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

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

Zbl 1210.60024
Full Text: