Convex Entropy Decay via the Bochner-Bakry-Emery approach

TL;DR

This paper introduces a novel method based on a Bochner-type identity to estimate the exponential decay rate of relative entropy in Markov processes, demonstrating its effectiveness on birth-death, zero-range, and Bernoulli-Laplace models.

Contribution

The paper develops a new approach using convex entropy decay via the Bochner-Bakry-Emery method, extending analysis to inhomogeneous models where previous methods were limited.

Findings

Effective entropy decay estimates for birth and death processes.

Extension of decay results to inhomogeneous zero-range and Bernoulli-Laplace models.

Demonstration of convex decay of relative entropy in the studied processes.

Abstract

We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli-Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear.