Gaussian integrability of distance function under the Lyapunov condition

In this note we give a direct proof of the Gaussian integrability of distance function as μe (x,x0) <∞ for some δ > 0 provided the Lyapunov condition holds for symmetric diffusion operators, which answers a question in Cattiaux-Guillin-Wu [6, Page 295]. The similar argument still works for diffusions processes with unbounded diffusion coefficients and for jump processes such as birth-death chains. An analogous discussion is also made under the Gozlan’s condition arising from [9, Proposition 3.5].