Lemma 4.9. We define

$$\mathcal{E}^N(t, m^N) = F^N(m^N) + \frac{1}{2} \int |\mathbf{v}^2|m^N + H(m^N) + at \int |\nabla_{\mathbf{v}} \log h^N|^2 m^N + 2bt^2 \int \nabla_{\mathbf{x}} \log h^N \cdot \nabla_{\mathbf{v}} \log h^N m^N + ct^3 \int |\nabla_{\mathbf{x}} \log h^N|^2 m^N$$
where $h^N := m^N/m_\infty^N$. By the computations in Lemma 4.9, we have $\frac{d}{dt}\mathcal{E}^N(t, m_t^N) \le -(Y_t^N)^T K_t'' Y_t^N$, where $K_t''$ is equal to

$$ \left( \begin{array}{cccc} 1 - a + 2at - 2(M_{mx}^F + M_{mm}^F)bt^2 & -2bt^2 & -2at - 4bt & 0 \\ 2at & -2(M_{mx}^F + M_{mm}^F)ct^3 & -4bt^2 & 0 \\ 2bt^2 - 3ct^2 & 0 & 2ct^3 & 0 \end{array} \right) $$

and $Y_t^N$ is defined by (4.23). We choose again the constants $a, b, c$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and $t \mapsto \mathcal{H}^N(t, m_t^N)$ is non-increasing on $[0, 1]$. Hence we have for every $t \in (0, 1]$,

$$ \begin{aligned} I(m_t^N | \hat{m}_\infty^N) &\le \frac{C}{t^3} (\mathcal{E}^N(t, m_t^N) - \mathcal{F}^N(m_t^N)) \\ &\le \frac{C}{t^3} (\mathcal{E}^N(t, m_t^N) - \mathcal{F}^N(m_\infty^N)) \\ &\le \frac{C}{t^3} (\mathcal{E}^N(0, m_0^N) - \mathcal{F}^N(m_\infty^N)) \\ &= \frac{C}{t^3} (\mathcal{F}^N(m_0^N) - \mathcal{F}^N(m_\infty^N)) \\ &= \frac{C}{t^3} H(m_0^N | m_\infty^N). \end{aligned} $$

Similarly, we use the fact that $\mathcal{F}^N(m_t^N) - \mathcal{F}^N(m_\infty^N) = H(m_t^N | m_\infty^N) \ge 0$ to get the second inequality. Here the difference is that the $N$-particle system is linear and this fact does not rely on the convexity of $F$. $\square$

## 5.4 Propagation of chaos

Using all the regularization results proved in Sections 5.2 and 5.3 we can finally give the proof of the main theorem.

*Proof of Theorem 2.3.* Let $m_0$ and $m_0^N$ be the respective initial values for the dynamics (2.12) and (2.13) and suppose they have finite second moment. The first claim of the theorem (2.22) can be written as two bounds on $W_2^2(m_t^N, m_t^{\otimes N})$, the first of which follows directly from the finite-time bound in Proposition 5.2. The second claim (2.23) is nothing but Lemma 5.3. It remains to find some $C_2, \kappa$ depending only on $\rho^x, M_{mx}^F, M_{mm}^F$ and prove

$$ W_2^2(m_t^N, m_t^{\otimes N}) \le \frac{C_2 N}{(t \wedge 1)^6} W_2^2(m_0, m_\infty) e^{-\kappa t} \\ + \frac{C_2}{(t \wedge 1)^6} W_2^2(m_0^N, m_\infty^{\otimes N}) e^{-(\kappa - C_2/N)t} + \frac{C_2 d}{\kappa - C_2/N} \quad (5.19) $$

for $t > 0$. Set $t_1 = \frac{t \wedge 1}{2}$ and $t_2 = t \wedge 1$. By the Wasserstein to entropy regularization result in Proposition 5.4, we can find a constant $C$ depending on $M_{mx}^F$ and $M_{mm}^F$ such that

$$ H(m_{t_1} | m_\infty) \le \frac{C}{t_1^3} W_2^2(m_0, m_\infty) \quad \text{and} \quad H(m_{t_1}^N | m_\infty^N) \le \frac{C}{t_1^3} W_2^2(m_0^N, m_\infty^N). $$

In the following $C$ may change from line to line and may depend additionally on the LSI constant $\rho$. Applying the regularization in Proposition 5.5 to the dynamics with $m_{t_1}$ and $m_{t_1}^N$ as respective initial values and noting that $t_2 - t_1 \le 1$ by definition, we obtain

$$ I(m_{t_2} | \hat{m}_{t_2}) \le \frac{C}{(t_2 - t_1)^3} (\mathcal{F}(m_{t_1}) - \mathcal{F}(m_\infty)), $$

$$ I(m_{t_2}^N | \hat{m}_{t_2}^N) \le \frac{C}{(t_2 - t_1)^3} H(m_{t_1}^N | m_\infty^N), $$
where $h^N := m^N/m_\infty^N$. By the computations in Lemma 4.9, we have $\frac{d}{dt}\mathcal{E}^N(t, m_t^N) \le -(Y_t^N)^T K_t'' Y_t^N$, where $K_t''$ is equal to

$$ \begin{pmatrix} 1 - a + 2at - 2(M_{mx}^F + M_{mm}^F)bt^2 & -2bt^2 & -2at - 4bt & 0 \\ 2at & -2(M_{mx}^F + M_{mm}^F)ct^3 & -4bt^2 & 0 \\ 2bt^2 - 3ct^2 & 0 & 2ct^3 & 0 \end{pmatrix} $$

and $Y_t^N$ is defined by (4.23). We choose again the constants $a, b, c$ depending only on $M_{mx}^F$ and $M_{mm}^F$ such that $ac > b^2$ and $t \mapsto \mathcal{H}^N(t, m_t^N)$ is non-increasing on $[0, 1]$. Hence we have for every $t \in (0, 1]$,

$$ \begin{aligned} I(m_t^N | \hat{m}_\infty^N) &\le \frac{C}{t^3} (\mathcal{E}^N(t, m_t^N) - \mathcal{F}^N(m_t^N)) \\ &\le \frac{C}{t^3} (\mathcal{E}^N(t, m_t^N) - \mathcal{F}^N(m_\infty^N)) \\ &\le \frac{C}{t^3} (\mathcal{E}^N(0, m_0^N) - \mathcal{F}^N(m_\infty^N)) \\ &= \frac{C}{t^3} (\mathcal{F}^N(m_0^N) - \mathcal{F}^N(m_\infty^N)) \\ &= \frac{C}{t^3} H(m_0^N | m_\infty^N). \end{aligned} $$

Similarly, we use the fact that $\mathcal{F}^N(m_t^N) - \mathcal{F}^N(m_\infty^N) = H(m_t^N | m_\infty^N) \ge 0$ to get the second inequality. Here the difference is that the $N$-particle system is linear and this fact does not rely on the convexity of $F$. $\square$

## 5.4 Propagation of chaos

Using all the regularization results proved in Sections 5.2 and 5.3 we can finally give the proof of the main theorem.

**Proof of Theorem 2.3.** Let $m_0$ and $m_0^N$ be the respective initial values for the dynamics (2.12) and (2.13) and suppose they have finite second moment. The first claim of the theorem (2.22) can be written as two bounds on $W_2^2(m_t^N, m_t^{\otimes N})$, the first of which follows directly from the finite-time bound in Proposition 5.2. The second claim (2.23) is nothing but Lemma 5.3. It remains to find some $C_2, \kappa$ depending only on $\rho^x, M_{mx}^F, M_{mm}^F$ and prove

$$ W_2^2(m_t^N, m_t^{\otimes N}) \le \frac{C_2 N}{(t \wedge 1)^6} W_2^2(m_0, m_\infty) e^{-\kappa t} \\ + \frac{C_2}{(t \wedge 1)^6} W_2^2(m_0^N, m_\infty^{\otimes N}) e^{-(\kappa - C_2/N)t} + \frac{C_2 d}{\kappa - C_2/N} \quad (5.19) $$

for $t > 0$. Set $t_1 = \frac{t \wedge 1}{2}$ and $t_2 = t \wedge 1$. By the Wasserstein to entropy regularization result in Proposition 5.4, we can find a constant $C$ depending on $M_{mx}^F$ and $M_{mm}^F$ such that

$$ H(m_{t_1} | m_\infty) \le \frac{C}{t_1^3} W_2^2(m_0, m_\infty) \quad \text{and} \quad H(m_{t_1}^N | m_\infty^N) \le \frac{C}{t_1^3} W_2^2(m_0^N, m_\infty^N). $$

In the following $C$ may change from line to line and may depend additionally on the LSI constant $\rho$. Applying the regularization in Proposition 5.5 to the dynamics with $m_{t_1}$ and $m_{t_1}^N$ as respective initial values and noting that $t_2 - t_1 \le 1$ by definition, we obtain

$$ I(m_{t_2} | \hat{m}_{t_2}) \le \frac{C}{(t_2 - t_1)^3} (\mathcal{F}(m_{t_1}) - \mathcal{F}(m_\infty)), $$

$$ I(m_{t_2}^N | \hat{m}_{t_2}^N) \le \frac{C}{(t_2 - t_1)^3} H(m_{t_1}^N | m_\infty^N), $$
whereas $\mathcal{F}(m_{t_1}) - \mathcal{F}(m_{\infty})$ is bounded by the entropy sandwich in Lemma 4.2:

$$ \mathcal{F}(m_{t_1}) - \mathcal{F}(m_{\infty}) \leq C H(m_{t_1} | m_{\infty}). $$

Consequently, both the measures $m_{t_2}$ and $m_{t_2}^N$ have finite entropy and finite Fisher information, and we can apply respectively Theorems 2.1 and 2.2 to the dynamics with initial values $m_{t_2}$ and $m_{t_2}^N$. We then obtain

$$ \mathcal{F}(m_t) - \mathcal{F}(m_{\infty}) \leq \frac{C}{t_1^6} W_2^2(m_0, m_{\infty}) e^{-\kappa(t-t_2)} $$

and

$$ \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_{\infty}) \leq \frac{C}{t_1^6} W_2^2(m_0^N, m_{\infty}^{\otimes N}) e^{-(\kappa - C/N)(t-t_2)} + \frac{Cd}{\kappa - C/N}. $$

Using consecutively the triangle inequality, Talagrand's inequality (2.9) for $m_{\infty}^{\otimes N}$ and the entropy inequalities in Lemmas 4.2 and 4.3, we have

$$ \begin{aligned} W_2^2(m_t^N, m_t^{\otimes N}) &\leq 2W_2^2(m_t^N, m_{\infty}^{\otimes N}) + 2N W_2^2(m_t, m_{\infty}) \\ &\leq \frac{2}{\rho} (H(m_t^N | m_{\infty}^{\otimes N}) + NH(m_t | m_{\infty})) \\ &\leq \frac{2}{\rho} (\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_{\infty}) + N(\mathcal{F}(m_t) - \mathcal{F}(m_{\infty}))) \end{aligned} $$

