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Self-improvement of the Bakry-Emery criterion for Poincaré inequalities and Wasserstein contraction using variable curvature bounds
Patrick Cattiauxa, Max Fathib c d e✉, Arnaud Guillinf
- aInstitut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex 09, France
- bUniversité Paris Cité and Sorbonne Université, CNRS Laboratoire Jacques-Louis Lions & Laboratoire de Probabilités, Statistique et Modélisation 8 place Aurélie Nemours, F-75013 Paris, France
- cLJLL (UMR_7598) - Laboratoire Jacques-Louis Lions (Sorbonne-Université, Boîte courrier 187 - 75252 Paris Cedex 05 - France)
- dLPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation (Campus Jussieu Tour 16-26, 1er étage 4, Place Jussieu 75005 Paris / Bâtiment Sophie Germain 5ème étage Avenue de France 75013 Paris - France)
- eUPCité - Université Paris Cité (85 boulevard Saint-Germain 75006 Paris - France)
- fLaboratoire de Mathématiques, CNRS UMR 6620, Université Clermont-Auvergne, avenue des Landais, F-63177 Aubière, France
- ✉Corresponding author
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Abstract
We study Poincare inequalities and long-time behavior for diffusion processes on R^n under a variable curvature lower bound, in the sense of Bakry-Emery. We derive various estimates on the rate of convergence to equilibrium in L^1 optimal transport distance, as well as bounds on the constant in the Poincare 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 Poincare inequalities when curvature is positive but not constant.