A new characterization of quadratic transportation-information inequalities

Yuan Liu

Published 2015-06-08Version 1

It is known that a quadratic transportation-information inequality $\mathrm{W_2I}$ interpolates between the Talagrand's inequality $\mathrm{W_2H}$ and the log-Sobolev inequality (LSI for short). Our aim of the present paper is threefold: (1) To prove $\mathrm{W_2I}$ through the Lyapunov condition, which fills a gap in the subject of transport inequalities according to Cattiaux-Guillin-Wu [8]. (2) To prove the stability of $\mathrm{W_2I}$ under bounded perturbations, which yields a transference principle in the sense of Holley-Stroock. (3) To prove $\mathrm{W_2H}$ through a restricted $\mathrm{W_2I}$, as a characterization of $\mathrm{W_2H}$ similar to the restricted LSI according to Gozlan-Roberto-Samson [14].