So the inequality (5.19) is proved by combining the above three inequalities. □

## A Lower-semicontinuities

**Lemma A.1.** The entropy $H(\cdot)$ and the Fisher information $I(\cdot)$ are lower-semicontinuous with respect to the weak topology of $P_2$. Consequently, under the assumption (2.2), if $(m_n)_{n \in \mathbb{N}}$ is a sequence converging to $m_*$ in $P_2(\mathbb{R}^{2d})$, then

$$ \liminf_{n \to +\infty} H(m_n | \hat{m}_n) \geq H(m_* | \hat{m}_*) \quad \text{and} \quad \liminf_{n \to +\infty} I(m_n | \hat{m}_n) \geq I(m_* | \hat{m}_*). $$

*Proof.* The lower semicontinuity of $m \mapsto H(m)$ is classical. We show the lower semi-continuity of the Fisher information. Let $(m_n)_{n \in \mathbb{N}}$ be a sequence converging to $m_*$ in $P_2(\mathbb{R}^d)$. Without loss of generality, we suppose $I(m_n) \leq M^2$ for every $n \in \mathbb{N}$. This implies in particular $\|\nabla m_n\|_{L^1} \leq M$ by Cauchy-Schwarz. For every function $\varphi$ belonging to $C_c^\infty(\mathbb{R}^d)$, we have

$$ |\langle \nabla \varphi, m_* \rangle| = \lim_{n \to +\infty} |\langle \nabla \varphi, m_n \rangle| \leq M \|\varphi\|_{\infty}. $$

Hence $\|\nabla m_*\|_{TV} \leq M$ as well. Moreover, for every $f \in C_c(\mathbb{R}^d)$ and every $\varepsilon > 0$, we can find $\varphi \in C_c^\infty(\mathbb{R}^d)$ such that $\|f - \varphi\|_{\infty} < \frac{\varepsilon}{4M}$ and $n \in \mathbb{N}$ such that $|\langle \nabla \varphi, m_n - m_* \rangle| < \frac{\varepsilon}{2}$. Then,

$$ \begin{aligned} |\langle f, \nabla(m_n - m_*) \rangle| &\leq |\langle f - \varphi, \nabla m_n \rangle| + |\langle f - \varphi, \nabla m_* \rangle| + |\langle \nabla \varphi, m_n - m_* \rangle| \\ &< 2 \cdot \frac{\varepsilon}{4M} \cdot M + \frac{\varepsilon}{2} = \varepsilon. \end{aligned} $$

Equivalently, the sequence of $\mathbb{R}^d$-valued Radon measures $(\nabla m_n)_{n \in \mathbb{N}}$ converges to $\nabla m$ locally weakly. We then apply [1, Theorem 2.34] to obtain

$$ \liminf_{n \to +\infty} I(m_n) \geq I(m_*). $$

Finally, the lower semicontinuity of $m \mapsto H(m|\hat{m})$ (resp. $m \mapsto I(m|\hat{m})$) follows from the lower semicontinuity of $m \mapsto H(m)$ (resp. $m \mapsto I(m)$) and the locally uniform quadratic growth of $x \mapsto \frac{\delta F}{\delta m}(m, x)$ (resp. the locally uniform linear growth of $x \mapsto D_m F(m, x)$). □
whereas $\mathcal{F}(m_{t_1}) - \mathcal{F}(m_{\infty})$ is bounded by the entropy sandwich in Lemma 4.2:

$$ \mathcal{F}(m_{t_1}) - \mathcal{F}(m_{\infty}) \leq CH(m_{t_1}|m_{\infty}). $$

Consequently, both the measures $m_{t_2}$ and $m_{t_2}^N$ have finite entropy and finite Fisher information, and we can apply respectively Theorems 2.1 and 2.2 to the dynamics with initial values $m_{t_2}$ and $m_{t_2}^N$. We then obtain

$$ \mathcal{F}(m_t) - \mathcal{F}(m_{\infty}) \leq \frac{C}{t_1^6} W_2^2(m_0, m_{\infty})e^{-\kappa(t-t_2)} $$

and

$$ \mathcal{F}^N(m_t^N) - N\mathcal{F}(m_{\infty}) \leq \frac{C}{t_1^6} W_2^2(m_0^N, m_{\infty}^{\otimes N})e^{-(\kappa-C/N)(t-t_2)} + \frac{Cd}{\kappa-C/N}. $$

Using consecutively the triangle inequality, Talagrand's inequality (2.9) for $m_{\infty}^{\otimes N}$ and the entropy inequalities in Lemmas 4.2 and 4.3, we have

$$ \begin{aligned} W_2^2(m_t^N, m_t^{\otimes N}) &\leq 2W_2^2(m_t^N, m_{\infty}^{\otimes N}) + 2NW_2^2(m_t, m_{\infty}) \\ &\leq \frac{2}{\rho}(H(m_t^N|m_{\infty}^{\otimes N}) + NH(m_t|m_{\infty})) \\ &\leq \frac{2}{\rho}(\mathcal{F}^N(m_t^N) - N\mathcal{F}(m_{\infty}) + N(\mathcal{F}(m_t) - \mathcal{F}(m_{\infty}))) \end{aligned} $$

