A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics

Technical Summary: A Sharp Hypocoercive Entropy Decay Estimate for Underdamped Langevin Dynamics

Problem Statement
The paper investigates the quantitative convergence rates of the underdamped (kinetic) Langevin dynamics toward its invariant measure $\mu(dx dv) \propto e^{-U(x)-|v|^2/2} dx dv$. The dynamics are governed by the stochastic differential equation:
$$dX_t = V_t dt, \quad dV_t = -\nabla U(X_t) dt - \gamma V_t dt + \sqrt{2\gamma} dW_t,$$
where $\gamma > 0$ is the friction coefficient. The system exhibits hypocoercivity: while the entropy dissipation (Fisher information) controls only velocity derivatives, spatial relaxation is achieved indirectly through the Hamiltonian transport term $v \cdot \nabla_x - \nabla U \cdot \nabla_v$.

The central problem is to establish explicit, sharp entropy decay rates for general convex potentials $U$ under the assumption that the position marginal $\mu_x \propto e^{-U}$ satisfies a Logarithmic Sobolev Inequality (LSI) with constant $\rho > 0$. Previous literature often yields constants dependent on auxiliary Sobolev norms or derivative bounds on coefficients, rather than solely on the LSI constant $\rho$. The natural benchmark, derived from the quadratic potential case $U(x) = \rho|x|^2/2$, suggests an optimal convergence rate of order $\sqrt{\rho}$. The goal is to recover this sharp $\sqrt{\rho}$ order for general convex potentials with constants depending only on the friction parameter $\Gamma$ (where $\gamma = \Gamma\sqrt{\rho}$) and $\rho$.

Methodology
The authors employ a modified entropy approach, extending the "modified $L^2$" hypocoercivity framework (Dolbeault, Mouhot, Schmeiser) to the nonlinear entropy setting. The core innovation is the introduction of a nonlinear Wasserstein entropy-current corrector.

1. Decomposition and Notation: The probability density ratio $g_t = p_t / \varrho_\infty$ is decomposed into its spatial marginal $q_t = \Pi_v g_t$ and conditional velocity density. The relative entropy is split into spatial and velocity components: $\text{Ent}(g) = \text{Ent}_x(q) + \text{Ent}_v(g)$.
2. The Corrector: The authors define a corrector term $COT(g)$ coupling the velocity current to the optimal transport displacement:
   $$COT(g) = \int j(x) \cdot (x - T_q(x)) \, \mu_x(dx),$$
   where $j = \Pi_v(vg)$ is the current, and $T_q = \nabla \phi$ is the Brenier optimal transport map pushing the marginal $q\mu_x$ to the equilibrium $\mu_x$. The term $(x - T_q(x))$ represents the Wasserstein gradient of the squared distance $W_2^2(q\mu_x, \mu_x)$.
3. Modified Functional: A modified entropy functional is constructed:
   $$H_\epsilon(g) = \text{Ent}(g) + \epsilon COT(g).$$
   The parameter $\epsilon$ is chosen proportional to $\sqrt{\rho}$.
4. Differential Inequality: The time derivative of $H_\epsilon$ is analyzed. The derivative of the standard entropy yields the negative Fisher information $-\gamma I_v$. The derivative of the corrector is estimated using:
   • The moment equations for $q$ and $j$.
   • A distributional Wasserstein acceleration inequality (second variation of $W_2^2$).
   • A stress estimate for Brenier maps (Section 5), which bounds the interaction between the Hessian of the transport potential and the stress tensor $\Theta$ (related to the conditional covariance of velocity). This step utilizes the convexity of $U$ and the LSI of $\mu_x$.
5. Closing the Inequality: By combining the entropy dissipation and the corrector derivative, and carefully balancing terms using Young's inequality and the specific choice of $\epsilon$, the authors derive a closed Lyapunov inequality of the form $\frac{d}{dt} H_\epsilon \leq -\Lambda H_\epsilon$.

Key Contributions

• Sharp Rate Dependence: The paper establishes an entropy decay rate $\Lambda$ that scales explicitly as $\sqrt{\rho}$. This matches the optimal rate known for quadratic potentials and demonstrates the "diffusive-to-ballistic" acceleration of underdamped dynamics compared to the overdamped rate $\rho$.
• Nonlinear Corrector: Unlike previous Hilbert-space methods that use linear correctors or projections, this work introduces a nonlinear corrector based on the Brenier map and Wasserstein geometry. This allows the constants in the final estimate to depend only on the LSI constant $\rho$ and the friction parameter $\Gamma$, avoiding dependence on higher-order derivatives of $U$ or auxiliary Sobolev norms.
• General Convex Potentials: The result holds for arbitrary convex potentials satisfying mild growth conditions (Assumption 2.2), provided the position marginal satisfies an LSI.
• Rigorous Approximation: The proof handles arbitrary finite-entropy initial data by establishing the result for "regular" solutions (smooth, strictly positive) and then using a density argument involving Hérau–Nier smoothing and lower semicontinuity of entropy and Fisher information.

Main Results
Theorem 2.3 states that for $\gamma = \Gamma\sqrt{\rho}$ and $\theta = \min\{\frac{\Gamma}{12}, \frac{1}{4\Gamma}\}$, the relative entropy decays as:
$$\text{Ent}(p_t | \mu) \leq \frac{1+\theta}{1-\theta} e^{-\Lambda t} \text{Ent}(p_0 | \mu),$$
with the explicit rate:
$$\Lambda = \frac{\theta}{2(1+\theta)} \sqrt{\rho}.$$
The constants are explicit and depend only on $\Gamma$ and $\rho$. The pre-factor $\frac{1+\theta}{1-\theta}$ arises from the equivalence between the modified entropy $H_\epsilon$ and the standard relative entropy.

Significance and Claims
The paper claims that this work provides the first explicit entropy hypocoercivity estimate for underdamped Langevin dynamics with a convergence rate of sharp order $\sqrt{\rho}$ for general convex potentials, where the constants depend solely on the LSI constant of the spatial marginal.

The authors position this result as a refinement of the hypocoercivity program initiated by Villani and others. While previous works (e.g., Villani's memoir, Dolbeault et al., Cao et al.) established hypocoercivity or sharp $L^2$ rates, they often relied on linear correctors, Hilbert projections, or required assumptions on the Hessian of $U$ (boundedness) that introduced constants beyond the LSI parameter. By utilizing a Wasserstein-based corrector and displacement convexity arguments, this approach isolates the LSI constant $\rho$ as the sole governing parameter for the rate, offering a more precise characterization of the relaxation mechanism in the entropy framework. The work bridges the gap between the spectral analysis of kinetic operators and optimal transport theory, specifically leveraging the geometry of the Brenier map to control the spatial coercivity missing in the standard entropy dissipation.