  We study the regularity of a Markov semigroup $(P_t)_{t>0}$, that is, when
$P_t(x,dy)=p_t(x,y)dy$ for a suitable smooth function $p_t(x,y)$. This is done
by transferring the regularity from an approximating Markov semigroup sequence
$(P^n_t)_{t>0}$, $n\in\mathbb{N}$, whose associated densities $p^n_t(x,y)$ are
smooth and can blow up as $n\to\infty$. We use an interpolation type result and
we show that if there exists a good equilibrium between the blow up and the
speed of convergence, then $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ has some
regularity properties.
