Reverse Transport Inequality for Langevin Dynamics

• The paper provides explicit, dimension-free upper bounds on KL and Rényi divergences between marginals of the overdamped Langevin SDE.
• It employs reflection coupling and a shifted Girsanov transform to control divergence in non-convex potential regions.
• It establishes uniform exponential decay rates and extends classical contraction properties beyond the log-concave regime.

A reverse transportation inequality for Langevin dynamics provides explicit, dimension-free, and time-uniform upper bounds on divergences (notably, Kullback-Leibler and Rényi) between marginals of an overdamped Langevin stochastic differential equation (SDE) started from different initial points, particularly under non-convex potentials. These inequalities, dual to classical Harnack inequalities, quantify exponential convergence, extend key contraction properties beyond the log-concave regime, and establish essential entropy-cost controls, all without requiring global convexity or uniform dissipativity of the potential. The framework leverages advanced coupling and interpolation techniques to ensure robustness and sharpness under minimal structural assumptions on the drift.

1. Overdamped Langevin SDE and Problem Formulation

The setting is the overdamped Langevin SDE in $\mathbb{R}^d$:

$$\mathrm{d} X_t = -\nabla V(X_t) \, \mathrm{d} t + \sqrt{2} \,\mathrm{d} W_t, \quad X_0 = x,$$

where $W_t$ is standard Brownian motion and $V:\mathbb{R}^d \to \mathbb{R}$ is twice-differentiable. The associated Markov semigroup acts on test functions via $P_t f(x) = \mathbb{E}[f(X_t) | X_0 = x]$, and $P_t^x$ denotes the law at time $t$ with initial value $x$.

Contrary to much of the literature on Langevin dynamics, the potential $V$ is allowed to be non-convex within a compact domain. Specifically, only a one-sided dissipativity/Lipschitz assumption is imposed outside a ball of radius $R$; no global convexity or uniform smoothness is mandated.

2. Assumptions on the Potential

The crucial structural condition (Assumption A) states:

• For some $R > 0$ and constants $0 < m \leq M$, for all $x, y \in \mathbb{R}^d$:

$$-(x-y) \cdot (\nabla V(x) - \nabla V(y)) \leq \begin{cases} -m\,|x-y|^2, & \text{if } |x-y| > R \ M\,|x-y|^2, & \text{if } |x-y| \leq R \end{cases}$$

• Outside the ball, the drift is strongly dissipative ($-m$), while inside only a one-sided Lipschitz bound $M$ holds.

This allows for arbitrary non-convex wells within the compact region, generalizing classical results that require strong convexity everywhere. No global Lipschitz constant for $\nabla V$ is required.

3. Main Reverse Transportation Inequalities

Dimension-free, uniform-in-time bounds on the divergences between $P_T^x$ and $P_T^y$ are established for arbitrary $T > 0$ and $x, y \in \mathbb{R}^d$.

3.1 Kullback-Leibler (KL) Case ($\alpha=1$)

Define:

• $\nu = \frac{1}{2} m e^{- \frac{1}{2} R^2 (m + M)}$
• $M_{x, y} = \max\{1, \sqrt{(R+1)/|x - y|}\}$
• $$C = \frac{8}{3} \exp\left(\frac{5}{4} R^2 (m+M)\right) \frac{2 e^{\nu T} + 1}{(e^{\nu T} + 1)^3}$$, which is uniform in $T \geq 0$

Then,

$$D_{\mathrm{KL}}(P_T^y \,\|\, P_T^x) \leq C M_{x, y} \nu (e^{\nu T} - 1)^{-1} |x - y|^2.$$

As $T \to \infty$, this decays as $\mathcal{O}(e^{-\nu T}) |x-y|^2$.

