Gaussian integrability of distance function under the Lyapunov condition

TL;DR

This paper provides a direct proof that the distance function has Gaussian integrability under the Lyapunov condition for certain Markov processes, extending results to unbounded diffusions and jump processes.

Contribution

It offers a new, straightforward proof of Gaussian integrability under Lyapunov conditions, applicable to a broad class of diffusion and jump processes.

Findings

Gaussian integrability of the distance function is established under Lyapunov conditions.

The proof extends to diffusions with unbounded coefficients and jump processes.

Analogous results are discussed under Gozlan's condition.

Abstract

In this note we give a direct proof of the Gaussian integrability of distance function as $\mu e^{\delta d^2(x,x_0)} < \infty$ for some $\delta>0$ provided the Lyapunov condition holds for symmetric diffusion Markov operators, which answers a question proposed 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].