# Transfer of regularity for Markov semigroups

## TL;DR

This paper investigates how regularity properties of Markov semigroups can be transferred from approximating sequences with smooth densities, under certain conditions balancing blow-up and convergence speed.

## Contribution

It introduces an interpolation approach to transfer regularity from smooth approximations to the limiting Markov semigroup.

## Key findings

- Established conditions for regularity transfer based on blow-up and convergence balance
- Proved the existence of a smooth density for the limiting semigroup under these conditions
- Provided a framework for analyzing regularity in Markov processes via approximation sequences

## 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 $(P^n_t)_{t>0}$, $n\in\mathbb{N}$, 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.