A new characterization of quadratic transportation-information inequalities

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 the present 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-Guillin-Wu [8]. (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-Roberto-Samson [15].