Space-Time Log-Sobolev Inequality and Hypocoercive Hypercontractivity for Underdamped Langevin Dynamics

Technical Summary: Space-Time Log-Sobolev Inequality and Hypocoercive Hypercontractivity for Underdamped Langevin Dynamics

1. Problem Statement
The paper investigates the hypercontractivity properties of the underdamped Langevin dynamics, a stochastic differential equation describing the evolution of position $X_t$ and velocity $V_t$ in a convex confining potential $U$. Unlike the overdamped case, where noise acts on the position variable, the underdamped dynamics feature degenerate diffusion acting solely on the velocity. This degeneracy prevents the direct application of the classical logarithmic Sobolev inequality (LSI), which typically links entropy decay to hypercontractivity in reversible diffusion processes. Consequently, standard arguments establishing that a semigroup improves integrability over time (hypercontractivity) fail for the underdamped case because the diffusion is not elliptic in the full phase space. The central challenge is to establish a hypercontractivity estimate that captures the "kinetic" time scale ($t \sim \rho^{-1/2}$), where dissipation in velocity is transferred to position via Hamiltonian transport, rather than the slower diffusive scale ($t \sim \rho^{-1}$) associated with the overdamped limit.

2. Methodology
The authors develop a novel analytical framework that bridges the gap between hypocoercive entropy decay and hypercontractivity without relying on a standard instantaneous LSI. The methodology proceeds in three main stages:

• Space-Time Functional Inequality: The core technical innovation is the derivation of a "space-time logarithmic Sobolev inequality." Instead of seeking an instantaneous LSI (which does not hold due to degeneracy), the authors lift the functional inequality to the space-time domain $[0, T] \times \mathbb{R}^{2d}$. This inequality bounds the time-averaged entropy of a density $g$ by the time-averaged velocity Fisher information and a term involving the negative Sobolev norm of the residual $(\partial_t + L_a)g_t$. This estimate is derived by controlling the evolution of a modified entropy functional that includes a "Wasserstein current corrector" (COT), a term coupling the spatial marginal and velocity current.
• Controlled Entropy Decay: The proof of the space-time LSI utilizes a controlled version of the entropy decay estimates previously established for hypocoercivity. By introducing a velocity control field, the authors derive a differential inequality for a modified entropy that absorbs the corrector term, effectively transferring dissipation from the velocity to the spatial variable.
• Forward/Backward Interpolation: To translate the space-time LSI into a hypercontractivity estimate, the authors employ a duality argument based on a forward/backward time interpolation. Since the underdamped Langevin semigroup $P_t$ is non-reversible, the interpolation pairs the forward semigroup with its $L^2(\mu)$-adjoint $P^*_t$. This construction creates a path of probability densities satisfying the regularity assumptions required for the space-time LSI.

3. Key Contributions and Results

• Space-Time LSI (Theorem 3.4): The paper establishes a space-time LSI for the underdamped Langevin dynamics. For a density $g$ on $[0, T] \times \mathbb{R}^{2d}$, the inequality states:
  $$\text{Ent}_{\bar{\mu}_T}(g) \lesssim I_{v,T}(g) + \frac{1}{\rho} \int_0^T \|(\partial_t + L_a)g_t\|_{-1, g_t}^2 \frac{dt}{T}$$
  This result holds provided the friction parameter $\gamma$ scales as $\Gamma\sqrt{\rho}$ and the time horizon $T$ is of the order $\rho^{-1/2}$.

• Hypocoercive Hypercontractivity (Theorem 5.2 & Corollary 5.3): Leveraging the space-time LSI and the interpolation argument, the authors prove that the underdamped Langevin semigroup improves integrability on the kinetic time scale. Specifically, for friction $\gamma = \Gamma\sqrt{\rho}$ and time $t$, the semigroup maps $L^p(\mu)$ to $L^{q(t)}(\mu)$ where the exponent grows exponentially:
  $$\|P_t f\|_{L^{q(t)}(\mu)} \leq \|f\|_{L^p(\mu)}, \quad q(t) - 1 = \alpha^{\lfloor \sqrt{\rho} t / \tau \rfloor} (p-1)$$
  Here, $\alpha > 1$ is a constant independent of dimension $d$ and the LSI constant $\rho$. This confirms that the integrability gain occurs on the ballistic time scale $t \sim \rho^{-1/2}$, matching the sharp hypocoercive decay rates established in prior work.

• Rényi Divergence Decay (Corollary 5.5): As a direct consequence, the paper derives the decay of the Rényi divergence $R_q(\nu_t | \mu)$ at the sharp hypocoercive rate $O(\sqrt{\rho})$. The result provides an explicit bound:
  $$R_q(\nu_t | \mu) \lesssim e^{-\lambda \sqrt{\rho} t} R_q(\nu_0 | \mu)$$
  where the rate $\lambda$ depends on the friction scaling parameter $\Gamma$.

4. Significance and Claims
The paper claims to resolve the difficulty of extending hypercontractivity to degenerate, non-reversible systems like the underdamped Langevin dynamics. By introducing the space-time LSI, the authors provide a mechanism to recover the "missing" instantaneous LSI in a time-averaged sense, allowing the application of duality arguments typically reserved for reversible diffusions.

The significance of the work lies in:

1. Sharpness: The derived hypercontractivity rate matches the sharp hypocoercive entropy decay rates previously known for this system, confirming that the kinetic time scale is the correct regime for integrability improvement.
2. Methodological Advance: The space-time LSI offers a new tool for analyzing hypocoercive systems, potentially applicable to other degenerate diffusions where standard LSI fails.
3. Algorithmic Implications: The results provide a theoretical foundation for the analysis of sampling algorithms, particularly Hamiltonian Monte Carlo (HMC). The paper notes that these sharp hypocoercive estimates could lead to improved complexity bounds for HMC, specifically regarding "warm starts" and convergence in Rényi divergence, although the paper leaves the detailed application to future work.

The authors emphasize that their approach avoids the high-friction regime assumptions of previous coupling-based methods (e.g., Wang [37]), thereby providing a kinetic analogue of the Gross estimate that is valid in the low-friction regime where the dynamics are genuinely underdamped.