3.2 Rényi Divergences ($\alpha = q > 1$)

For $q>1$, with auxiliary constants:

• $\alpha(T) = e^{\nu T}/(\nu(e^{2\nu T}-1))$
• $$\beta(T) = 4\nu^2 e^{2\nu T}/(C_0^2 (e^{2\nu T}-1)^2)$$
• $C_0 = \frac{1}{2} e^{-\frac{1}{2} R^2(m+M)}$
• $C_1 = \exp(\frac{1}{4} R^2(m+M))$

one has

$$\mathcal{R}_q(P_T^y \,\|\, P_T^x) \leq \frac{1}{2(q-1)} \log\left[1 + C_1 \alpha(T) T^{-1} M_{x, y}^{-1} \left(\exp\left((q-1)+2(q-1)^2) T |x-y|^2 M_{x,y}^2 \beta(T)\right) - 1\right)\right].$$

Short- and long-time behavior:

• As $T \to 0$, $\mathcal{R}_q \lesssim q |x-y|^2 / T$.
• As $T \to \infty$, $$\mathcal{R}_q \lesssim q\,e^{-3\nu T}M_{x, y}|x-y|^2$$.

All constants are independent of dimension $d$ and are bounded uniformly in $T \geq 0$.

4. Exponential Decay and Uniformity

Unlike classical results requiring global convexity, these bounds hold uniformly for all $T$, exhibit explicit exponential rates in $T$, and constants remain independent of the underlying dimension. For the KL case, the convergence rate is $\mathcal{O}(e^{-\nu T})$, while for Rényi divergences of higher order, the decay is even faster, at $\mathcal{O}(e^{-3\nu T})$.

5. Extensions Beyond Log-Concave Potentials

Traditional contraction and transportation inequalities for Langevin dynamics (e.g., [Altschuler–Chewi ’24]) critically depend on $V$ being globally $m$-strongly convex. The present reverse transportation inequalities extend this by permitting $V$ to be non-convex within any compact set of radius $R$, provided strong dissipativity is restored outside. A canonical example is $V(x) = |x|^4 - |x|^2$. The approach combines reflection coupling (to navigate the non-convex region) with a shifted Girsanov transform, enabling explicit and stable divergence control for this wider class of potentials (Lu et al., 21 Dec 2025).

6. Duality with Harnack Inequalities

By standard variational representations,

$$D_{\mathrm{KL}}(P_T^y \,\|\, P_T^x) = \sup_{\varphi} [P_T \varphi(y) - \log P_T(e^{\varphi})(x)],$$

implying the log-Harnack inequality

$$P_T \varphi(y) \leq \log P_T(e^{\varphi})(x) + C M_{x, y} \nu (e^{\nu T} - 1)^{-1}|x-y|^2.$$

For $q>1$, the corresponding power-Harnack inequality is

$$P_T \varphi(y) \leq \left(P_T (\varphi^{q'})(x)\right)^{1/q'} \exp\left(\frac{q-1}{q}\, \mathcal{R}_q(P_T^y \,\|\, P_T^x)\right),$$

demonstrating a dual relationship between reverse transportation and Harnack inequalities.

7. Proof Architecture and Key Techniques

The main proof framework incorporates:

• Reflection coupling + shifted interpolation: Constructs three coupled processes—standard, reflected, and drift-shifted—so that controlled coalescence enforces meeting at finite time, even in non-convex regions.
• $L^1$-contraction via Lyapunov function: Utilizes a carefully chosen Lyapunov function $f(r)$, concave and linear beyond $R$, to quantify contraction and ensure exponential decay of the mean displacement between coupled processes.
• Girsanov transform for divergences: Applies the Girsanov theorem to relate differences in drift between coupled processes to upper bounds on KL and Rényi divergences.
• Explicit drift optimization: Selects a time-dependent control $\eta_t$ to ensure that the controlled drift process meets the reference process exactly at time $T$, enabling sharp, explicit divergence estimates.

This synthesis yields uniform-in-time, dimension-free reverse entropy-cost inequalities, generalizing log-concave contraction theory to encompass Langevin dynamics with non-globally dissipative, non-convex potentials (Lu et al., 21 Dec 2025).