Gaussian integrability of distance function under the Lyapunov condition. (English) Zbl 1308.26023

Liu, Yuan

Electron. Commun. Probab. 20, Paper No. 9, 10 p. (2015).

Summary: 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 operators, which answers a question by P. Cattiaux et al. [Probab. Theory Relat. Fields 148, No. 1–2, 285–304 (2010; Zbl 1210.60024), p. 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.

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
60E15	Inequalities; stochastic orderings
60J60	Diffusion processes

Keywords:

Gaussian integrability; Lyapunov condition; diffusion process; jump process

Citations:

Zbl 1210.60024