So the inequality (5.19) is proved by combining the above three inequalities. □

## A Lower-semicontinuities

**Lemma A.1.** The entropy $H(\cdot)$ and the Fisher information $I(\cdot)$ are lower-semicontinuous with respect to the weak topology of $\mathcal{P}_2$. Consequently, under the assumption (2.2), if $(m_n)_{n \in \mathbb{N}}$ is a sequence converging to $m_*$ in $\mathcal{P}_2(\mathbb{R}^{2d})$, then

$$ \liminf_{n \to +\infty} H(m_n|\hat{m}_n) \geq H(m_*|\hat{m}_*) \quad \text{and} \quad \liminf_{n \to +\infty} I(m_n|\hat{m}_n) \geq I(m_*|\hat{m}_*). $$

*Proof.* The lower semicontinuity of $m \mapsto H(m)$ is classical. We show the lower semi-continuity of the Fisher information. Let $(m_n)_{n \in \mathbb{N}}$ be a sequence converging to $m_*$ in $\mathcal{P}_2(\mathbb{R}^d)$. Without loss of generality, we suppose $I(m_n) \leq M^2$ for every $n \in \mathbb{N}$. This implies in particular $\|\nabla m_n\|_{L^1} \leq M$ by Cauchy-Schwarz. For every function $\varphi$ belonging to $C_c^\infty(\mathbb{R}^d)$, we have

$$ |\langle \nabla \varphi, m_* \rangle| = \lim_{n \to +\infty} |\langle \nabla \varphi, m_n \rangle| \leq M \|\varphi\|_{\infty}. $$

Hence $\|\nabla m_*\|_{TV} \leq M$ as well. Moreover, for every $f \in C_c(\mathbb{R}^d)$ and every $\varepsilon > 0$, we can find $\varphi \in C_c^\infty(\mathbb{R}^d)$ such that $\|f - \varphi\|_{\infty} < \frac{\varepsilon}{4M}$ and $n \in \mathbb{N}$ such that $|\langle \nabla \varphi, m_n - m_* \rangle| < \frac{\varepsilon}{2}$. Then,

$$ |\langle f, \nabla(m_n - m_*) \rangle| \leq |\langle f - \varphi, \nabla m_n \rangle| + |\langle f - \varphi, \nabla m_* \rangle| + |\langle \nabla \varphi, m_n - m_* \rangle| \\ < 2 \cdot \frac{\varepsilon}{4M} \cdot M + \frac{\varepsilon}{2} = \varepsilon. $$

Equivalently, the sequence of $\mathbb{R}^d$-valued Radon measures $(\nabla m_n)_{n \in \mathbb{N}}$ converges to $\nabla m$ locally weakly. We then apply [1, Theorem 2.34] to obtain

$$ \liminf_{n \to +\infty} I(m_n) \geq I(m_*) $$

Finally, the lower semicontinuity of $m \mapsto H(m|\hat{m})$ (resp. $m \mapsto I(m|\hat{m})$) follows from the lower semicontinuity of $m \mapsto H(m)$ (resp. $m \mapsto I(m)$) and the locally uniform quadratic growth of $x \mapsto \frac{\delta F}{\delta m}(m, x)$ (resp. the locally uniform linear growth of $x \mapsto D_m F(m, x)$). □
## B Convergence of non-linear functional of empirical measures

Let $\phi: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a (non-linear) mean field functional and $m$ be a probability measure with finite second moment. We suppose the first and second-order functional derivatives $\frac{\delta\phi}{\delta m}$, $\frac{\delta^2\phi}{\delta m^2}$ exist and that $(\phi, m)$ satisfies

$$ \forall m' \in \mathcal{P}_2(\mathbb{R}^d), \forall x \in \mathbb{R}^d, \quad |D_m\phi(m', x)| \le M_1, \qquad (B.1) $$

$$ \forall m' \in \mathcal{P}_2(\mathbb{R}^d), \quad \left|\iint \left[ \frac{\delta^2 \phi}{\delta m^2}(m', x, x) - \frac{\delta^2 \phi}{\delta m^2}(m', x, y) \right] m(dx)m(dy) \right| \le M_2 \qquad (B.2) $$

for some constants $M_1$ and $M_2$.

Remark B.1. The condition (B.2) is a modified version of the condition [39, (p-LFD)]. Our version has the advantage of being intrinsic: the left hand side of (B.2) stays invariant under the change $\frac{\delta^2\phi}{\delta m^2}(m, x, y) \to \frac{\delta^2\phi}{\delta m^2}(m, x, y) + \frac{\delta\phi_1}{\delta m}(m, y) + \phi_2(m)$ for regular enough $\phi_1$ and $\phi_2$.

**Lemma B.1.** If the mean field functional $\phi$ and the measure $m$ satisfy (B.1) and (B.2), then for $N$ i.i.d. random variables $\xi_1, \dots, \xi_N \sim \mu$, we have

$$ \mathrm{E}[|\phi(\mu_{\xi}) - \phi(m)|^2] \le \frac{M_1^2 \mathrm{Var} m}{N} + \frac{M_2^2}{4N^2}. \qquad (B.3) $$

*Proof.* We have the decomposition

$$ \mathrm{E}[|\phi(\mu_{\xi}) - \phi(m)|^2] = \mathrm{Var} \phi(\mu_{\xi}) + (\mathrm{E}[\phi(\mu_{\xi})] - \phi(m))^2. $$

Thanks to (B.1), the mapping $\xi^i \mapsto \phi(\mu_{\xi})$ is $\frac{M_1}{N}$-Lipschitz continuous, so by the Efron-Stein inequality we have

$$ \mathrm{Var} \phi(\mu_{\xi}) \le \frac{M_1^2}{N} \mathrm{Var} m. $$

For the second term we apply the argument of [39, Theorem 4.2.9 (i)] and obtain

$$ |\mathrm{E}[\phi(\mu_{\xi})] - \phi(m)| \le \frac{M_2}{2N}. \qquad \square $$

## C Validity of Girsanov transforms

We prove a lemma similar to [18, Lemma A.1] which allows us to justify Girsanov transforms.

