Transfer of regularity for Markov semigroups

Vlad Bally (1, 2) , Lucia Caramellino (3)

1 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
2 MATHRISK - Mathematical Risk Handling
3 Dipartimento di Matematica [Rome]

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 ∈ N, whose associated densities p n t (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 P t (x, dy) = p t (x, y)dy and p t has some regularity properties.

Mots clés: Interpolation spaces; Markov semigroups; Regularity of probability laws

Domaines: Probabilités [math.PR]

Vlad Bally, Lucia Caramellino. Transfer of regularity for Markov semigroups. Journal of Stochastic Analysis , 2021, 2 (3), pp.Article 13. ⟨hal-02429530⟩