# Logarithmic Sobolev inequalities for non-equilibrium steady states

Pierre Monmarché, Songbo Wang

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Pierre Monmarché, Songbo Wang. Logarithmic Sobolev inequalities for non-equilibrium steady states. Potential Analysis, In press, 10.1007/s11118-025-10211-6. hal-05004079v2

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Distributed under a Creative Commons Attribution 4.0 International License
Logarithmic Sobolev inequalities
for non-equilibrium steady states

Pierre Monmarché¹ and Songbo Wang²

¹LJLL & LCT, Sorbonne Université, Paris, France

²CMAP, École polytechnique, IP Paris, Palaiseau, France

March 24, 2025

Abstract

We consider two methods to establish log-Sobolev inequalities for the in-
variant measure of a diffusion process when its density is not explicit and
the curvature is not positive everywhere. In the first approach, based on
the Holley–Stroock and Aida–Shigekawa perturbation arguments [J. Stat.
Phys., 46(5-6):1159–1194, 1987, J. Funct. Anal., 126(2):448–475, 1994], the
control on the (non-explicit) perturbation is obtained by stochastic control
methods, following the comparison technique introduced by Conforti [Ann.
Appl. Probab., 33(6A):4608–4644, 2023]. The second method combines the
Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941–973,
2023] to prove a Poincaré inequality in some non-equilibrium cases, with
Wang’s hypercontractivity results [Potential Anal., 53(3):1123–1144, 2020].

# 1 Introduction

1.1 Overview

A probability measure $\mu$ on $\mathbb{R}^d$ is said to satisfy a logarithmic Sobolev inequality (LSI) with constant $C_{LS} > 0$ if for all smooth and compactly supported function $f$ on $\mathbb{R}^d$ with $\int f^2 d\mu = 1$, we have

$$
\int_{\mathbb{R}^d} f^2 \ln f^2 \, d\mu \le 2C_{\text{LS}} \int_{\mathbb{R}^d} |\nabla f|^2 \, d\mu.
$$

It is related to the long-time convergence of diffusion processes and concentration inequalities, see [3] and references therein for general considerations on this topic. The main question addressed here is to establish such an LSI in cases where $\mu$ has no explicit density but is defined as the invariant measure of a diffusion process $(Z_t)_{t \ge 0}$ satisfying

$$
dZ_t = b(Z_t) dt + \sigma dB_t, \quad (1)
$$

where $b \in C^1(\mathbb{R}^d, \mathbb{R}^d)$, $\sigma$ is a constant matrix and $B$ is a $d$-dimensional Brownian motion. Among other applications, this is motivated by non-equilibrium statistical physics models, such as [18, 19, 9]. In this literature, these non-explicit invariant measures are referred to as Non-Equilibrium Steady States (NESS).
Many criteria to establish LSI are known, but most of them require an explicit expression for $\mu$ (for instance in order to use bounded or Lipschitz perturbation arguments) or the process (1) to be reversible with respect to $\mu$ (for instance the Lyapunov-based results of [2, 6]). In fact, denoting by

$$ \mathcal{L} = b \cdot \nabla + \Sigma : \nabla^2 $$

the generator of (1) (where $\Sigma = \sigma\sigma^T/2$ and $\Sigma : \nabla^2 = \sum_{i,j} \Sigma_{ij}\partial_{z_i}\partial_{z_j}$), some arguments based on reversibility (such as those of [2, 6]) may sometimes be extended to non-reversible cases when the dual $\mathcal{L}^*$ of $\mathcal{L}$ in $L^2(\mu)$ is known. Since

$$ \mathcal{L}^* = (2\Sigma\nabla \ln \mu - b) \cdot \nabla + \Sigma : \nabla^2, $$

knowing $\mathcal{L}^*$ requires an explicit expression of $\mu$. A notable exception is the use of Bakry-Émery curvature conditions: if there exists $\rho > 0$ such that, for all $x$, $y \in \mathbb{R}^d$,

$$ (b(x) - b(y)) \cdot (x - y) \leq -\rho|x - y|^2, \quad (2) $$

then the diffusion (1) admits a unique invariant measure that satisfies an LSI with constant $|\Sigma|/\rho$, see [5, 24], even when the process is neither reversible nor elliptic. However, such a contraction condition is very restrictive. If (2) holds only for $x, y$ outside some compact set, we can decompose $b = b_0 + b_1$ where $b_0$ satisfies a similar condition on the whole $\mathbb{R}^d$ and $b_1$ is compactly supported. Then, we know that the invariant measure $\mu_0$ of the process with generator $\mathcal{L}_0 = b_0 \cdot \nabla + \Sigma : \nabla^2$ satisfies an LSI, but to the best of our knowledge it is not known how to transfer the result to $\mu$ in this general case.

In this work, we will consider two cases:

* In the first one, $\mu$ is a perturbation of an explicit measure $\mu_0$, invariant for $\mathcal{L}_0 = b_0 \cdot \nabla + \Sigma : \nabla^2$ for some $b_0$. Our method relies on the bounded perturbation result of Holley and Stroock [16] and the Lipschitz perturbation result of Aida and Shigekawa [1]. In other words, the key point is to prove that $\ln(\mu/\mu_0)$ is the sum of a bounded function and a Lipschitz function. This is done by seeing this quantity as the long-time limit of the solution of a (parabolic) Hamilton-Jacobi-Bellman (HJB) equation and using a stochastic control representation for the solution together with a coupling argument, following the method introduced by Conforti in [7]. This approach is applied to the elliptic case and to a non-elliptic kinetic case.

* In the second one, we consider the high-diffusivity elliptic framework of [26], namely (2) holds for every $y$ once $x$ lies outside some compact set and $\sigma = \bar{\sigma}Id$ where $\bar{\sigma} > 0$ is large enough. Under these conditions, on the one hand, it is proven in [26] that $\mu$ satisfies a Poincaré inequality, using the large-time contraction of the Wasserstein-2 distance along the diffusion semi-group. On the other hand, as established by Wang in [35] (in the reversible case but we will see that the proof applies without any change in the non-reversible case), the semi-group is hypercontractive, which implies a so-called defective LSI (which is well known in the reversible case and turns out to be true in general). The Poincaré inequality together with the defective LSI is equivalent to the LSI.

In fact, thanks to the powerful [32, Corollary 1.2], the defective LSI alone is already equivalent to a tight LSI for irreducible diffusion processes. This has been
used in subsequent works [34, 33, 17] for both elliptic and kinetic processes. However this argument is non-constructive and thus does not provide explicit constants, similarly to the tightening argument based on weak Poincaré inequalities in [30, Proposition 1.3]. This is in our contrast to our approach, as illustrated in [28] which is based on the present work.

The rest of this work is organized as follows. The results are stated in the remainder of Section 1. The results based on perturbation, Theorems 1 and 2, are proven in Section 2. Section 3 is devoted to the proofs for the defective LSI. A coupling construction for the kinetic Langevin process, used in the proof of Theorem 2, is postponed to Appendix A.

## 1.2 Perturbation approach: the elliptic case

In the elliptic case where $\Sigma = \text{Id}$, we get the following, proven in Section 2.1.

**Theorem 1.** Assume that $\Sigma = \text{Id}$ and $b = b_0 + b_1$ for some $b_0, b_1 \in C^1(\mathbb{R}^d, \mathbb{R}^d)$ with bounded derivatives such that the generator $\mathcal{L}_0 = b_0 \cdot \nabla + \Delta$ admits a unique $C^2$ and positive invariant probability density $\mu_0$ satisfying an LSI with constant $C_0 > 0$. Write $\tilde{b} := 2\nabla \ln \mu_0 - b$ and $\varphi := -\nabla \cdot b_1 + b_1 \cdot \nabla \ln \mu_0$. Assume that there exist $L, R, M^\varphi, L^\varphi \ge 0$ and $\rho > 0$ such that $\varphi = \varphi_1 + \varphi_2$ with $\varphi_1$ being $M^\varphi$-bounded and $\varphi_2$ being $L^\varphi$-Lipschitz, and for all $x, y \in \mathbb{R}^d$,

$$ (\tilde{b}(x) - \tilde{b}(y)) \cdot (x - y) \le \begin{cases} -\rho|x - y|^2 & \text{if } |x - y| \ge R, \\ L|x - y|^2 & \text{otherwise.} \end{cases} \quad (3) $$

Finally, assume that the law of $Z_t$ solving (1) converges weakly for all initial condition as $t \to \infty$ to a unique invariant measure $\mu$ on $\mathbb{R}^d$. Then $\mu$ satisfies an LSI with constant $C_{\text{LS}} = C_{\text{LS}}(C_0, \rho, L, R, d; M^\varphi, L^\varphi)$.

Notice that, when $\Sigma = \text{Id}$, the carré du champ operator $\Gamma(f) = \frac{1}{2}\mathcal{L}(f^2) - f\mathcal{L}f$ is equal to $|\nabla f|^2$ and is the same for the dual operator $\mathcal{L}^*$. In particular, the LSI is equivalent to the constant-rate decay of the relative entropy,

$$ \forall \nu \ll \mu, \quad \mathcal{H}(P_t^* \nu | \mu) \le e^{-t/C_{\text{LS}}} \mathcal{H}(\nu | \mu), \quad (4) $$

where $P_t = \exp(t\mathcal{L})$ is the semi-group generated by $\mathcal{L}$, and the relative entropy $\mathcal{H}$ is defined by $\mathcal{H}(\nu|\mu) = \int \ln(d\nu/d\mu) d\nu$ (see e.g. [3, Theorem 5.2.1]; reversibility is not used in the proof).

**Example 1.** Consider on $\mathbb{R}^2$ the SDE

$$ dX_t = (f(|X_t|)X_t^\perp - X_t - \nabla V(X_t)) dt + \sqrt{2} dB_t, $$

where $(u, v)^\perp = (v, -u)$, $f \in C^1(\mathbb{R}_+, \mathbb{R})$ and $V \in C^2(\mathbb{R}^2)$. We can decompose the drift $b = b_0 + b_1$ as

$$ b_0(x) = x^\perp f(|x|) - x, \quad b_1(x) = -\nabla V(x), \quad (5) $$

or, alternatively,

$$ b_0(x) = -\nabla V(x) - x, \quad b_1(x) = f(|x|)x^\perp. \quad (6) $$
In the first case (5), the invariant measure of $b_0 \cdot \nabla + \Delta$ is the standard Gaussian measure $\mu_0(x) = (2\pi)^{-d/2} \exp(-|x|^2/2)$ and, using the notations of Theorem 1,

$$ \tilde{b}(x) = -f(|x|)x^{\perp} - x + \nabla V(x), \quad \varphi(x) = \Delta V(x) + \nabla V(x) \cdot x. $$

For instance, if $f(|x|)$ is constant for $|x|$ large enough, then the rotating part does not intervene in the condition (3) outside some compact set, which means that this condition is satisfied as soon as $(x-y) \cdot (\nabla V(x) - \nabla V(y)) \le \eta|x-y|^2$ outside some compact set for some $\eta < 1$. In this situation, ergodicity for (1) is easily shown using Harris Theorem. Then Theorem 1 applies as soon as $\varphi$ is Lipschitz, which is for instance the case if $V$ is compactly supported (which implies the previous condition). Notice that here we do not use the fact that the perturbative term $b_1 = -\nabla V$ is a gradient.

If, alternatively, we use the decomposition (6), then $\mu_0$ is the probability density proportional to $\exp(-|x|^2/2 - V(x))$ and

$$ \tilde{b}(x) = -f(|x|)x^{\perp} - x - \nabla V(x), \quad \varphi(x) = f(|x|)x^{\perp} \cdot \nabla V(x). $$

Then, Theorem 1 applies for instance if $f$ is compactly supported and $(x-y) \cdot (\nabla V(x) - \nabla V(y)) \ge -\eta|x-y|^2$ for some $\eta < 1$ outside some compact set.

