Shifted Composition II: Shift Harnack Inequalities and Curvature Upper Bounds

TL;DR

This work develops sharp forward regularity results for Langevin diffusions by extending the shifted composition rule to the forward problem and deriving optimal shift Harnack inequalities. The authors connect these inequalities to a local gradient-entropy bound and to curvature upper bounds, mirroring Bakry–Émery theory but in the upper-curvature regime. They provide two continuous-time proofs (Girsanov-based and divergence-differentiation) and a discrete-time reduction that yields tight constants, enabling precise consequences for stationary distributions. The results culminate in a duality framework with reverse transport inequalities and yield sub-Gaussian concentration for the score under the stationary measure, highlighting the quantitative regularity and concentration properties of Langevin dynamics.

Abstract

We apply the shifted composition rule -- an information-theoretic principle introduced in our earlier work [AC23] -- to establish shift Harnack inequalities for the Langevin diffusion. We obtain sharp constants for these inequalities for the first time, allowing us to investigate their relationship with other properties of the diffusion. Namely, we show that they are equivalent to a sharp "local gradient-entropy" bound, and that they imply curvature upper bounds in a compelling reflection of the Bakry-Emery theory of curvature lower bounds. Finally, we show that the local gradient-entropy inequality implies optimal concentration of the score, a.k.a. the logarithmic gradient of the density.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Rényi divergence
• Theorem 2.2
• Theorem 2.3: Shifted composition rule
• Lemma 2.4: Convexity principle
• proof
• Remark 2.5: Refinements
• Theorem 3.1: One-step-to-multi-step for forward regularity
• proof : Proof of Theorem \ref{['thm:one-to-multi']}