Generalized $\Gamma$ calculus and application to interacting particles on a graph

TL;DR

This paper extends the classical Gamma calculus to analyze degenerate Markov processes, providing new insights into convergence rates, functional inequalities, and interacting particle systems on graphs.

Contribution

It introduces a generalized Gamma calculus framework applicable to non-elliptic, non-reversible processes and derives optimal convergence and inequality results for interacting particles.

Findings

Optimal convergence speed for ergodic Ornstein-Uhlenbeck processes.

Log-Sobolev inequalities for invariant measures of interacting particles.

Constants of order N^2 for N interacting particles' inequalities.

Abstract

The classical Bakry-\'Emery calculus is extended to study, for degenerated (non-elliptic, non-reversible, or non-diffusive) Markov processes, questions such as hypoellipticity, hypocoercivity, functional inequalities or Wasserstein contraction. In particular we obtain the optimal speed of convergence to equilibrium for any ergodic Ornstein-Uhlenbeck process, which is given by the spectral gap of the drift matrix and the size of the corresponding Jordan blocks. We also study chains of $N$ interacting overdamped particles and establish for their invariant measures log-Sobolev inequalities with constants of order $N^2$, which is optimal.