**Example 2.** Let us check how Theorem 1 reads in the classical reversible case, namely taking

$$ b_0(x) = -\nabla U(x), \quad b_1(x) = -\nabla W(x) $$

for some $U, W \in C^2(\mathbb{R}^d)$, so that $\mu_0 \propto e^{-U}$, $\mu \propto e^{-U-W}$ and, with the notations of Theorem 1,

$$ \tilde{b}(x) = -\nabla U(x) + \nabla W(x), \quad \varphi(x) = \Delta W(x) + \nabla U(x) \cdot \nabla W(x). $$

Hence, the conditions in Theorem 1 does not seem to be similar to those of classical perturbation results in the reversible case. However, for instance, if we take $U(x) = |x|^2/2$ outside some compact set, then to get that $x \mapsto \nabla U(x) \cdot \nabla W(x)$ is Lipschitz one will typically require $\nabla W$ to be bounded, in which case the LSI for $\mu$ follows from [1].

**Example 3.** We now consider a non-linear McKean-Vlasov equation on $\mathbb{R}^d$:

$$ \partial_t \mu_t = \nabla \cdot ((\nabla V + \lambda b_{\mu_t}) \mu_t + \nabla \mu_t) \quad (7) $$

where $x \mapsto \mu_t(x)$ is a probability density on $\mathbb{R}^d$ (which we identify with the corresponding probability measure), $V \in C^2(\mathbb{R}^d)$ is a confining potential, $\lambda \in \mathbb{R}$ encodes the non-linearity amplitude, $\nabla \cdot$ stands for the divergence operator and, for all probability measure $\nu$, $b_\nu \in C^1(\mathbb{R}^d)$. If $\mu_*$ is a stationary measure for (7), it is the invariant measure of the diffusion process with generator $\mathcal{L}_{\mu_*}$ where

$$ \mathcal{L}_{\mu} = -(\nabla V + \lambda b_{\mu}) \cdot \nabla + \Delta. $$

Among other examples of interest where, for a given $\mu$, $b_\mu$ is not the gradient of some potential, we can mention the competition models considered in [22], where $x = (x_1, x_2) \in \mathbb{R}^{2p}$ and

$$ b_{\mu}(x_1, x_2) = \left( \begin{array}{c} \int_{\mathbb{R}^p} \nabla_{x_1} K(x_1, y_2) \mu(y_1, y_2) \, dy_1 \, dy_2 \\ - \int_{\mathbb{R}^p} \nabla_{x_2} K(y_1, x_2) \mu(y_1, y_2) \, dy_1 \, dy_2 \end{array} \right) \quad (8) $$
for some $K \in C^2(\mathbb{R}^{2p})$. In other words, the population is divided in two types of individuals, the first type (resp. second) tends to maximize (resp. minimize) its value of K averaged with respect to the population distribution of the other type.

In order to get an LSI for $\mu_*$, for instance, it is straightforward to check that the assumptions of Theorem 1 are met (with $\mu_0 \propto e^{-V}$) under the following condition:

**Assumption 1.** The potential $V$ is strictly convex outside a compact, and its hessian is bounded. There exists $L', C' > 0$ such that, for all $\mu$, $b_\mu$ is $L'$-Lipschitz and for all $x \in \mathbb{R}^d$,

$$b_{\mu}(x) \le C' \frac{1 + \int_{\mathbb{R}^d} |y|\mu(dy)}{1 + |x|}. \qquad (9)$$

In particular, the condition (9) is used to get that, for a given $\mu$ with finite expectation, $\varphi$ is bounded (as $|\nabla \ln \mu_0(x)| = |\nabla V(x)| \le C(1+|x|)$ for some $C > 0$). The condition that $\nabla^2 V$ is bounded can be lifted if (9) is replaced by a stronger decay of $b_\mu(x)$.

For instance, in the case (8), the condition (9) holds when $|\nabla_x K(x, y)| \le C/(1+|x-y|^2)$ for some constant $C > 0$ (and similarly for $\nabla_y K$). Indeed, then, considering the first coordinate of (8) (the second one being similar), we bound

$$\left| \int_{\mathbb{R}^p} \nabla_{x_1} K(x_1, y_2) \mu(y_1, y_2) \, dy_1 \, dy_2 \right| \le C \int_{\mathbb{R}^p} \frac{1}{1 + |x - y_2|^2} \mu(y_1, y_2) \, dy_1 \, dy_2 \\ \le \frac{C}{1 + |x|} + C \mathbb{P}_{\mu} \left( |Y_2| \ge |x| - \sqrt{|x|} \right)$$

and the Markov inequality concludes. The fact that a stationary solution of (7) has a finite expectation is implied by Assumption 1 for $\lambda$ small enough, as e.g. $w(x) = |x|^2$ is a Lyapunov function for (7).

Contrary to the linear case, obtaining an LSI for $\mu_*$ is however not sufficient to get the exponential convergence of the solution of (7) toward $\mu_*$, as, even under Assumption 1, several stationary solutions may exist [15]. However, this is sufficient to conclude in the weak interaction regime (i.e. when $\lambda$ is small enough), provided the interaction drift is Lipschitz in terms of the non-linearity:

$$\exists B > 0 \text{ s.t. } \forall \nu, \nu', \quad \|b_{\nu} - b_{\nu'}\|_{\infty} \le BW_2(\nu, \nu'), \qquad (10)$$

where the $W_2$-Wasserstein distance between two probability measures $\nu, \nu'$ is defined as

$$W_2(\nu, \nu') = \inf_{\pi \in C(\nu, \nu')} \left( \int_{(\mathbb{R}^d)^2} |x - x'|^2 \pi(dx, dx') \right)^{1/2},$$

with $C(\nu, \nu')$ the set of probability measures on $(\mathbb{R}^d)^2$ with marginal $\nu$ and $\nu'$.
Indeed, in that case, by a classical computation,

$$
\begin{align*}
\partial_t \mathcal{H}(\mu_t | \mu_*) &= \int_{\mathbb{R}^d} \partial_t (\ln \mu_t) \mu_t + \ln \frac{\mu_t}{\mu_*} \partial_t \mu_t \\
&= \partial_t \int_{\mathbb{R}^d} \mu_t + \int_{\mathbb{R}^d} \mathcal{L}_{\mu_t} \left( \ln \frac{\mu_t}{\mu_*} \right) \mu_t \\
&= \int_{\mathbb{R}^d} \mathcal{L}_{\mu_*} \left( \ln \frac{\mu_t}{\mu_*} \right) \mu_t + \int_{\mathbb{R}^d} (\mathcal{L}_{\mu_t} - \mathcal{L}_{\mu_*}) \left( \ln \frac{\mu_t}{\mu_*} \right) \mu_t \\
&= - \int_{\mathbb{R}^d} \left| \nabla \ln \frac{\mu_t}{\mu_*} \right|^2 \mu_t + \lambda \int_{\mathbb{R}^d} \nabla \ln \frac{\mu_t}{\mu_*} \cdot (b_{\mu_*} - b_{\mu_t}) \mu_t \\
&\le -\frac{1}{2} \int_{\mathbb{R}^d} \left| \nabla \ln \frac{\mu_t}{\mu_*} \right|^2 \mu_t + \frac{1}{2} B^2 \lambda^2 \mathcal{W}_2^2(\mu_t, \mu_*),
\end{align*}
$$

where we used Cauchy–Schwarz and (10). Now, the LSI satisfied by $\mu_*$ implies the
Talagrand inequality

$$
\mathcal{W}_2^2(\mu_t, \mu_*) \le C \mathcal{H}(\mu_t | \mu_*)
$$

(where $C$ is the LSI constant of $\mu_*$), see [29]. As a consequence, using that the LSI constant of $\mu_*$ is uniformly bounded over small values of $\lambda$ (since Theorem 1 can be applied with the same constants $L^\varphi$ and $M^\varphi$ for all values of $\lambda \in [0, \lambda_0]$ for any $\lambda_0 > 0$), we get that

$$
\partial_t \mathcal{H}(\mu_t | \mu_*) \le -\varepsilon \mathcal{H}(\mu_t | \mu_*)
$$

for some $\varepsilon > 0$ for $\lambda$ small enough. As a conclusion, we obtain that $\mu_*$ is the unique
stationary solution of (7) and globally attractive.

Notice that (10) holds for the model (8) as soon as $\nabla^2 K$ is bounded, since in
that case,

$$
\left| \int_{\mathbb{R}^p} \nabla_{x_1} K(x_1, y_2) \nu(y_1, y_2) \, dy_1 \, dy_2 - \int_{\mathbb{R}^p} \nabla_{x_1} K(x_1, y_2) \nu'(y_1, y_2) \, dy_1 \, dy_2 \right| \\
\le \|\nabla^2 K\|_{\infty} \int_{(\mathbb{R}^p)^2} |y_2 - y_2'| \pi(dy_1 \, dy_2, dy_1 \, dy_1'),
$$

where $\pi$ is any coupling of $\nu$ and $\nu'$, so that conclusion follows by Cauchy–Schwarz
and taking the infimum over all couplings (the second coordinate of $b_\nu - b_{\nu'}$ being
treated similarly).

1.3 Perturbation approach: the kinetic case

We consider in this section a non-equilibrium Langevin diffusion $Z = (X, V)$ on
$\mathbb{R}^d \times \mathbb{R}^d$ solving

$$
\left\{
\begin{array}{l}
dX_t = V_t dt \\
dV_t = -\nabla U(X_t) dt + G(X_t, V_t) dt - \gamma V_t dt + \sqrt{2\gamma} dB_t
\end{array}
\right.
\qquad (11)
$$

for some $\gamma > 0$, $U \in C^2(\mathbb{R}^d)$, $G \in C^1(\mathbb{R}^{2d}, \mathbb{R}^d)$, as studied in [18, 27]. The particular case where $G$ depends only on $V$ correspond to non-linear friction models, see e.g. [23, 21].

Contrary to the elliptic case, now, the LSI is not equivalent to the entropy decay
(4). However, from the LSI, the (hypocoercive) decay of the entropy along $(P_t)_{t \ge 0}$
(rather than $(P_t^*)_{t \ge 0}$) can be obtained applying Theorems 9 and 10 of [25] even
without the explicit knowledge of the invariant measure.
**Theorem 2.** Assume that $e^{-U}$ is integrable and that the probability measure with density proportional to $e^{-U}$ satisfies an LSI with constant $C_0$. Let

$$ \varphi(x, v) = -\nabla_v G(x, v) + G(x, v) \cdot v. $$

Assume that $\varphi$ is $L^\varphi$-Lipschitz and the drift writes

$$ -\nabla U(x) + G(x, -v) = -Kx + g(x, v) $$

for a positive-definite matrix $K$ whose smallest eigenvalue is $k > 0$, and a function $g: \mathbb{R}^{2d} \to \mathbb{R}$ satisfying

$$ |g(x, v) - g(x', v')| \le \begin{cases} L_1|z - z'| & \text{if } |x - x'| + |v - v'| \le R, \\ L_2|z - z'| & \text{otherwise,} \end{cases} $$

where $|z - z'| = \sqrt{|x - x'|^2 + |v - v'|^2}$ is the Euclidean distance, for some constants $R$, $L_1$, $L_2 \ge 0$. If additionally $19 \max(1, \gamma)L_2 \le \min(1, k)$ and the law of $(X_t, V_t)$ solving (11) converges weakly for all initial condition as $t \to \infty$ to a unique invariant measure $\mu$ on $\mathbb{R}^{2d}$, then $\mu$ satisfies an LSI with a constant $C_{LS} = C_{LS}(C_0, K, L_1, L_2, R, \gamma; L^\varphi)$.

The proof of this result is given in Section 2.2.

**Remark 1.** The assumptions of the kinetic perturbation theorem seem to be more restrictive than the elliptic one. First, the drift in the kinetic case must be the sum of a positive linear transform plus a perturbation term whose oscillation “grows slowly enough” compared to the linear term, while in the elliptic case it only needs to satisfy a weak convexity condition. This is because our proof is based on $W_1$-contraction of diffusion processes and in the kinetic case such contraction is harder to establish (e.g. compare Theorem 13 to [11]). Second, the function $\varphi$ in the kinetic case must be Lipschitz while in the elliptic case it can be the sum of a Lipschitz and a bounded function, due to the fact that the coupling of Theorem 13 does not allow us to obtain total variation bounds (dual to bounded functions) as is done for the elliptic case.

