Time dependent Lyapunov functions for some Kolmogorov semigroups perturbed by unbounded potentials

We study global regularity properties of transition kernels associated to second order differential operators in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}^N}$$\end{document} with unbounded drift and potential terms. Under suitable conditions, we prove pointwise upper bounds. We use time dependent Lyapunov function techniques allowing us to gain a better time behaviour of such kernels.