Transfer of regularity for Markov semigroups

Abstract

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 $(P_t^n{)}_{t>0}$, $n\in \mathbb{N}$, whose associated densities $p_t^n(x,y)$ 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.

References in corpus (1)

• Convergence and regularity of probability laws by using an interpolation method