## 1.4 Defective LSI approach

A measure $\mu \in \mathcal{P}(\mathbb{R}^d)$ is said to satisfy a defective log-Sobolev inequality if for all $f \ge 0$ with $\int f d\mu = 1$,

$$ \int_{\mathbb{R}^d} f \ln f \, d\mu \le A \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu + B, \quad (12) $$

for some constants $A, B \ge 0$. From [3, Proposition 5.1.3], such a defective LSI, together with a Poincaré inequality

$$ \forall f \in L^2(\mu), \quad \int_{\mathbb{R}^d} \left( f - \int_{\mathbb{R}^d} f \, d\mu \right)^2 \, d\mu \le C \int_{\mathbb{R}^d} |\nabla f|^2 \, d\mu \quad (13) $$

for some constant $C > 0$, implies an LSI for $\mu$ with constant $A' = A + C(B + 2)/4$ (i.e. (12) but with $A$ replaced by $A'$ and $B$ replaced by $0$).

In some non-reversible elliptic cases, a Poincaré inequality has been established in [26] (see Proposition 6 below). To improve this result into an LSI, it is thus enough to obtain a defective LSI.
In the following for $\alpha, \beta \ge 1$ we write $\|f\|_{\alpha} = (\int |f|^{\alpha} d\mu)^{1/\alpha}$ and

$$ \|P_t\|_{\alpha \to \beta} = \sup\{\|P_t f\|_{\beta} : f \in L^{\alpha}(\mu), \|f\|_{\alpha} = 1\}, $$

where $\mu$ is the invariant measure of the semi-group $P_t$ considered. The semi-group is said to be hypercontractive if there exist $t_0 > 0$, $\alpha < \beta$ such that $\|P_{t_0}\|_{\alpha \to \beta} < \infty$. In that case without loss of generality we can assume that $\alpha = 1$, as the following easily follows from Hölder's inequality (the proof is given in Section 3.1 for completeness):

**Lemma 3.** For all $\alpha, \gamma > 1$,

$$ \|P_t\|_{1 \to \alpha} \le \|P_t\|_{\alpha \to (\gamma\alpha-1)/(\gamma-1)}. $$

In the reversible settings, it is well-known that hypercontractivity implies a defective LSI, see [3]. We show that it is also true in the non-reversible case. For simplicity, we only consider the case where the diffusion matrix $\Sigma$ is constant, since this is anyway the case in [26]. In the reversible case, the proof relies on [3, Proposition 5.2.6], whose proof requires reversibility. In the non-reversible case, we replace this result by the following (proven in Section 3):

**Proposition 4.** Let $(P_t)_{t \ge 0}$ be a diffusion semi-group with invariant measure $\mu$ and generator $b \cdot \nabla + \Sigma : \nabla^2$ where $\Sigma$ is a constant diffusion matrix and $b$ satisfies the one-sided Lipschitz condition

$$ \forall x, y \in \mathbb{R}^d, \quad (b(x) - b(y)) \cdot (x - y) \le L|x - y|^2. \quad (14) $$

Then, for all $f \ge 0$ with $\int_{\mathbb{R}^d} f d\mu = 1$, all $\alpha > 0$ and all $t \ge 0$,

$$ \int_{\mathbb{R}^d} f \ln f \, d\mu \le \frac{\alpha+1}{\alpha} \ln \|P_t\|_{1 \to 1+\alpha} + |\Sigma| \frac{e^{2Lt}-1}{2L} \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu. $$

In the remaining of this section, we focus on the following elliptic case.

**Assumption 2.** The semi-group $(P_t)_{t \ge 0}$, whose generator reads $b \cdot \nabla + \sigma\Delta$ for $\sigma > 0$, admits an invariant measure $\mu$ and there exist $L, \rho, R > 0$ such that

$$ \forall x, y \in \mathbb{R}^d, \quad (b(x) - b(y)) \cdot (x - y) \le \begin{cases} -\rho|x - y|^2 & \text{if } |x| \ge R, \\ L|x - y|^2 & \text{otherwise.} \end{cases} \quad (15) $$

Note that this assumption is different from (3), which we imposed for the perturbation result in the elliptic case. Under this assumption, hypercontractivity follows from the Harnack inequality established by Wang in [35] (originally stated in the reversible case but the proof, recalled in Section 3.2, is unchanged in the non-reversible one). More specifically, we get the following.

**Proposition 5.** Let $\beta > \alpha > 1$. Under Assumption 2, set

$$ t_0 = \frac{2\beta}{\sigma^2 \rho(\alpha - 1)}. $$

Then, for all $t > t_0$,

$$ \|P_t\|_{\alpha \to \beta} \le (1 + 4d + 2(L + \rho)R^2) \exp \left( \frac{\beta LRt}{2\sigma^2(\alpha - 1)} + \frac{1}{8} \max \left( \frac{1 + 4d}{t/t_0 - 1}, 2\rho R^2 \right) \right). $$
Combining Propositions 4 and 5 gives a defective LSI (see Corollary 12). As a conclusion, we recall the following result from [26, Theorems 1 and 2].

**Proposition 6.** Under Assumption 2, assume furthermore that

$$ \sigma \ge \sigma_0 := (2L + \rho) \frac{(2L + \rho/2)R_*^2 + 2 \sup\{-x \cdot b(x), |x| \le R_*\}}{\rho d}, \quad (16) $$

where $R_* = R(2 + 2L/\rho)^{1/d}$. Then $\mu$ satisfies the Poincaré inequality (13) with constant

$$ C = \frac{4\sigma}{\rho} \left( 1 + \frac{\alpha(2L + \rho)R_*^2}{4d\sigma} \right). \quad (17) $$

Thanks to [3, Proposition 5.1.3], the defective LSI of Corollary 12 and the Poincaré inequality of Proposition 6 yields the following.

**Corollary 7.** Under Assumption 2, provided furthermore (16), $\mu$ satisfies an LSI with constant $C_{LS} = A + C(B+2)/4$ where A, B, C are respectively given in (27), (28), (17).

As in Section 1.2, in the present elliptic case, the LSI is equivalent to the entropy decay (4).

# 2 Proofs

## 2.1 The elliptic case

Before proving the theorem, let us first show a key lemma on the value function of stochastic optimal control problems.

**Lemma 8.** Let $U \subset \mathbb{R}^d$. Under the conditions of Theorem 1, consider the stochastic optimal control problem,

$$ V(T, x) = \sup_{\nu} \sup_{\alpha: \alpha_t \in U} \mathbb{E} \left[ \int_0^T \left( \varphi(X_t^{\alpha, x}) - |\alpha_t|^2 \right) dt \right], $$

where $\nu = (\Omega, F, (\mathcal{F}_t), \mathbb{P}, (B_t))$ stands for a filter probability space with the usual conditions and an $(\mathcal{F}_t)$-Brownian motion, $\alpha$ is an $\mathbb{R}^d$-valued progressively measurable process such that $\int_0^T \mathbb{E}[|\alpha_t|^m] dt$ is finite for every $m \in \mathbb{N}$, and $X^{\alpha, x}$ solves

$$ X_0^{\alpha, x} = x, \quad dX_t^{\alpha, x} = (\tilde{b}(X_t^{\alpha, x}) + 2\alpha_t) dt + \sqrt{2} dB_t. $$

Then there exists $C' > 0$, depending only on $\rho$, $L$, $R$, such that for every $x, y \in \mathbb{R}^d$, every $T > 0$ and every $t \in (0, T]$, we have

$$ |V(T, x) - V(T, y)| \le 2M^\varphi t + C' \left( \frac{2M^\varphi}{t} + L^\varphi \right) |x - y|. $$

*Proof.* The method has been demonstrated in [7] and we give a proof for the sake of completeness. Fix $\varepsilon > 0$ and $(T, x) \in (0, +\infty) \times \mathbb{R}^d$. Take an $\varepsilon$-optimal control $(\nu^\varepsilon, \alpha^\varepsilon)$ such that

$$ V(T, x) \le \mathbb{E} \left[ \int_0^T \left( \varphi(X_t) - |\alpha_t^\varepsilon|^2 \right) dt \right] + \varepsilon, $$
where we denote $X = X^{\alpha^{\varepsilon},x}$. Construct the process Y solving

$$Y_0 = y, \quad dY_t = (\tilde{b}(Y_t) + 2\alpha_t^{\varepsilon}) dt + \sqrt{2}(1 - 2e_t e_t^{\mathrm{T}}) dB_t,$$

until $\tau := \inf\{t : X_t = Y_t\}$ and $Y_t = X_t$ henceforth, where $e_t$ is defined by

$$e_t = \begin{cases} \frac{X_t - Y_t}{|X_t - Y_t|} & \text{if } X_t \neq Y_t, \\ (1, 0, \dots, 0)^{\mathrm{T}} & \text{otherwise.} \end{cases}$$

Then the difference process $\delta X_t := X_t - Y_t$ solves

$$d\delta X_t = (\tilde{b}(X_t) - \tilde{b}(Y_t)) dt + 2\sqrt{2}e_t e_t^{\mathrm{T}} dB_t.$$

Thanks to the weak convexity condition (3), there exist $C \ge 0$ and $\kappa > 0$ such that

$$\mathbb{E}[|X_t - Y_t|] \le Ce^{-\kappa t}|x - y|,$$

$$\mathbb{P}[X_t \neq Y_t] \le \frac{Ce^{-\kappa t}}{t} |x - y|,$$

where the first inequality is due to Eberle [11] and the second to the sticky coupling [13, Theorem 3]. By the definition of $V$, we have

$$V(T, y) \ge \mathbb{E} \left[ \int_{0}^{T} \left( \varphi(Y_t) - |\alpha_t^{\varepsilon}|^2 \right) dt \right].$$

Hence by subtracting the expressions for $V(T, x)$ and $V(T, y)$, we obtain

$$\begin{aligned} V(T, x) - V(T, y) &\le \left( \int_{0}^{t} + \int_{t}^{T} \right) \mathbb{E}[\varphi_{1}(X_{s}) - \varphi_{1}(Y_{s})] ds + \int_{0}^{T} \mathbb{E}[\varphi_{2}(X_{s}) - \varphi_{2}(Y_{s})] ds \\ &\le 2M^{\varphi}t + \frac{C}{\kappa} \left( \frac{2M^{\varphi}}{t} + L^{\varphi} \right) |x - y| + \varepsilon. \end{aligned}$$

Taking $\varepsilon \to 0$ gives the desired upper bound for $V(T, x) - V(T, y)$. The lower bound follows by exchanging $x$ and $y$. $\square$

Now we present the proof for the perturbation result in the elliptic case. We will use the notion of viscosity solution and we refer readers to [8, Section 8] for its definition.

*Proof of Theorem 1.* Under the conditions of Theorem 1, consider $m_t = \text{Law}(Z_t)$ where $Z$ solves (1) with initial distribution $m_0 = \mu_0$. Then $m_t, \mu_0$ solve respectively

$$\begin{aligned} \partial_t m_t &= -\nabla \cdot (bm_t) + \Delta m_t, \\ 0 &= \partial_t \mu_0 = -\nabla \cdot (b_0 \mu_0) + \Delta \mu_0 \end{aligned}$$

where the first equation holds in the sense of distributions a priori. By approximation arguments, we can show that $(t, x) \mapsto m_t(x)$ is continuous and a viscosity solution to the first equation. Define the relative density $h_t = m_t/\mu_0$. Then, it is a viscosity solution to

$$\partial_t h_t = \Delta h_t + \tilde{b}_t \cdot \nabla h_t + \varphi h_t, \quad (18)$$
where $\varphi = -\nabla \cdot b_1 + b_1 \cdot \nabla \ln \mu_0$ and $\tilde{b} = -b + 2\nabla \ln \mu_0$. Notice that the value of $h_t$ can be given by the Feynman-Kac formula

$$h_t(x) = \mathbb{E} \left[ \exp \left( \int_0^t \varphi(X_s^{t,x}) \mathrm{d}s \right) h_0(X_t^{t,x}) \right],$$

where $X^{t,x}$ solves

$$X_0^{t,x} = x, \quad dX_s^{t,x} = \tilde{b}_{t-s}(X_s^{t,x}) \mathrm{d}s + \sqrt{2} dB_t \quad \text{for } s \in [0, t].$$

Suppose additionally that $\varphi$ is bounded and Lipschitz continuous. Then applying synchronous coupling to the Feynman-Kac formula above, we obtain a constant $M > 0$ such that

$$M^{-1} \le h(t, x) \le M \quad \text{and} \quad |h(t, x) - h(s, y)| \le M(|t - s|^{1/2} + |x - y|)$$

for every $t, s \in [0, T]$ and every $x, y \in \mathbb{R}^d$. Taking the logarithm $u_t := \ln h_t$ and using the fact that $h \mapsto \ln h$ is a strictly increasing and $C^2$ mapping, we obtain that $u_t$ is a bounded and uniformly continuous viscosity solution to the HJB equation,

$$\partial_t u_t = \Delta u_t + |\nabla u_t|^2 + \tilde{b}_t \cdot \nabla u_t + \varphi. \quad (19)$$

The rest of the proof then amounts to linking the HJB equation to the stochastic optimal control problem considered in Lemma 8.

For $N \in \mathbb{N}$, consider the approximative HJB equation,

$$\partial_t u_t^N = \Delta u_t^N + \sup_{\alpha:|\alpha| \le N} \{2\alpha \cdot \nabla u_t^N - |\alpha|^2\} + \tilde{b} \cdot \nabla u_t^N + \varphi, \quad (20)$$

and the associated control problem,

$$V^N(T, x) = \sup_{\nu} \sup_{\alpha:|\alpha_t| \le N} \mathbb{E} \left[ \int_0^T \left( \varphi(X_t^{\alpha,x}) - |\alpha_t|^2 \right) \mathrm{d}t \right], \quad (21)$$

where $\nu, \alpha$ satisfy the conditions in the statement of Lemma 8. By Theorem IV.7.1 and the results in Sections V.3 and V.9 of [14], the value function $V^N$ defined by (21) is a bounded and uniformly continuous viscosity solution to (20). Applying Lemma 8 to the approximative problem (21), we obtain a constant $C' > 0$ such that

$$|V^N(t, x) - V^N(t, y)| \le C' \|\varphi\|_{\mathrm{Lip}} |x - y|, \\ |V^N(T, x) - V^N(T, y)| \le 2M^\varphi t + C' \left( \frac{2M^\varphi}{t} + L^\varphi \right) |x - y|$$

for every $t \in (0, T]$ and every $x, y \in \mathbb{R}^d$. Hence if $w \in C^{1,2}([0, T) \times \mathbb{R}^d)$ is such that $V^N - w$ attains a local maximum or a local minimum at $(t, x) \in [0, T) \times \mathbb{R}^d$, then $|\nabla w(t, x)| \le C' \|\varphi\|_{\mathrm{Lip}}$ by the first inequality above. This implies that $V^N$ is actually a viscosity solution to the original (19) for $N \ge C' \|\varphi\|_{\mathrm{Lip}}$. Since both $u$ and $V^N$ are bounded and uniformly continuous on $[0, T] \times \mathbb{R}^d$, we can apply the parabolic comparison for viscosity solutions on the whole space [10, Theorem 1] to obtain $V^N(T, x) = u_T(x)$ for $N$ sufficiently large. Therefore, for every $T > 0$, every $t \in (0, T]$ and every $x, y \in \mathbb{R}^d$, we have

$$|u_T(x) - u_T(y)| \le 2M^\varphi t + C' \left( \frac{2M^\varphi}{t} + L^\varphi \right) |x - y|. \quad (22)$$
Now we remove the additional assumption on $\varphi$ and take a sequence of $\varphi^n = \varphi^{n,1} + \varphi^{n,2}$ such that each of $\varphi^n$ is bounded and Lipschitz continuous, $\|\varphi^{n,1}\|_{\infty} \le M\varphi$, $\|\varphi^{n,2}\|_{\text{Lip}} \le L\varphi$ for all $n \in \mathbb{N}$, and $\varphi^n \to \varphi$ locally uniformly. For each $n$, consider the equation

$$ \partial_t h_t^n = \Delta h_t^n + \dot{b} \cdot \nabla h_t^n + \varphi^n h_t^n $$

and let $h^n$ be the solution given by the Feynman-Kac formula with the initial condition $h_0^n = 1$. Taking the limit $n \to +\infty$ in the Feynman-Kac formulas and using the dominated convergence theorem, we obtain that $h_T^n \to h_T$ pointwise. Yet, each $u_T^n := \ln h_T^n$ satisfies the bound (22) when $u$ is replaced by $u^n$. So taking the limit, we obtain that (22) still holds without the additional assumption on $\varphi$.

Denote the Gaussian kernel in $\mathbb{R}^d$ by $g^\varepsilon = (2\pi\varepsilon)^{-d/2} \exp(-|x|^2/2\varepsilon)$. We decompose $u_T$ in the following way:

$$ u_T = u_T \star g^\varepsilon + (u_T - u_T \star g^\varepsilon). $$

Thanks to (22), we find that the first term is uniformly Lipschitz, and the second term is uniformly bounded. Then we apply successively the Holley-Stroock and Aida-Shigekawa perturbation lemmas [16, 1], and obtain that the flow of measures

$$ (m_T)_{T \ge 1} = (\mu_0 \exp u_T)_{T \ge 1} $$

satisfies a uniform log-Sobolev inequality (see [4, Theorem 2.7] for an explicit constant for Aida-Shigekawa). Noticing that the LSI is stable under the weak convergence of measures, we take the limit $T \to +\infty$ and conclude. □

**Remark 2.** We exploit the properties of viscosity solution to the HJB equation (19) instead of classical solution, contrary to what is done by Conforti [7]. The main reason for this is that we wish to be able to treat the kinetic, therefore degenerate elliptic, case in the same framework, for which the existence of classical solution, despite the system's hypoellipticity, is lacking in classical literatures of stochastic optimal control to our knowledge.

## 2.2 The kinetic case

As in the previous section, we first establish a lemma on the kinetic stochastic optimal control problem.

**Lemma 9.** Let $U \subset \mathbb{R}^d$. Under the conditions of Theorem 2, consider the stochastic optimal control problem,

$$ V(T, z) = \sup_{\nu} \sup_{\alpha: \alpha_t \in U} \mathbb{E} \left[ \int_0^T (\varphi(Z_t^{\alpha, z}) - \gamma |\alpha_t|^2) \, dt \right], $$

where $\nu = (\Omega, F, (\mathcal{F}), \mathbb{P}, (B))$ stands for a filter probability space with the usual conditions and an $(\mathcal{F})$-Brownian motion, $\alpha$ is an $\mathbb{R}^d$-valued progressively measurable process such that $\int_0^T E[|\alpha_t|^m] \, dt$ is finite for every $m \in \mathbb{N}$, and $Z^{\alpha, z} = (X^{\alpha, z}, Y^{\alpha, z})$ solves

$$ Z_0^{\alpha, x} = z, \quad \begin{cases} dX_t^{\alpha, z} = V_t^{\alpha, z} \, dt, \\ dV_t^{\alpha, z} = (-\gamma V_t^{\alpha, z} - \nabla U(X_t^{\alpha, z}) + G(X_t^{\alpha, z}, -V_t^{\alpha, z}) + 2\gamma \alpha_t) \, dt + \sqrt{2\gamma} dB_t. \end{cases} $$
Then there exists $C' > 0$, depending only on $K, L_1, L_2, R, \gamma$, such that for every $z, z' \in \mathbb{R}^d$, we have

$$|V(T, z) - V(T, z')| \le C' L^{\varphi} |z - z'|.$$

*Proof.* Fix $\varepsilon > 0$ and $(T, z) \in (0, +\infty) \times \mathbb{R}^{2d}$. Take an $\varepsilon$-optimal control $(\nu^{\varepsilon}, \alpha^{\varepsilon})$ such that

$$V(T, z) \le \mathbb{E} \left[ \int_0^T (\varphi(Z_t) - \gamma |\alpha_t^{\varepsilon}|^2) dt \right] + \varepsilon,$$

where we denote $(X, V) = Z = Z^{\alpha^{\varepsilon}, z}$. Using $\gamma t$ as the new time variable and $\gamma^{-1}X$ as the new space variable, and noticing that $-\nabla U(x) + G(x, -v) = -Kx + g(x, v)$, we can apply Theorem 13 in the appendix to construct the processes $Z'_n = (X'_n, Y'_n)$ solving

$$Z'_{n,0} = z', \quad \begin{cases} \mathrm{d}X'_{n,t} = V'_{n,t} \mathrm{d}t, \\ \mathrm{d}V'_{n,t} = (-\gamma V'_{n,t} - KX'_{n,t} + g(Z'_{n,t}) + 2\gamma\alpha_t) \mathrm{d}t + \sqrt{2\gamma} \mathrm{d}B'_{n,t}, \end{cases}$$

where $B'_n$ are Brownian motions, and there exist constant $C_1 \ge 1$, $\kappa > 0$ such that

$$\limsup_{n \to +\infty} \mathbb{E}[|Z_t - Z'_{n,t}|] \le C_1 e^{-\kappa t} \rho(z, z') \quad \text{for } t \ge 0.$$

By the definition of $V$, we have

$$V(T, z') \ge \mathbb{E} \left[ \int_0^T (\varphi(Z'_{n,t}) - \gamma |\alpha_t^{\varepsilon}|^2) dt \right].$$

Hence by subtracting the expressions for $V(T, z)$ and $V(T, z')$, we obtain

$$V(T, z) - V(T, z') \le \int_0^T \mathbb{E}[\varphi(Z_s) - \varphi(Z'_{n,s})] ds + \varepsilon \le L^{\varphi} \int_0^T \mathbb{E}[|Z_s - Z'_{n,s}|] ds + \varepsilon,$$

By Fatou's lemma we have

$$\liminf_{n \to +\infty} \int_0^T \mathbb{E}[|Z_s - Z'_{n,s}|] ds \le \int_0^T \liminf_{n \to +\infty} \mathbb{E}[|Z_s - Z'_{n,s}|] ds \le \int_0^T C_1 e^{-\kappa t} \rho(z, z') ds \\ < \frac{C_1}{\kappa} \rho(z, z').$$

So taking $\varepsilon \to 0$ and exchanging $x$ and $y$, we obtain

$$|V(T, z) - V(T, z')| \le \frac{C_1 L^{\varphi}}{\kappa} \rho(z, z').$$

Setting the interpolation $z_s = (1-s)z + sz'$, by the previous inequality we have

$$\begin{aligned} |V(T, z) - V(T, z')| &= \sum_{i=0}^{N-1} |V(T, z_{(i+1)/N}) - V(T, z_{i/N})| \\ &\le \frac{C_1 L^{\varphi}}{\kappa} \sum_{i=0}^{N-1} \rho(z_{(i+1)/N}, z_{i/N}) \\ &= \frac{C_1 L^{\varphi} |z - z'|}{\kappa N} \sum_{i=0}^{N-1} \frac{1}{N} \frac{\rho(z_{(i+1)/N}, z_{i/N})}{|z_{(i+1)/N} - z_{i/N}|}. \end{aligned}$$
As the uniform convergence $\limsup_{z \to z'} \rho(z, z')/|z - z'| \le C_2$ holds, we have

$$ \limsup_{N \to +\infty} \frac{1}{N} \sum_{i=0}^{N-1} \frac{\rho(z_{(i+1)/N}, z_i/N)}{|z_{(i+1)/N} - z_i/N|} \le C_2. $$

Thus taking the limit $N \to +\infty$ in the inequality above concludes the proof. □

*Proof of Theorem 2.* Under the conditions of Theorem 2, consider $m_t = \text{Law}(Z_t)$ where $Z$ solves (11) with the initial distribution

$$ m_0(dx \, dv) = \mu_0(dx \, dv) \propto \exp\left(-U(x) - \frac{1}{2}|v|^2\right) dx \, dv, $$

which is the unique invariant measure of the diffusion (11) when $G = 0$. By the tensorization property, $\mu_0$ satisfies an LSI with constant $\max(1, C_0)$. The measures $m_t$, $\mu_0$ solve respectively

$$ \begin{aligned} \partial_t m_t &= \gamma \Delta_v m_t + \nabla_v \cdot [m_t(\gamma v + \nabla U - G(x, v))] - v \cdot \nabla_x m_t, \\ 0 &= \partial_t \mu_0 = \gamma \Delta_v \mu_0 + \nabla_v \cdot [\mu_0(\gamma v + \nabla U)] - v \cdot \nabla_x \mu_0, \end{aligned} $$

where the first equation holds in the sense of viscosity. Define the relative density $h_t = m_t/\mu_0$. Then, it is a viscosity solution to

$$ \partial_t h_t = \gamma \Delta_v h_t + (-\gamma v + \nabla U(x) - G(x, v)) \cdot \nabla_v h - v \cdot \nabla_x h + \varphi h, \quad (23) $$

where $\varphi = -\nabla_v G(x, v) + G(x, v) \cdot v$. Taking the logarithm $u_t := \ln h_t$ and using the fact that $h \mapsto \ln h$ is a strictly increasing and $C^2$ mapping, we obtain that $u_t$ is a viscosity solution to the kinetic HJB equation,

$$ \partial_t u_t = \gamma \Delta_v u_t + \gamma |\nabla_v u_t|^2 + (-\gamma v + \nabla U(x) - G(x, v)) \cdot \nabla_v u - v \cdot \nabla_x u + \varphi. \quad (24) $$

