Generalized Γ Calculus and Application to Interacting Particles on a Graph

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 N2, which is optimal.