A new characterization of quadratic transportation-information inequalities

Abstract

It is known that a quadratic transportation-information inequality $$\mathrm{W}_2\mathrm{I}$$ interpolates between the Talagrand’s inequality $$\mathrm{W}_2\mathrm{H}$$ and the log-Sobolev inequality (LSI for short). The aim of this paper is threefold: (1) To prove the equivalence of $${\mathrm {W}}_2\mathrm{I}$$ and the Lyapunov condition, which gives a new characterization inspired by Cattiaux et al. (Probab Theory Relat Fields 148(1–2):285–304, 2010). (2) To prove the stability of $${\mathrm {W}}_2\mathrm{I}$$ under bounded perturbations, which gives a transference principle in the sense of Holley–Stroock. (3) To prove $$\mathrm{W}_2\mathrm{H}$$ through a restricted $$\mathrm{W}_2\mathrm{I}$$ , which gives a counterpart of the restricted LSI presented by Gozlan et al. (Ann Probab 39(3):857–880, 2011).