Now, on the formal level the kinetic HJB equation (24) is related to the optimal control problem considered in Lemma 9: if the domain of control in the lemma is unrestricted, i.e. $U = \mathbb{R}^d$, then we expect to have

$$ u_T(x, v) = V(T, x, -v). $$

We then argue as in the proof of Theorem 1 (suppose $\varphi$ is regular enough, then restrict the domain of control, finally approximate for general $\varphi$) to validate this claim. Then by Lemma 9, for every $z, z' \in \mathbb{R}^{2d}$, we have

$$ |u_T(z) - u_T(z')| \le C' L^\varphi |z - z'|. $$

We conclude as in the end of the proof of Theorem 1. □

# 3 Defective log-Sobolev inequality

## 3.1 From Hypercontractivity to defective LSI

*Proof of Proposition 4.* From [24, Theorem 1], the one-sided Lipschitz condition (14) implies that $|\nabla P_t f| \le e^{tL} P_t |\nabla f|$ for all $t \ge 0$ for all $f$. Then, classically, for
$f \ge 0$ with $\int f d\mu = 1$,

$$
\begin{align*}
\int_{\mathbb{R}^d} P_t f \ln P_t f \, d\mu &= \int_{\mathbb{R}^d} f \ln f \, d\mu - \int_0^t \int_{\mathbb{R}^d} \frac{|\Sigma^{1/2} \nabla P_s f|^2}{P_s f} \, d\mu \, ds \\
&\geq \int_{\mathbb{R}^d} f \ln f \, d\mu - |\Sigma| \int_0^t \int_{\mathbb{R}^d} \frac{e^{2sL} (P_s |\nabla f|)^2}{P_s f} \, d\mu \, ds \\
&\geq \int_{\mathbb{R}^d} f \ln f \, d\mu - |\Sigma| \int_0^t \int_{\mathbb{R}^d} \frac{e^{2sL} |\nabla f|^2}{f} \, d\mu \, ds,
\end{align*}
$$

where $(P_s(|\nabla f|))^2 \le P_s(|\nabla f|^2/f)P_s(f)$ (by Cauchy-Schwarz) and the invariance
of $\mu$ by $P_s$ were used in the last inequality. In other words,

$$
\begin{align*}
\int_{\mathbb{R}^d} f \ln f \, d\mu &\le \int_{\mathbb{R}^d} P_t f \ln P_t f \, d\mu + |\Sigma| \frac{e^{2Lt} - 1}{2L} \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu \\
&= \frac{1}{\alpha} \int_{\mathbb{R}^d} P_t f \ln(P_t f)^\alpha \, d\mu + |\Sigma| \frac{e^{2Lt} - 1}{2L} \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu,
\end{align*}
$$

for every $\alpha > 0$. By Jensen's inequality applied to the probability measure $P_t f \mu$,

$$
\int_{\mathbb{R}^d} P_t f \ln(P_t f)^\alpha \, d\mu \le \ln \int_{\mathbb{R}^d} (P_t f)^{1+\alpha} \, d\mu \le (1+\alpha) \ln \|P_t\|_{1 \to 1+\alpha},
$$

since $\|P_t f\|_1 = 1$.

<figure><img src="image_2.png" /></figure>

Proof of Lemma 3. Using Hölder's inequality, for $f \ge 0$ with $\int f d\mu = 1$ (so that $\|P_t f\|_1 = 1$ since $\mu$ is invariant by $P_t$),

$$
\begin{align*}
\|P_t f\|_{\alpha}^{\alpha} &\le \|P_t f\|_{1}^{1/\gamma} \|P_t f\|_{(\gamma\alpha-1)/(\gamma-1)}^{(\gamma\alpha-1)/\gamma} \\
&\le \|P_t\|_{\alpha \to (\gamma\alpha-1)/(\gamma-1)}^{(\gamma\alpha-1)/\gamma} \|P_t f\|_{\alpha}^{(\gamma\alpha-1)/\gamma}.
\end{align*}
$$

Dividing by $\|P_t f\|_{\alpha}^{(\gamma\alpha-1)/\gamma}$ concludes.

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

3.2 Hypercontractivity in the elliptic case

Next, we recall (here in a non-reversible settings – which doesn’t change the proof
– and only in the flat space ; also with explicit constants) the Harnack inequality
of [35].

**Proposition 10.** *Assume that there exists K > 0 such that*

$$
\forall x, y \in \mathbb{R}^d, \quad (x - y) \cdot (b(x) - b(y)) \le K |x - y|. \qquad (25)
$$

Then, for all $t \ge 0$, all $x$, $y \in \mathbb{R}^d$ and all $\alpha > 1$,

$$
(P_t f(y))^\alpha \le (P_t f^\alpha)(x) \exp\left(\frac{\alpha}{2\sigma^2(\alpha-1)}\left(K^2t + \frac{|x-y|^2}{t}\right)\right).
$$

Notice that, in particular, Assumption 2 implies (25) with $K = LR$.
Proof. For two initial conditions $x \neq y$ and a final time $T > 0$, let $X, Y$ solve

$$ 
\begin{aligned}
X_0 &= x, & \mathrm{d}X_t &= b(X_t) \mathrm{d}t + \sigma \mathrm{d}B_t, \\
Y_0 &= y, & \mathrm{d}Y_t &= b(Y_t) \mathrm{d}t + \sigma \mathrm{d}B_t + \xi e_t \mathrm{d}t,
\end{aligned}
$$

where $e_t = (X_t - Y_t)/|X_t - Y_t|$ for $t < \tau := \inf\{s \ge 0, X_s = Y_s\}$ and $e_t = 0$ for $t \ge \tau$ (so that in particular $X_t = Y_t$ for $t \ge \tau$) and $\xi = K + |x - y|/T$.

Since the norm is $C^2$ outside the origin, we can apply Itô's formula up to time $\tau$ to get, for $t < \tau$,

$$ 
\begin{aligned}
\mathrm{d}|X_t - Y_t| &= e_t \cdot (b(X_t) - b(Y_t)) \mathrm{d}t - \xi_t \mathrm{d}t \\
&\leq -\frac{|x - y|}{T} \mathrm{d}t.
\end{aligned}
$$

This implies that $\tau \le T$, and thus $X_T = Y_T$. By Girsanov's theorem,

$$ P_T f(y) = \mathbb{E}[f(Y_T)R], \quad \text{with} \quad R = e^{\frac{\xi}{\sigma} \int_0^{\tau} e_t \cdot \mathrm{d}B_t - \frac{\xi^2}{2\sigma^2} \tau}, $$

so that, by Hölder's inequality, for $f \ge 0$ and $\alpha > 1$, using that $Y_T = X_T$,

$$ (P_T f(y))^\alpha \le (P_T f^\alpha)(x) (\mathbb{E} R^{\alpha/(\alpha-1)})^{\alpha-1} $$

with

$$ (\mathbb{E} R^{\alpha/(\alpha-1)})^{\alpha-1} \le \exp\left(\frac{\alpha}{4\sigma^2(\alpha-1)}\xi^2 T\right) \le \exp\left(\frac{\alpha}{2\sigma^2(\alpha-1)}\left(K^2 T + \frac{|x-y|^2}{T}\right)\right). \quad \square $$

**Lemma 11.** Under Assumption 2,

$$ \int_{\mathbb{R}^d} e^{\delta|x-y|^2} \mu(dx) \mu(dy) \le (1 + 4d + (2L + 8\delta)R^2) \exp\left(\delta \max\left(\frac{1+4d}{2(\rho-4\delta)}, R^2\right)\right), $$

for all $\delta \in (0, \rho/4)$.

Proof. Let $\mathcal{L}_2$ be the generator of two independent diffusion processes satisfying (1), so that $\mu \otimes \mu$ is invariant by $\mathcal{L}_2$. For $\delta \in (0, \rho/4)$, consider the function $V(x, y) = e^{\delta|x-y|^2}$. Then

$$ 
\begin{aligned}
\frac{\mathcal{L}_2 V(x, y)}{V(x, y)} &= 2\delta(x - y) \cdot (b(x) - b(y)) + 4\delta d + 8\delta^2 |x - y|^2 \\
&\le \begin{cases} 4\delta d + (8\delta^2 - 2\delta\rho)|x - y|^2 & \text{if } |x - y| \ge R \\ 4\delta d + (2\delta L + 8\delta^2)R^2 & \text{otherwise} \end{cases} \\
&\le -\delta \mathbb{1}_{|x-y| \ge R_*} + \delta (4d + (2L + 8\delta)R^2) \mathbb{1}_{|x-y| < R_*}
\end{aligned}
$$

with

$$ R_*^2 = \max\left(\frac{1+4d}{2(\rho-4\delta)}, R^2\right). $$

Hence

$$ 
\begin{aligned}
\mathcal{L}_2 V(x, y) &\le -\delta V(x, y) + \delta(1 + 4d + (2L + 8\delta)R^2) \mathbb{1}_{|x-y| \le R_*} V(x, y) \\
&\le -\delta V(x, y) + \delta(1 + 4d + (2L + 8\delta)R^2) e^{\delta R_*^2}.
\end{aligned}
$$
Integrating with respect to $\mu \otimes \mu$, the left hand side vanishes and we get

$$ \int_{\mathbb{R}^d} V(x, y)\mu(dx)\mu(dy) \le (1 + 4d + (2L + 8\delta)R^2)e^{\delta R_*^2}, $$

as announced.

<figure><img src="image_4.png" /></figure>

*Proof of Proposition 5.* Let $f \ge 0$ be such that $\mu(f^\alpha) = 1$. By Proposition 10 (with $K = LR$), for any $y \in \mathbb{R}^d$,

$$ \begin{aligned} 1 &= \int_{\mathbb{R}^d} P_t(f^\alpha)(x)\mu(dx) \\ &\ge (P_t f(y))^\alpha \int_{\mathbb{R}^d} \exp\left(\frac{-\alpha}{2\sigma^2(\alpha-1)}\left(K^2t + \frac{|x-y|^2}{t}\right)\right)\mu(dx). \end{aligned} $$

As a consequence, for $\beta > \alpha$,

$$ \begin{aligned} \int_{\mathbb{R}^d} (P_t f(y))^\beta \mu(dy) &\le \int_{\mathbb{R}^d} \left[ \int_{\mathbb{R}^d} \exp\left(\frac{-\alpha}{2\sigma^2(\alpha-1)}\left(Kt + \frac{|x-y|^2}{t}\right)\right) \mu(dx) \right]^{-\beta/\alpha} \mu(dy) \\ &\le \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \exp\left(\frac{\beta}{2\sigma^2(\alpha-1)}\left(Kt + \frac{|x-y|^2}{t}\right)\right) \mu(dx)\mu(dy). \end{aligned} $$

Conclusion follows from (11) and using that $t/t_0 \le 1$. □

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

To conclude, gathering Propositions 4 and 5, we get the following:

**Corollary 12.** Assume (15) for some $L$, $R \ge 0$ and $\rho > 0$. Then for all $f \ge 0$ with $\int_{\mathbb{R}^d} f \, d\mu = 1$, we have

$$ \int_{\mathbb{R}^d} f \ln f \le A \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu + B \quad (26) $$

with

$$ A = \frac{\sigma^2}{2L} \left( \exp\left(\frac{24L}{\sigma^2\rho}\right) - 1 \right), \quad (27) $$

$$ B = 6 \ln(1 + 4d + 2(L + \rho)R^2) + \frac{108LR}{\sigma^4\rho} + \frac{3}{4} \max(1 + 4d, 2\rho R^2) \quad (28) $$

(taking for A the limit as $L \to 0$ of this expression if $L = 0$).

