Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature
Francesco Pedrotti
TL;DR
This work relaxes Ollivier's coarse Ricci curvature to a weak Wasserstein contraction with additive defect, encoded as $W_p(\mu P,\nu P) \le K W_p(\mu,\nu) + M$, and develops two key mechanisms: defective local concentration (defective transport–entropy) and entropy–Wasserstein regularization. It establishes propagation and combination results under a general weak-curvature framework, then specializes to Langevin dynamics and the Proximal Sampler with perturbations of log-concave targets, yielding finite-time concentration/ergodicity and reverse-transport–entropy bounds. These tools lead to explicit mixing-window and cutoff criteria in negatively curved regimes, including bounds that depend on the perturbation magnitude $L$, curvature strength $\alpha$, and Poincaré constants, and provide practical estimates for convergence to equilibrium in sampling algorithms. Overall, the paper extends cutoff analysis beyond nonnegative curvature by quantifying how additive defects in Wasserstein contraction influence concentration, entropy decay, and mixing behavior in continuous- and discrete-time Markov chains.
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 $W_p(μP, νP) \leq K W_p(μ,ν)+M$ for constants $M\ge 0, K\in [0,1], p \ge 1$. 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.