Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature

2026/01/19 by Francesco Pedrotti
#math.PR #math.FA #math.ST #stat.TH

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Abstract

Notions of positive curvature have been shown to imply many remarkable properties for Markov processes, in terms, e.g., of regularization effects, functional inequalities, mixing time bounds and, more recently, the cutoff phenomenon. In this work, we are interested in a relaxed variant of Ollivier's coarse Ricci curvature, where a Markov kernel P satisfies only a weaker Wasserstein bound W_p(μP, νP) ≤ K W_p(μ,ν)+M for constants M≥ 0, K∈ [0,1], p ≥ 1. Under appropriate additional assumptions on the one-step transition measures δ_x P, we establish (i) a form of local concentration, given by a defective Talagrand inequality, and (ii) an entropy-transport regularization effect. We consider as illustrative examples the Langevin dynamics and the Proximal Sampler when the target measure is a log-Lipschitz perturbation of a log-concave measure. As an application of the above results, we derive criteria for the occurrence of the cutoff phenomenon in some negatively curved settings.

Citations

• Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon
• Ricci curvature of Markov chains on metric spaces
• Ricci curvature for metric-measure spaces via optimal transport
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Related

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• On cutoff via rigidity for high dimensional curved diffusions
• A transport approach to the cutoff phenomenon
• Projected Langevin dynamics and a gradient flow for entropic optimal transport
• Coarse Ricci curvature for continuous-time Markov processes