*Proof.* For simplicity, take $\alpha = 1$ in Proposition 4, $\alpha = \gamma = 2$ in Lemma 3 and $t = 2t_0$ in Proposition 5, we end up with

$$ \begin{aligned} \int_{\mathbb{R}^d} f \ln f &\le 2 \ln \|P_t\|_{1 \to 2} + \sigma^2 \frac{e^{2Lt} - 1}{2L} \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu \\ &\le 6 \ln \|P_t\|_{2 \to 3} + \sigma^2 \frac{e^{2Lt} - 1}{2L} \int_{\mathbb{R}^d} \frac{|\nabla f|^2}{f} \, d\mu, \end{aligned} $$

and conclusion follows the expression given in Proposition 5. □

<figure><img src="image_6.png" /></figure>
# A Reflection coupling for kinetic diffusions

**Theorem 13** (Coupling by reflection for kinetic diffusions). Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions. Let $X, V$ be $\mathbb{R}^d$-valued continuous and adapted processes, and $\alpha$ be an $\mathbb{R}^d$-valued progressively measurable process solving

$$ \begin{aligned} dX_t &= V_t dt, \\ dV_t &= \alpha_t dt + (-V_t - KX_t + g(X_t, V_t)) dt + \sqrt{2} dB_t, \end{aligned} \quad (29) $$

for $t \ge 0$, where $K$ is a $d \times d$ symmetric and positive-definite matrix, $g: \mathbb{R}^{2d} \to \mathbb{R}^d$ is a Lipschitz continuous function, and $(B_t)_{t \ge 0}$ is an $(\mathcal{F})$-Brownian motion in $d$ dimensions. Let $X'_0, V'_0$ be $\mathbb{R}^d$-valued and $\mathcal{F}_0$-measurable random variables. Denote by $k$ the smallest eigenvalue of $K$. Suppose that $\int_0^T \mathbb{E}[|\alpha_t|^2] dt$ is finite for every $T > 0$, and $X_0, V_0, X'_0, V'_0$ are all square-integrable. If there exist nonnegative constants $R, L_1, L_2$ such that for every $z, z' \in \mathbb{R}^{2d}$, we have

$$ |g(z) - g(z')| \le \begin{cases} L_1 |z - z'| & \text{if } |x - x'| + |v - v'| \le R, \\ L_2 |z - z'| & \text{otherwise,} \end{cases} $$

with $L_2 < \frac{1}{19} \min(1, k)$ and $L_2 \le L_1$, then upon enlarging the probability space, we can construct a sequence of continuous and adapted processes $X'_n, V'_n$ such that

1. their initial values are given by $X'_0, V'_0$, that is, $X'_{n,0} = X'_0$ and $V'_{n,0} = V'_0$;

2. they solve

$$ \begin{aligned} dX'_{n,t} &= V'_{n,t} dt, \\ dV'_{n,t} &= \alpha_t dt + (-V'_{n,t} - KX'_{n,t} + b(X'_{n,t}, V'_{n,t})) dt + \sqrt{2} dB'_{n,t}, \end{aligned} \quad (30) $$

for $(\mathcal{F})$-Brownians $(B'_{n,t})_{t \ge 0}$;

3. and finally, there exists constants $C_1, C_2 \ge 1$, $\kappa > 0$ and a continuous function of quadratic growth $\rho: \mathbb{R}^{2d} \times \mathbb{R}^{2d} \to \mathbb{R}$, all explicitly expressible by $K$, $R, L_1, L_2$, such that

$$ \limsup_{n \to +\infty} \mathbb{E}[|Z_t - Z'_{n,t}|] \le C_1 e^{-\kappa t} \mathbb{E}[\rho(Z_0, Z'_0)] \quad \text{for } t \ge 0, \quad (31) $$

and uniformly in $z'$, we have

$$ \limsup_{z \to z'} \frac{\rho(z, z')}{|z - z'|} \le C_2. $$

**Remark 3.** We develop a translation-invariant version of the additive metric constructed in [12] under which the difference processes, $\delta X = X - X'$, $\delta V = V - V'$, are contractive in average. Our assumptions are an improvement over [20, Theorem 2.16] although we do not elaborate on mean field dependence. Also, we can recover the contraction in $W_1$ distance from the transport cost $\rho$ by a limiting procedure (as is done in the proof of Lemma 9). So our approach can be used to achieve the $W_1$-contraction of [31, Theorem 5] with simpler calculations.
Proof. For $x, x', v, v' \in \mathbb{R}^d$, introduce the variables $q = x + v, q' = x' + v'$ and denote $\delta x = x - x', \delta v = v - v', \delta q = q - q'$. Define

$$r(z, z') = \theta|\delta x| + |\delta q| = \theta|x - x'| + |q - q'|$$

where

$$\theta := 2 \max(|K| + L_1, 1).$$

Denote $r_0 = (\theta + 1)R$.

**Reflection-synchronous coupling.** Fix an $n \in \mathbb{N}$. Let us construct the desired processes $Z'_n = (X'_n, V'_n)$. Find Lipschitz-continuous $rc_n, sc_n : \mathbb{R}^{2d} \times \mathbb{R}^{2d} \to \mathbb{R}$ satisfying $rc_n^2 + sc_n^2 = 1$ and

$$\mathrm{rc}_n(z, z') = \begin{cases} 0 & \text{if } r(z, z') \ge r_0 + n^{-1} \text{ or } |\delta q| \le n^{-1}, \\ 1 & \text{if } r(z, z') \le r_0 \text{ and } |\delta q| \ge 2n^{-1}. \end{cases}$$

Upon enlarging the filtered probability space, we can also find another ($\mathcal{F}$)-Brownian motion $B''$ that is independent from $B$. 1 Let $Z'_n = (X'_n, V'_n)$ solve

$$\begin{aligned} dX'_{n,t} &= V'_{n,t} dt, \\ dV'_{n,t} &= \alpha_t dt + (-V'_{n,t} - KX'_{n,t} + g(X'_{n,t}, V'_{n,t})) dt \\ &\quad + \mathrm{rc}_n(Z_t, Z'_{n,t})(1 - 2e_{n,t}e^T_{n,t})\sqrt{2} dB^{\mathrm{rc}}_{n,t} + \mathrm{sc}_n(Z_t, Z'_{n,t})\sqrt{2} dB^{\mathrm{sc}}_{n,t}, \end{aligned}$$

with initial value $Z'_{n,0} = (X'_0, V'_0)$, where $e_{n,t}$ is defined by

$$e_{n,t} = \begin{cases} \frac{Q_t - Q'_{n,t}}{|Q_t - Q'_{n,t}|} & \text{if } Q_t \ne Q'_{n,t}, \\ (1, 0, \dots, 0)^\mathrm{T} & \text{otherwise,} \end{cases}$$

for $Q_t := X_t + V_t$ and $Q'_{n,t} := X'_{n,t} + V'_{n,t}$, and $B^{\mathrm{rc}}_n, B^{\mathrm{sc}}_n$ are defined by

$$\begin{aligned} dB^{\mathrm{rc}}_{n,t} &= \mathrm{rc}_n(Z_t, Z'_{n,t}) dB_t + \mathrm{sc}_n(Z_t, Z'_{n,t}) dB''_t, \\ dB^{\mathrm{sc}}_{n,t} &= \mathrm{sc}_n(Z_t, Z'_{n,t}) dB_t - \mathrm{rc}_n(Z_t, Z'_{n,t}) dB''_t. \end{aligned}$$

The solution $Z'_n$ to this system of equations is well defined: they have Lipschitz-continuous coefficients in $Z'_n$, or in $X'_n$ and $V'_n$ (although the Lipschitz constants explode when $n \to +\infty$), so the existence and uniqueness of (strong) solution follow from Cauchy-Lipschitz arguments. By Lévy's characterization, $B^{\mathrm{rc}}_n, B^{\mathrm{sc}}_n$ are independent Brownian motions and therefore if we define $B'_n$ by

$$dB'_{n,t} = \mathrm{rc}_n(Z_t, Z'_{n,t})(1 - 2e_{n,t}e^T_{n,t}) dB^{\mathrm{rc}}_{n,t} + \mathrm{sc}_n(Z_t, Z'_{n,t}) dB^{\mathrm{sc}}_{n,t},$$

then $B'_n$ is also a Brownian motion. Hence $Z'_n$ satisfies indeed (30) and it remains only to verify the last claim.

**Difference process.** Denote $\mathrm{rc}_{n,t} = \mathrm{rc}_n(Z_t, Z'_{n,t})$, $\mathrm{sc}_{n,t} = \mathrm{sc}_n(Z_t, Z'_{n,t})$, $\delta X_{n,t} = X_t - X'_{n,t}$, $\delta V_{n,t} = V_t - V'_{n,t}$, $\delta Q_{n,t} = Q_t - Q'_{n,t}$, $\delta g_{n,t} = g(Z_t) - g(Z'_{n,t})$, $dW_{n,t} = e^T_{n,t} dB^{\mathrm{rc}}_{n,t}$ and $r_{n,t} = r(Z_t, Z'_{n,t}) = \theta|\delta X_{n,t}| + |\delta Q_{n,t}|$. In the following we will omit the subscript $n$ in the variables defined above to simplify the notation and recover

<footnote n="1">¹The additional Brownian motion $B''$ can be shared between stochastic processes $Z'_n$ with different index $n$, so we do not need to extend the probability space infinitely.</footnote>
it if necessary. By our construction of Brownian motions, the original Brownian B admits the decomposition

$$dB_t = \mathrm{rc}(Z_t, Z'_t) dB_t^{\mathrm{rc}} + \mathrm{sc}(Z_t, Z'_t) dB_t^{\mathrm{sc}}.$$

Therefore, by taking the difference between the two systems of equations (29) and (30), we find the difference process $\delta Z = (\delta X, \delta V)$ satisfy

$$\begin{aligned} d\delta X_t &= \delta V_t dt, \\ d\delta V_t &= (-\delta V_t - K\delta X_t + \delta g_t) dt + 2\sqrt{2}rc_t e_t dW_t. \end{aligned}$$

We note that the process $\alpha$ disappears in the equations above. Using Itō's formula, we further obtain

$$\begin{aligned} d(\delta X_t)^2 &= 2\delta X_t \cdot \delta V_t dt, \\ d(\delta V_t)^2 &= 2\delta V_t \cdot (-\delta V_t - K\delta X_t + \delta g_t) dt + 8(\mathrm{rc}_t)^2 dt + 4\sqrt{2}\mathrm{rc}_t \delta V_t \cdot e_t dW_t, \\ d(\delta X_t \cdot \delta V_t) &= |\delta V_t|^2 dt + \delta X_t \cdot (-\delta V_t - K\delta X_t + \delta g_t) dt + 2\sqrt{2}\mathrm{rc}_t \delta X_T \cdot e_t dW_t. \end{aligned}$$

We have the semimartingle decomposition $dr_t = dA_t^r + dM_t^r$ where $A^r$ is absolutely continuous with

$$dA_t^r \le (|K| + L_1 - \theta)|\delta X_t| dt + \theta|\delta Q_t| dt,$$

and $M_t^r = 2\sqrt{2}\int_0^t \mathrm{rc}_s dW_s$ is a martingale. Consequently, if $f: [0, +\infty) \to \mathbb{R}$ is a piecewise $C^2$, non-decreasing and concave function, then the Itō–Tanaka formula yields $df(r_t) = dA_t^f + dM_t^f$ where $A^f$ is absolutely continuous with

$$dA_t^f \le f_-(r_t) [(|K| + L_1 - \theta)|\delta X_t| dt + \theta|\delta Q_t|] dt + 4f''(r_t)(\mathrm{rc}_t)^2 dt,$$

and $M^f$ is a martingale.

