Asymptotic estimates on the time derivative of .-entropy on Riemannian manifolds

In this note, we obtain an asymptotic estimate for the time derivative of the .-entropy in terms of the lower bound of the Bakry-Emery .2 curvature. In the cases of hyperbolic space and the Heisenberg group (more generally, the nilpotent Lie group of rank two), we show that the time derivative of the .-entropy is non-increasing and concave in time t, also we get a sharp asymptotic bound for the time derivative of the entropy in these cases.