Gaussian Integrability of Distance Function under the Lyapunov Condition

Yuan Liu

Published 2014-09-30Version 1

In this note, we give a direct proof of the Gaussian integrability as $\mu e^{\delta d^2(x,x_0)} < \infty$ for some $\delta>0$ provided the Lyapunov condition for symmetric diffusion Markov operators, which answers a question proposed in Cattiaux-Guillin-Wu [3, Page 295]. The similar argument also works under the Gozlan's condition arising from [6, Proposition 3.5].