*Choice of the Lyapunov function G.* Define $G(z, z') = \frac{1}{2}\delta x^T K \delta x + \frac{1}{2}|\delta v|^2 + \eta \delta x \cdot \delta v$, where we set $\eta = \frac{1}{2}\min(1, k)$, and denote $G_t = G(Z_t, Z'_t)$. The function G satisfies the mutual bound

$$\lambda(|\delta x|^2 + |\delta v|^2) \le G(z, z') \le \frac{\theta}{2} (|\delta x|^2 + |\delta v|^2),$$

where

$$\lambda := \frac{1}{4} \min(1, k).$$

We have also

$$\begin{aligned} dG_t = &-((1 - \eta)|\delta V_t|^2 + \eta \delta X_t \cdot \delta V_t + \eta \delta X_t^T K \delta X_t) dt \\ &+ (\delta V_t + \eta \delta X_t) \cdot (\delta g_t dt + 2\sqrt{2}\mathrm{rc}_t e_t dW_t) + 4(\mathrm{rc}_t)^2 dt := dA_t^G + dM_t^G, \end{aligned}$$

where $A^G$, $M^G$ are the finite-variation and the martingale part respectively. In particular, if $|\delta X_t| + |\delta V_t| \ge R$, then

$$\begin{aligned} dA_t^G \le &-(\delta V_t, \delta X_t) \begin{pmatrix} 1 - L_2 - \eta & -L_2 - (1 + L_2)\eta \\ 0 & \eta(K - L_2) \end{pmatrix} \begin{pmatrix} \delta V_t \\ \delta X_t \end{pmatrix} dt + 4(\mathrm{rc}_t)^2 dt \\ = &:-\delta Z_t^T M_G \delta Z_t dt + 4(\mathrm{rc}_t)^2 dt. \end{aligned}$$
We choose $\eta = \frac{1}{2} \min(1, k)$ and the symmetric part of the matrix $M_G$ is positive-definite as its determinant is lower bounded by

$$ (1 - L_2 - \eta) \cdot \eta(k - L_2) - \frac{1}{4}(L_2 + (1 + L_2)\eta)^2 \geq \frac{(\min(1, k) - 19L_2)k}{16}. $$

By the same computation we obtain

$$ M_G \geq \frac{(\min(1, k) - 19L_2)k}{16 \max(1 - L_2 - \eta, \eta(k - L_2))} \geq \frac{(\min(1, k) - 19L_2)k}{8 \max(1 - L_2, k - L_2)}. $$

As a result,

$$ \begin{aligned} dA_t^G - 4(\mathrm{rc}_t)^2 \mathrm{d}t &\leq -\delta Z_t^\mathrm{T} M_G \delta Z_t \mathrm{d}t \leq -\frac{(\min(1, k) - 19L_2)k}{8 \max(1 - L_2, k - L_2)} |\delta Z_t|^2 \mathrm{d}t \\ &\leq -\frac{(\min(1, k) - 19L_2)k}{8 \max(1 - L_2, k - L_2) \max(1, |K|)} G_t \mathrm{d}t := -\kappa_2 G_t \mathrm{d}t. \end{aligned} $$

To summarize, if $|\delta X_t| + |\delta V_t| \geq R$, then $dA_t^G \leq -\kappa_2 G_t \mathrm{d}t + 4(\mathrm{rc}_t)^2 \mathrm{d}t$. In the general case, we have

$$ \begin{aligned} dA_t^G &\leq (|\delta V_t|, |\delta X_t|) \begin{pmatrix} |1 - \eta| + L_1 & L_1 + (1 + L_1)\eta \\ 0 & \eta|K| + L_1 \end{pmatrix} \begin{pmatrix} |\delta V_t| \\ |\delta X_t| \end{pmatrix} \mathrm{d}t + 4(\mathrm{rc}_t)^2 \mathrm{d}t \\ &\leq \theta(|\delta X_t|^2 + |\delta Y_t|^2) \mathrm{d}t + 4(\mathrm{rc}_t)^2 \mathrm{d}t. \end{aligned} $$

Choice of $\rho$ and $f$. Now recover the subscript $n$. Set

$$ \rho_n(z, z') = \varepsilon_n G(z, z') + f_n(\theta|\delta x| + |\delta q|) $$

for $\varepsilon_n > 0$, and piecewise $C^2$, non-decreasing and concave $f_n: [0, +\infty) \to [0, +\infty)$, to be determined below. Denote $\rho_{n,t} = \rho_n(Z_{n,t}, Z'_{n,t})$. Then by the previous computations $d\rho_{n,t} = dA_t^{\rho_n} + dM_t^{\rho_n}$ where $M^{\rho_n}$ is a martingale and $A^{\rho_n}$ is absolutely continuous with

$$ dA_t^{\rho_n} \leq \varepsilon_n dA_t^G + [(|K| + L_1 - \theta)|\delta X_{n,t}| + \theta|\delta Q_{n,t}|] f'_{n,-}(r_{n,t}) \mathrm{d}t + 4f''_n(r_{n,t})(\mathrm{rc}_{n,t})^2 \mathrm{d}t. $$

Define the functions

$$ \varphi(r) = \exp\left(-\frac{\theta r^2}{8}\right), $$

$$ \Phi(r) = \int_{0}^{r} \varphi(u) \mathrm{d}u, $$

$$ g_n(r) = 1 - \frac{\kappa_{n,1}}{2} \int_0^r \Phi(u) \varphi(u)^{-1} \mathrm{d}u - \frac{\varepsilon_n}{2} \int_0^r \left[ \left(1 + \frac{\kappa_1}{2}\right) \theta u^2 + 4 \right] \varphi(u)^{-1} \mathrm{d}u $$

for $r \geq 0$, where $\kappa_{n,1}$, $\varepsilon_n$ are positive constants defined by

$$ \kappa_{n,1} = \frac{1}{2} \left( \int_{0}^{r_0+n^{-1}} \Phi(u) \varphi(u)^{-1} \mathrm{d}u \right)^{-1}, $$

$$ \varepsilon_n = \frac{1}{2} \left( \int_0^{r_0+n^{-1}} \left[ \left(1 + \frac{\kappa_{n,1}}{2}\right) \theta u^2 + 4 \right] \varphi(u)^{-1} \mathrm{d}u \right)^{-1} \wedge \frac{4}{9R}. $$
We choose

$$f_n(r) = \int_0^{r \wedge (r_0 + n^{-1})} \varphi(u)g(u) \, du.$$

Note that $\kappa_{n,1}$, $\varepsilon_n$, $g_n$ and $f_n$ all converge when $n \to +\infty$, and we denote their limit by

$$(\kappa_{*,1}, \varepsilon_*, g_*, f_*) = \lim_{n \to +\infty} (\kappa_1, \varepsilon, g, f).$$

Denote also

$$\rho_*(z, z') = \varepsilon_* G(z, z') + f_* (\theta |\delta x| + |\delta q|).$$

Since $G$ is a quadratic form and $f_*(0) = \varphi(0)g(0) = 1$, we have the uniform upper limit

$$\limsup_{z \to z'} \frac{\rho_*(z, z')}{|z - z'|} = \limsup_{z \to z'} \frac{\theta|x - x'| + |x - x' + v - v'|}{\sqrt{|x - x'|^2 + |v - v'|^2}} \le \theta + \sqrt{2} =: C_2,$$

validating the last property of the last claim. By elementary calculations, we also have

$$\lim_{n \to +\infty} \sup_{z, z' \in \mathbb{R}^{2d}} \frac{|\rho_n(z, z') - \rho_*(z, z')|}{1 + |z - z'|^2} = 0. \quad (32)$$

The function $f_n$ is $C^2$ on $[0, r_0 + n^{-1})$ and $(r_0 + n^{-1}, +\infty)$, non-decreasing, concave, and satisfies

$$4f_n''(r) + \theta f_n'(r)r + \kappa_{n,1}f_n(r) + \varepsilon_n \left[ \left( 1 + \frac{\kappa_{n,1}}{2} \right) \theta r^2 + 4 \right] \le 0$$

for $r \in [0, r_0 + n^{-1}]$. Moreover if $r \in [0, r_0 + n^{-1}]$, then $\frac{1}{2} \le g_n(r) \le 1$. Therefore for every $r \ge 0$, we have

$$\frac{r}{2} \le \frac{\Phi(r)}{2} \le f_n(r) \le \Phi(r).$$

*Proof of contraction.* We now prove the contraction by investigating the following three cases. We temporarily omit the subscript $n$.

1. Suppose $r_t > r_0$. Then we have $|\delta X_t| + |\delta V_t| \ge R$ and therefore

$$\begin{aligned} |\delta X_t|^2 + |\delta V_t|^2 &\ge \frac{R^2}{2} = \frac{R^2}{2f(r_0 + n^{-1})} f(r_0 + n^{-1}) \\ &\ge \frac{R^2}{2\Phi(r_0 + n^{-1})} f(r_0 + n^{-1}) \\ &\ge \frac{R^2}{2\Phi(r_0 + n^{-1})} f(r_t). \end{aligned}$$

As a result, $G_t \ge \lambda R^2 (2\Phi(r_0 + n^{-1}))^{-1} f(r_t)$. Hence,

$$\begin{aligned} dA_t^\rho &\le -\varepsilon\kappa_2 G_t \, dt + (4f''(r_t) + \theta f'_-(r_t)r_t + 4\varepsilon)(rc_t)^2 \, dt \le -\varepsilon\kappa_2 G_t \, dt \\ &\le -\left(\varepsilon + \frac{2\Phi(r_0 + n^{-1})}{\lambda R^2}\right)^{-1} \varepsilon\kappa_2 (f(r_t) + \varepsilon G_t) \, dt \\ &= -\frac{\lambda R^2 \varepsilon}{\lambda R^2 \varepsilon + 2\Phi(r_0 + n^{-1})} \kappa_2 \rho_t \, dt. \end{aligned}$$
2. Suppose $r_t \le r_0$ and $|\delta Q_t| \ge 2n^{-1}$. Then we have $rc_t = 1$ and therefore

$$
\begin{align*}
\mathrm{d}A_t^\rho &\le \varepsilon\theta(|\delta V_t|^2 + |\delta X_t|^2) \mathrm{d}t + (4f''(r_t) + \theta f'_-(r_t)r_t + 4\varepsilon) \mathrm{d}t \\
&\le \varepsilon\theta r_t^2 \mathrm{d}t + (4f''(r_t) + \theta f'_-(r_t)r_t + 4\varepsilon) \mathrm{d}t \\
&\le (4f''(r_t) + \theta f'_-(r_t)r_t + \varepsilon(\theta r_t^2 + 4)) \mathrm{d}t \\
&\le -\kappa_1 \left(f(r_t) + \frac{\varepsilon\theta}{2}r_t^2\right) \mathrm{d}t \\
&\le -\kappa_1 \left(f(r_t) + \varepsilon G_t\right) \mathrm{d}t = -\kappa_1 \rho_t \mathrm{d}t.
\end{align*}
$$

3. Suppose $r_t \le r_0$ and $|\delta Q_t| < 2n^{-1}$. Then we have

$$
|\delta V_t|^2 \le 2|\delta X_t|^2 + 2|\delta Q_t|^2 \le 2|\delta X_t|^2 + 8n^{-2}.
$$

and

$$
|\delta X_t| \le \theta^{-1} r_t \le \theta^{-1} r_0 = \frac{\theta + 1}{\theta} R \le \frac{3}{2} R.
$$

Consequently,

$$
\begin{align*}
\mathrm{d}A_t^\rho &\le \varepsilon\theta(|\delta V_t|^2 + |\delta X_t|^2) \mathrm{d}t - \frac{\theta}{2}|\delta X_t| \mathrm{d}t + 2\theta n^{-1} \mathrm{d}t + (4f''(r_t) + 4\varepsilon)(\mathrm{rc}_t)^2 \mathrm{d}t \\
&\le \varepsilon\theta(3|\delta X_t|^2 + 8n^{-2}) \mathrm{d}t - \frac{\theta}{2}|\delta X_t| \mathrm{d}t + 2\theta n^{-1} \mathrm{d}t \\
&\le -\frac{\theta}{4}|\delta X_t| \mathrm{d}t + (2\theta n^{-2} + 8\theta\varepsilon n^{-2}) \mathrm{d}t.
\end{align*}
$$