**Lemma C.1.** Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0, T]}, \mathbb{P})$ be a filtered probability space. If $\beta, \gamma, X, Y : \Omega \times [0, T] \to \mathbb{R}^d$ are $\mathcal{F}_t$-adapted continuous stochastic processes satisfying

$$ |\beta_t| + |\gamma_t| \le C(1 + |X_t| + |V_t|) $$

almost surely for some constant $C$, and if the tuple $(X, V, \beta)$ solves

$$ dX_t = V_t dt, \\ dV_t = \beta_t dt + \sqrt{2} dW_t $$

for an $\mathcal{F}_t$-adapted Brownian $W_t$ with $\mathrm{E}[|X_0|^2 + |V_0|^2] < +\infty$, then the exponential local martingale

$$ R. := \exp\left(\int_0^t \gamma_s \cdot dW_s - \frac{1}{2} \int_0^t |\gamma_s|^2 ds\right) $$

is uniformly integrable.
## B Convergence of non-linear functional of empirical measures

Let $\phi: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ be a (non-linear) mean field functional and $m$ be a probability measure with finite second moment. We suppose the first and second-order functional derivatives $\frac{\delta\phi}{\delta m}$, $\frac{\delta^2\phi}{\delta m^2}$ exist and that $(\phi, m)$ satisfies

$$ \forall m' \in \mathcal{P}_2(\mathbb{R}^d), \forall x \in \mathbb{R}^d, \quad |D_m\phi(m', x)| \le M_1, \qquad (B.1) $$

$$ \forall m' \in \mathcal{P}_2(\mathbb{R}^d), \quad \left|\iint \left[ \frac{\delta^2 \phi}{\delta m^2}(m', x, x) - \frac{\delta^2 \phi}{\delta m^2}(m', x, y) \right] m(dx)m(dy) \right| \le M_2 \qquad (B.2) $$

for some constants $M_1$ and $M_2$.

**Remark B.1.** The condition (B.2) is a modified version of the condition [39, (p-LFD)]. Our version has the advantage of being intrinsic: the left hand side of (B.2) stays invariant under the change $\frac{\delta^2\phi}{\delta m^2}(m, x, y) \to \frac{\delta^2\phi}{\delta m^2}(m, x, y) + \frac{\delta\phi_1}{\delta m}(m, y) + \phi_2(m)$ for regular enough $\phi_1$ and $\phi_2$.

**Lemma B.1.** If the mean field functional $\phi$ and the measure $m$ satisfy (B.1) and (B.2), then for $N$ i.i.d. random variables $\xi_1, \dots, \xi_N \sim \mu$, we have

$$ E[|\phi(\mu_{\xi}) - \phi(m)|^2] \le \frac{M_1^2 \mathrm{Var} m}{N} + \frac{M_2^2}{4N^2}. \qquad (B.3) $$

*Proof.* We have the decomposition

$$ E[|\phi(\mu_{\xi}) - \phi(m)|^2] = \mathrm{Var} \phi(\mu_{\xi}) + (E[\phi(\mu_{\xi})] - \phi(m))^2. $$

Thanks to (B.1), the mapping $\xi^i \mapsto \phi(\mu_{\xi})$ is $\frac{M_1}{N}$-Lipschitz continuous, so by the Efron-Stein inequality we have

$$ \mathrm{Var} \phi(\mu_{\xi}) \le \frac{M_1^2}{N} \mathrm{Var} m. $$

For the second term we apply the argument of [39, Theorem 4.2.9 (i)] and obtain

$$ |E[\phi(\mu_{\xi})] - \phi(m)| \le \frac{M_2}{2N}. \qquad \square $$

## C Validity of Girsanov transforms

We prove a lemma similar to [18, Lemma A.1] which allows us to justify Girsanov transforms.

**Lemma C.1.** Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0, T]}, \mathbb{P})$ be a filtered probability space. If $\beta, \gamma, X, Y : \Omega \times [0, T] \to \mathbb{R}^d$ are $\mathcal{F}_t$-adapted continuous stochastic processes satisfying

$$ |\beta_t| + |\gamma_t| \le C(1 + |X_t| + |V_t|) $$

almost surely for some constant $C$, and if the tuple $(X, V, \beta)$ solves

$$ dX_t = V_t dt, \\ dV_t = \beta_t dt + \sqrt{2} dW_t $$

for an $\mathcal{F}_t$-adapted Brownian $W_t$ with $E[|X_0|^2 + |V_0|^2] < +\infty$, then the exponential local martingale

$$ R. := \exp\left(\int_0^t \gamma_s \cdot dW_s - \frac{1}{2} \int_0^t |\gamma_s|^2 ds\right) $$

is uniformly integrable.
Proof. It suffices to verify $E[R_T] = 1$. Put $M_t = R_t(1 + |X_t|^2 + |V_t|^2)$. By Itō's formula, the local semimartingale satisfies

$$dM_t = 2R_t(X_t \cdot Y_t + (\beta_t + \sqrt{2}\gamma_t) \cdot Y_t + 1)dt + R_t((1 + |X_t|^2 + |Y_t|^2)\gamma_t + 2\sqrt{2}Y_t) \cdot dW_t.$$

