The entropy method under curvature-dimension conditions in the spirit of Bakry-\'Emery in the discrete setting of Markov chains

TL;DR

This paper introduces a new curvature-dimension inequality for Markov chains called $CD_\Upsilon(\kappa,\infty)$, extending classical concepts to discrete spaces and establishing its properties and applications.

Contribution

It defines and analyzes the $CD_\Upsilon$ inequality for Markov chains, demonstrating its tensorization, relation to other curvature notions, and applicability to various examples and entropy types.

Findings

$CD_\Upsilon$ bounds are preserved under tensorization.

Links established between $CD_\Upsilon$ and other discrete curvature notions.

Examples include complete graphs, hypercube, and birth-death processes.

Abstract

We consider continuous-time (not necessarily finite) Markov chains on discrete spaces and identify a curvature-dimension inequality, the condition $CD_\Upsilon(\kappa,\infty)$, which serves as a natural analogue of the classical Bakry-\'Emery condition $CD(\kappa,\infty)$ in several respects. In particular, it is tailor-made to the classical approach of proofing the modified logarithmic Sobolev inequality via computing and estimating the second time derivative of the entropy along the heat flow generated by the generator of the Markov chain. We prove that curvature bounds in the sense of $CD_\Upsilon$ are preserved under tensorization, discuss links to other notions of discrete curvature and consider a variety of examples including complete graphs, the hypercube and birth-death processes. We further consider power type entropies and determine, in the same spirit, a natural CD condition which leads to Beckner inequalities. The $CD_\Upsilon$ condition is also shown to be compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property.