Self-improvement of the Bakry-Emery criterion for Poincaré inequalities and Wasserstein contraction using variable curvature bounds

Abstract

We study Poincaré inequalities and long-time behavior for diffusion processes on Rn under a variable curvature lower bound, in the sense of Bakry-Emery. We derive various estimates on the rate of convergence to equilibrium in L1 optimal transport distance, as well as bounds on the constant in the Poincaré inequality in several situations of interest, including some where curvature may be negative. In particular, we prove a self-improvement of the Bakry-Emery estimate for Poincaré inequalities when curvature is positive but not constant.