Using the uniform linear growth condition of $\beta, \gamma$, we can find a constant $C$ such that $t \mapsto e^{-Ct}M_t$ is a local supermartingale. But $e^{-Ct}M_t \ge 0$. So by Fatou's lemma $t \mapsto e^{-Ct}M_t$ is really a supermartingale and this yields $E[M_t] \le e^{Ct}E[M_0]$. The Itō's formula for $R_t$ writes

$$dR_t = R_t \gamma_t \cdot dW_t.$$

So for $\varepsilon > 0$ the bounded supermartingale $\frac{R_t}{1+\varepsilon R_t}$ satisfies

$$d\frac{R_t}{1+\varepsilon R_t} = -\frac{\varepsilon R_t^2 \gamma_t^2}{(1+\varepsilon R_t)^3} dt + \frac{R_t}{(1+\varepsilon R_t)^2} \gamma_t \cdot dW_t.$$

Taking expectations on both sides, we obtain

$$E\left[\frac{R_T}{1 + \varepsilon R_T}\right] = \frac{1}{1 + \varepsilon} - E\left[\int_0^T \frac{\varepsilon R_t^2 \gamma_t^2}{(1 + \varepsilon R_t)^3} dt\right].$$

Using the bound $\frac{\varepsilon R_t^2 \gamma_t^2}{(1+\varepsilon R_t)^3} \le R_t \gamma_t^2 \le CM_t$, we take the limit $\varepsilon \to 0$ by the dominated convergence theorem and obtain $E[R_T] = 1$. $\square$

## References

[1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Oxford: Clarendon Press, 2000. MR1857292

[2] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser, 2nd ed. edition, 2008. MR2401600

[3] François Bolley, Arnaud Guillin, and Florent Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM, Math. Model. Numer. Anal., 44(5):867–884, 2010. MR2731396

[4] René Carmona and François Delarue. Probabilistic theory of mean field games with applications I. Mean field FBSDEs, control, and games, volume 83 of Probab. Theory Stoch. Model. Cham: Springer, 2018. MR3752669

[5] Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. I: Models and methods. Kinet. Relat. Models, 15(6):895–1015, 2022. MR4489768

[6] Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. II: Applications. Kinet. Relat. Models, 15(6):1017–1173, 2022. MR4489769

[7] Fan Chen, Zhenjie Ren, and Songbo Wang. Uniform-in-time propagation of chaos for mean field Langevin dynamics. arXiv preprint arXiv:2212.03050, 2022.

[8] Lénaïc Chizat and Francis Bach. On the global convergence of gradient descent for over-parameterized models using optimal transport. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.

[9] Antonin Chodron de Courcel, Matthew Rosenzweig, and Sylvia Serfaty. Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows. To appear in Ann. Inst. Henri Poincaré, Anal. Non Linéaire.

[10] François Delarue and Alvin Tse. Uniform in time weak propagation of chaos on the torus. arXiv preprint arXiv:2104.14973, 2021.

[11] Andreas Eberle, Arnaud Guillin, and Raphael Zimmer. Couplings and quantitative contraction rates for Langevin dynamics. Ann. Probab., 47(4):1982–2010, 2019. MR3980913
Proof. It suffices to verify $E[R_T] = 1$. Put $M_t = R_t(1 + |X_t|^2 + |V_t|^2)$. By Itō's formula, the local semimartingale satisfies

$$dM_t = 2R_t(X_t \cdot Y_t + (\beta_t + \sqrt{2}\gamma_t) \cdot Y_t + 1)dt + R_t((1 + |X_t|^2 + |Y_t|^2)\gamma_t + 2\sqrt{2}Y_t) \cdot dW_t.$$

Using the uniform linear growth condition of $\beta, \gamma$, we can find a constant $C$ such that $t \mapsto e^{-Ct}M_t$ is a local supermartingale. But $e^{-Ct}M_t \ge 0$. So by Fatou's lemma $t \mapsto e^{-Ct}M_t$ is really a supermartingale and this yields $E[M_t] \le e^{Ct}E[M_0]$. The Itō's formula for $R_t$ writes

$$dR_t = R_t\gamma_t \cdot dW_t.$$

So for $\varepsilon > 0$ the bounded supermartingale $\frac{R_t}{1+\varepsilon R_t}$ satisfies

$$d\frac{R_t}{1+\varepsilon R_t} = -\frac{\varepsilon R_t^2 \gamma_t^2}{(1+\varepsilon R_t)^3}dt + \frac{R_t}{(1+\varepsilon R_t)^2}\gamma_t \cdot dW_t.$$

Taking expectations on both sides, we obtain

$$E\left[\frac{R_T}{1+\varepsilon R_T}\right] = \frac{1}{1+\varepsilon} - E\left[\int_0^T \frac{\varepsilon R_t^2 \gamma_t^2}{(1+\varepsilon R_t)^3}dt\right].$$

Using the bound $\frac{\varepsilon R_t^2 \gamma_t^2}{(1+\varepsilon R_t)^3} \le R_t \gamma_t^2 \le CM_t$, we take the limit $\varepsilon \to 0$ by the dominated convergence theorem and obtain $E[R_T] = 1$. $\square$

