Transfer of regularity for Markov semigroups

Vlad Bally; Lucia Caramellino

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arXiv · 1905.06051

Transfer of regularity for Markov semigroups

Vlad Bally·Lucia Caramellino

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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.

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Vlad Bally, Lucia Caramellino. 2019-05-15. Transfer of regularity for Markov semigroups. https://arxiv.org/abs/1905.06051

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