Transfer of Regularity for Markov Semigroups by Using an Interpolation Technique

We study the regularity of a Markov semigroup (Pt)t>0, that is, when Pt(x, dy) = pt(x, y)dy for a suitable smooth function pt(x, y). This is done by transferring the regularity from an approximating Markov semigroup sequence (Pn t )t>0, n ∈ N, whose associated densities pt (x, y) are smooth and can blow up as n → ∞. 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 Pt(x, dy) = pt(x, y)dy and pt has some regularity properties.