## References

[1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Oxford: Clarendon Press, 2000. MR1857292

[2] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser, 2nd ed. edition, 2008. MR2401600

[3] François Bolley, Arnaud Guillin, and Florent Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM, Math. Model. Numer. Anal., 44(5):867–884, 2010. MR2731396

[4] René Carmona and François Delarue. Probabilistic theory of mean field games with applications I. Mean field FBSDEs, control, and games, volume 83 of Probab. Theory Stoch. Model. Cham: Springer, 2018. MR3752669

[5] Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. I: Models and methods. Kinet. Relat. Models, 15(6):895–1015, 2022. MR4489768

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[11] Andreas Eberle, Arnaud Guillin, and Raphael Zimmer. Couplings and quantitative contraction rates for Langevin dynamics. Ann. Probab., 47(4):1982–2010, 2019. MR3980913
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[13] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché. Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems. To appear in *J. Eur. Soc. Math.*

[14] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché. Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases. *Electron. J. Probab.*, 27:44, 2022. Id/No 124. MR4489825

[15] Arnaud Guillin, Wei Liu, Liming Wu, and Chaoen Zhang. The kinetic Fokker-Planck equation with mean field interaction. *J. Math. Pures Appl.* (9), 150:1-23, 2021. MR4248461

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[17] Geoffrey Hinton, Nitish Srivastava, and Kevin Swersky. Neural networks for machine learning: Lecture 6a: Overview of mini-batch gradient descent. http://www.cs.toronto.edu/~hinton/coursera/lecture6/lec6.pdf, 2012. Accessed: 2023-06-23.

[18] Kaitong Hu, Zhenjie Ren, David Šiška, and Łukasz Szpruch. Mean-field Langevin dynamics and energy landscape of neural networks. *Ann. Inst. Henri Poincaré, Probab. Stat.*, 57(4):2043-2065, 2021. MR4328560

[19] Xing Huang. Coupling by change of measure for conditional McKean-Vlasov SDEs and applications. *arXiv preprint arXiv:2303.04369*, 2023.

[20] Pierre-Emmanuel Jabin and Zhenfu Wang. Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels. *Invent. Math.*, 214(1):523-591, 2018. MR3858403

[21] Anna Kazeykina, Zhenjie Ren, Xiaolu Tan, and Junjian Yang. Ergodicity of the underdamped mean-field Langevin dynamics. *To appear in Ann. Appl. Probab.*

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[23] Nikola B. Kovachki and Andrew M. Stuart. Continuous time analysis of momentum methods. *Journal of Machine Learning Research*, 22(17):1-40, 2021. MR4253710

[24] Daniel Lacker and Luc Le Flem. Sharp uniform-in-time propagation of chaos. *Probability Theory and Related Fields*, pages 1-38, 2023. MR4634344

[25] Yann LeCun, Corinna Cortes, and Christopher J.C. Burges. The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/, 1998. Accessed: 2023-06-23.

[26] Yanli Liu, Yuan Gao, and Wotao Yin. An improved analysis of stochastic gradient descent with momentum. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, *Advances in Neural Information Processing Systems*, volume 33, pages 18261-18271. Curran Associates, Inc., 2020.

[27] Yi-An Ma, Niladri S. Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L. Bartlett, and Michael I. Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC? *Bernoulli*, 27(3):1942-1992, 2021. MR4278799

[28] Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of two-layer neural networks. *Proc. Natl. Acad. Sci. USA*, 115(33):e7665-e7671, 2018. MR3845070

[29] Pierre Monmarché. Long-time behaviour and propagation of chaos for mean field kinetic particles. *Stochastic Processes Appl.*, 127(6):1721-1737, 2017. MR3646428

[30] Boris T. Polyak. Some methods of speeding up the convergence of iteration methods. *USSR Computational Mathematics and Mathematical Physics*, 4(5):1-17, 1964. MR0169403

[31] Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar. On the convergence of Adam and beyond. *In International Conference on Learning Representations*, 2018.

[32] Panpan Ren and Feng-Yu Wang. Exponential convergence in entropy and Wasserstein for McKean-Vlasov SDEs. *Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods*, 20:6:21-25, 2021. Id/No 112259. MR4206077

[33] Matthew Rosenzweig and Sylvia Serfaty. Global-in-time mean-field convergence for singular Riesz-type diffusive flows. *Ann. Appl. Probab.*, 33(2):954-998, 2023. MR4564418
[12] Nicolas Fournier and Arnaud Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. *Probab. Theory Relat. Fields*, 162(3-4):707-738, 2015. MR3383341

[13] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché. Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems. To appear in *J. Eur. Soc. Math.*

[14] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché. Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases. *Electron. J. Probab.*, 27:44, 2022. Id/No 124. MR4489825

[15] Arnaud Guillin, Wei Liu, Liming Wu, and Chaoen Zhang. The kinetic Fokker-Planck equation with mean field interaction. *J. Math. Pures Appl.* (9), 150:1-23, 2021. MR4248461

[16] Arnaud Guillin and Pierre Monmarché. Uniform long-time and propagation of chaos estimates for mean field kinetic particles in non-convex landscapes. *J. Stat. Phys.*, 185(2):20, 2021. Id/No 15. MR4333408

[17] Geoffrey Hinton, Nitish Srivastava, and Kevin Swersky. Neural networks for machine learning: Lecture 6a: Overview of mini-batch gradient descent. http://www.cs.toronto.edu/~hinton/coursera/lecture6/lec6.pdf, 2012. Accessed: 2023-06-23.