Since

$$
\begin{align*}
f(r_t) &\le \sup_{r \in (0, r_0]} \frac{f(r)}{r} r_t = r_t \le \theta |\delta X_t| + 2n^{-1}, \\
G_t &\le \theta (|\delta X_t|^2 + |\delta V_t|^2) \le \theta |\delta X_t|^2 + 4\theta n^{-2} \le \frac{3\theta R}{2} |\delta X_t| + 4\theta n^{-2},
\end{align*}
$$

we have

$$
\begin{align*}
\mathrm{d}A_t^\rho &\le -\frac{1}{4 + 6\varepsilon R} (f(r_t) + \varepsilon G_t - 2n^{-1} - 4\theta\varepsilon n^{-1}) \mathrm{d}t + (2\theta n^{-2} + 8\theta\varepsilon n^{-2}) \mathrm{d}t \\
&= -\frac{\rho_t}{4 + 6\varepsilon R} \mathrm{d}t + O(n^{-1}) \mathrm{d}t.
\end{align*}
$$

Recovering the subscript *n* and combining the three cases above, we obtain

$$
\mathbb{E}[\rho_n(Z_t, Z'_{n,t})] = \mathbb{E}[\rho_{n,t}] \le e^{-\kappa_n t} \mathbb{E}[\rho_{n,0}] + O(n^{-1}) = e^{-\kappa_n t} \mathbb{E}[\rho_n(Z_0, Z'_0)] + O(n^{-1}) \\
\text{for}
$$

$$
\kappa_n := \min \left( \kappa_{n,1}, \frac{\lambda R^2 \varepsilon_n \kappa_2}{\lambda R^2 \varepsilon_n + 2\Phi(r_0 + n^{-1})}, \frac{1}{4 + 6\varepsilon_n R} \right).
$$

Thanks to the uniform convergence of $\rho_n$ to $\rho_*$ in (32) and the square-integrability
of the processes $Z_t, Z'_{n,t}$, we can take the limit $n \to +\infty$ to derive

$$
\limsup_{n \to +\infty} \mathbb{E}[\rho_*(Z_t, Z'_{n,t})] = \limsup_{n \to +\infty} \mathbb{E}[\rho_{*,t}] \le e^{-\kappa_* t} \mathbb{E}[\rho_{*,0}] = e^{-\kappa_* t} \mathbb{E}[\rho_*(Z_0, Z'_0)]
$$
for

$$ \begin{aligned} \kappa &:= \min \left( \kappa_{*,1}, \frac{\lambda R^2 \varepsilon_* \kappa_2}{\lambda R^2 \varepsilon_* + 2\Phi((\theta+1)R)}, \frac{1}{4+6\varepsilon_* R} \right) \\ &\ge \min \left( \kappa_{*,1}, \frac{\lambda R^2 \varepsilon_* \kappa_2}{\lambda R^2 \varepsilon_* + 2\Phi((\theta+1)R)}, \frac{3}{20} \right), \end{aligned} $$

where the inequality is due to $\varepsilon_* \le \frac{4}{9R}$. If $r(z, z') \le r_0$, then

$$ \begin{aligned} |z-z'| &\le \sqrt{2}(|x-x'| + |v-v'|) \\ &\le \sqrt{2}(2|x-x'| + |q-q'|) \\ &\le \sqrt{2}r \le \frac{2\sqrt{2}r_0}{\Phi(r_0)} f_*(r); \end{aligned} $$

otherwise $|\delta x| + |\delta v| \ge R$, and then

$$ |z - z'| = \frac{|\delta x|^2 + |\delta v|^2}{|z - z'|} \le \frac{\sqrt{2}}{\lambda R} G(z, z'). $$

Therefore for every $z, z' \in \mathbb{R}^{2d}$, we have

$$ |z - z'| \le C_1 [\varepsilon_* G(z, z') + f_*(\theta|x-x'| + |q-q'|)] = C_1 \rho_*(z, z') $$

with

$$ C_1 = \sqrt{2} \max \left( \frac{2(\theta+1)R}{\Phi((\theta+1)R)}, \frac{1}{\lambda \varepsilon_* R} \right). $$

As a consequence,

$$ \limsup_{n \to +\infty} \mathbb{E}[|Z_t - Z'_{n,t}|] \le \limsup_{n \to +\infty} \mathbb{E}[\rho_*(Z_t, Z'_{n,t})] \le C_1 e^{-\kappa t} \mathbb{E}[\rho(Z_0, Z'_0)]. \quad \square $$

## Acknowledgements

The authors thank Zhenjie Ren for fruitful discussions. The research of P. M. is supported by the French ANR grant SWIDIMS (ANR-20-CE40-0022).

## References

[1] Shigeki Aida and Ichiro Shigekawa. Logarithmic Sobolev inequalities and spectral gaps: Perturbation theory. *J. Funct. Anal.*, 126(2):448–475, 1994. DOI: 10.1006/jfan.1994.1154.

[2] Dominique Bakry, Patrick Cattiaux and Arnaud Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. *J. Funct. Anal.*, 254(3):727–759, 2008. DOI: 10.1016/j.jfa.2007.11.002.

[3] Dominique Bakry, Ivan Gentil and Michel Ledoux. *Analysis and geometry of Markov diffusion operators*, volume 348 of *Grundlehren Math. Wiss*. Springer, Cham, 2014. DOI: 10.1007/978-3-319-00227-9.

[4] Patrick Cattiaux and Arnaud Guillin. Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems. *Bernoulli*, 28(4):2294–2321, 2022. DOI: 10.3150/21-BEJ1419.
[5] Patrick Cattiaux and Arnaud Guillin. Semi log-concave Markov diffusions. In *Séminaire de Probabilités XLVI*, pages 231–292. Springer, Cham, 2014. DOI: 10.1007/978-3-319-11970-0_9.

[6] Patrick Cattiaux, Arnaud Guillin and Pierre-André Zitt. Poincaré inequalities and hitting times. *Ann. Inst. Henri Poincaré, Probab. Stat.*, 49(1):95–118, 2013. DOI: 10.1214/11-AIHP447.

[7] Giovanni Conforti. Coupling by reflection for controlled diffusion processes: turnpike property and large time behavior of Hamilton–Jacobi–Bellman equations. *Ann. Appl. Probab.*, 33(6A):4608–4644, 2023. DOI: 10.1214/22-AAP1927.

[8] Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions. User's guide to viscosity solutions of second order partial differential equations. *Bull. Am. Math. Soc., New Ser.*, 27(1):1–67, 1992. DOI: 10.1090/S0273-0979-1992-00266-5.

[9] Noé Cuneo and Jean-Pierre Eckmann. Non-equilibrium steady states for chains of four rotors. *Commun. Math. Phys.*, 345(1):185–221, 2016. DOI: 10.1007/s00220-015-2550-2.

[10] Joscha Diehl, Peter K. Friz and Harald Oberhauser. Parabolic comparison revisited and applications. 2011. arXiv: 1102.5774 [math.AP].

[11] Andreas Eberle. Reflection couplings and contraction rates for diffusions. *Probab. Theory Relat. Fields*, 166(3-4):851–886, 2016. DOI: 10.1007/s00440-015-0673-1.

[12] Andreas Eberle, Arnaud Guillin and Raphael Zimmer. Couplings and quantitative contraction rates for Langevin dynamics. *Ann. Probab.*, 47(4):1982–2010, 2019. DOI: 10.1214/18-AOP1299.

[13] Andreas Eberle and Raphael Zimmer. Sticky couplings of multidimensional diffusions with different drifts. *Ann. Inst. Henri Poincaré, Probab. Stat.*, 55(4):2370–2394, 2019. DOI: 10.1214/18-AIHP951.

[14] Wendell H. Fleming and H. Mete Soner. *Controlled Markov processes and viscosity solutions*, volume 25 of *Stoch. Model. Appl. Probab*. Springer, New York, NY, 2nd edition, 2006. DOI: 10.1007/0-387-31071-1.

[15] Samuel Herrmann and Julian Tugaut. Non-uniqueness of stationary measures for self-stabilizing processes. *Stochastic Process. Appl.*, 120(7):1215–1246, 2010. DOI: 10.1016/j.spa.2010.03.009.

[16] Richard Holley and Daniel Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. *J. Stat. Phys.*, 46(5-6):1159–1194, 1987. DOI: 10.1007/BF01011161.

[17] Xing Huang, Eva Kopfer and Panpan Ren. Log-Sobolev inequality for decoupled and McKean-Vlasov SDEs and application on exponential ergodicity. 2025. arXiv: 2501.16092 [math.PR]. URL: https://arxiv.org/abs/2501.16092.

[18] Alessandra Iacobucci, Stefano Olla and Gabriel Stoltz. Convergence rates for nonequilibrium Langevin dynamics. *Ann. Math. Qué.*, 43(1):73–98, 2019. DOI: 10.1007/s40316-017-0091-0.
[19] Alessandra Iacobucci, Stefano Olla and Gabriel Stoltz. Thermo-mechanical transport in rotor chains. *J. Stat. Phys.*, 183(2), 26, 2021. doi: 10.1007/s10955-021-02748-8.

[20] Anna Kazeykina, Zhenjie Ren, Xiaolu Tan and Junjian Yang. Ergodicity of the underdamped mean-field Langevin dynamics. *Ann. Appl. Probab.*, 34(3):3181–3226, 2024. doi: 10.1214/23-aap2036.

[21] Alexei Kulik and Ilya Pavlyukevich. Non-Gaussian limit theorem for non-linear Langevin equations driven by Lévy noise. *Ann. Inst. Henri Poincaré, Probab. Stat.*, 55(3):1278–1315, 2019. doi: 10.1214/18-AIHP919.

[22] Yulong Lu. Two-scale gradient descent ascent dynamics finds mixed Nash equilibria of continuous games: a mean-field perspective. In *Proceedings of the 40th International Conference on Machine Learning*, volume 202 of *Proceedings of Machine Learning Research*, pages 22790–22811. PMLR, 2023. URL: https://proceedings.mlr.press/v202/lu23b.html.

[23] Fabio Marchesoni. A nonlinear langevin equation for fluid relaxation. *Prog. Theor. Phys.*, 74(6):1339–1342, 1985. doi: 10.1143/PTP.74.1339.

[24] Pierre Monmarché. Almost sure contraction for diffusions on $\mathbb{R}^d$. Application to generalized Langevin diffusions. *Stochastic Process. Appl.*, 161:316–349, 2023. doi: 10.1016/j.spa.2023.04.006.

[25] Pierre Monmarché. Generalized $\Gamma$ calculus and application to interacting particles on a graph. *Potential Anal.*, 50(3):439–466, 2019. doi: 10.1007/s11118-018-9689-3.

[26] Pierre Monmarché. Wasserstein contraction and Poincaré inequalities for elliptic diffusions with high diffusivity. *Ann. Henri Lebesgue*, 6:941–973, 2023. doi: 10.5802/ahl.182.

[27] Pierre Monmarché and Mouad Ramil. Overdamped limit at stationarity for non-equilibrium Langevin diffusions. *Electron. Commun. Probab.*, 27, 3, 2022. doi: 10.1214/22-ECP447.

[28] Pierre Monmarché, Zhenjie Ren and Songbo Wang. Time-uniform log-Sobolev inequalities and applications to propagation of chaos. *Electron. J. Probab.*, 29, 154, 2024. doi: 10.1214/24-EJP1217.

[29] Felix Otto and Cédric Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. *J. Funct. Anal.*, 173(2):361–400, 2000. doi: 10.1006/jfan.2000.3557.

[30] Michael Röckner and Feng-Yu Wang. Weak Poincaré inequalities and $L^2$-convergence rates of Markov semigroups. *J. Funct. Anal.*, 185(2):564–603, 2001. doi: 10.1006/jfan.2001.3776.

[31] Katharina Schuh. Global contractivity for Langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos. *Ann. Inst. Henri Poincaré Probab. Stat.*, 60(2):753–789, 2024. doi: 10.1214/22-aihp1337.

[32] Feng-Yu Wang. Criteria of spectral gap for Markov operators. *J. Funct. Anal.*, 266(4):2137–2152, 2014. doi: 10.1016/j.jfa.2013.11.016.

[33] Feng-Yu Wang. Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature. *Potential Anal.*, 53(3):1123–1144, 2020. doi: 10.1007/s11118-019-09800-z.
[34] Feng-Yu Wang. Functional inequalities for non-symmetric stochastic differential equations. Chinese. *Sci. Sin.*, Math., 50(1):167-178, 2020. DOI: 10.1360/N012018-00150.

[35] Feng-Yu Wang. Log-Sobolev inequalities: Different roles of Ric and Hess. *Ann. Probab.*, 37(4):1587-1604, 2009. DOI: 10.1214/08-AOP444.