[18] Kaitong Hu, Zhenjie Ren, David Šiška, and Łukasz Szpruch. Mean-field Langevin dynamics and energy landscape of neural networks. *Ann. Inst. Henri Poincaré, Probab. Stat.*, 57(4):2043-2065, 2021. MR4328560

[19] Xing Huang. Coupling by change of measure for conditional McKean-Vlasov SDEs and applications. *arXiv preprint arXiv:2303.04369*, 2023.

[20] Pierre-Emmanuel Jabin and Zhenfu Wang. Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels. *Invent. Math.*, 214(1):523-591, 2018. MR3858403

[21] Anna Kazeykina, Zhenjie Ren, Xiaolu Tan, and Junjian Yang. Ergodicity of the underdamped mean-field Langevin dynamics. *To appear in Ann. Appl. Probab.*

[22] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. *arXiv preprint arXiv:1412.6980*, 2014.

[23] Nikola B. Kovachki and Andrew M. Stuart. Continuous time analysis of momentum methods. *Journal of Machine Learning Research*, 22(17):1-40, 2021. MR4253710

[24] Daniel Lacker and Luc Le Flem. Sharp uniform-in-time propagation of chaos. *Probability Theory and Related Fields*, pages 1-38, 2023. MR4634344

[25] Yann LeCun, Corinna Cortes, and Christopher J.C. Burges. The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/, 1998. Accessed: 2023-06-23.

[26] Yanli Liu, Yuan Gao, and Wotao Yin. An improved analysis of stochastic gradient descent with momentum. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, *Advances in Neural Information Processing Systems*, volume 33, pages 18261-18271. Curran Associates, Inc., 2020.

[27] Yi-An Ma, Niladri S. Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L. Bartlett, and Michael I. Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC? *Bernoulli*, 27(3):1942-1992, 2021. MR4278799

[28] Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of two-layer neural networks. *Proc. Natl. Acad. Sci. USA*, 115(33):e7665-e7671, 2018. MR3845070

[29] Pierre Monmarché. Long-time behaviour and propagation of chaos for mean field kinetic particles. *Stochastic Processes Appl.*, 127(6):1721-1737, 2017. MR3646428

[30] Boris T. Polyak. Some methods of speeding up the convergence of iteration methods. *USSR Computational Mathematics and Mathematical Physics*, 4(5):1-17, 1964. MR0169403

[31] Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar. On the convergence of Adam and beyond. *In International Conference on Learning Representations*, 2018.

[32] Panpan Ren and Feng-Yu Wang. Exponential convergence in entropy and Wasserstein for McKean-Vlasov SDEs. *Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods*, 20:6:21-25, 2021. Id/No 112259. MR4206077

[33] Matthew Rosenzweig and Sylvia Serfaty. Global-in-time mean-field convergence for singular Riesz-type diffusive flows. *Ann. Appl. Probab.*, 33(2):954-998, 2023. MR4564418
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[37] Othmane Sebbouh, Robert M. Gower, and Aaron Defazio. Almost sure convergence rates for stochastic gradient descent and stochastic heavy ball. In Mikhail Belkin and Samory Kpotufe, editors, *Proceedings of Thirty Fourth Conference on Learning Theory*, volume 134 of *Proceedings of Machine Learning Research*, pages 3935–3971. PMLR, 15–19 Aug 2021.

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[40] Cédric Villani. *Hypocoercivity*, volume 950 of *Mem. Am. Math. Soc*. Providence, RI: American Mathematical Society (AMS), 2009. MR2562709
[34] Grant Rotskoff and Eric Vanden-Eijnden. Trainability and accuracy of artificial neural networks: an interacting particle system approach. *Commun. Pure Appl. Math.*, 75(9):1889–1935, 2022. MR4465905

[35] Sebastian Ruder. An overview of gradient descent optimization algorithms. *arXiv preprint arXiv:1609.04747*, 2016.

[36] Katharina Schuh. Global contractivity for Langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos. *To appear in Ann. Inst. Henri Poincaré, Probab. Stat.*

[37] Othmane Sebbouh, Robert M. Gower, and Aaron Defazio. Almost sure convergence rates for stochastic gradient descent and stochastic heavy ball. In Mikhail Belkin and Samory Kpotufe, editors, *Proceedings of Thirty Fourth Conference on Learning Theory*, volume 134 of *Proceedings of Machine Learning Research*, pages 3935–3971. PMLR, 15–19 Aug 2021.

[38] Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In Sanjoy Dasgupta and David McAllester, editors, *Proceedings of the 30th International Conference on Machine Learning*, volume 28 of *Proceedings of Machine Learning Research*, pages 1139–1147, Atlanta, Georgia, USA, 17–19 Jun 2013. PMLR.

[39] Alvin Tsz Ho Tse. *Quantitative propagation of chaos of McKean-Vlasov equations via the master equation*. PhD thesis, The University of Edinburgh, 2019.

[40] Cédric Villani. *Hypocoercivity*, volume 950 of *Mem. Am. Math. Soc*. Providence, RI: American Mathematical Society (AMS), 2009. MR2562709
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Electronic Journal of Probability
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Advantages of publishing in EJP-ECP

• Very high standards

• Free for authors, free for readers

• Quick publication (no backlog)

• Secure publication (LOCKSS<fnref n="1" />)

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