Convergence and regularity of probability laws by using an interpolation method

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

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

Fournier and Printems [Bernoulli, 2010] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Hölder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. By employing some Besov space techniques, Debussche and Romito [Probab. Theory Related Fields, 2014] have substantially improved the result of Fournier and Printems. In our paper we show that this kind of problem naturally fits in the framework of interpolation spaces: we prove an interpolation inequality (see Proposition 2.5) which allows to state (and even to slightly improve) the above absolute continuity result. Moreover it turns out that the above interpolation inequality has applications in a completely different framework: we use it in order to estimate the error in total variance distance in some convergence theorems.

Mots clés: spaces; interpolation; Regularity of probability laws; Orlicz spaces; Hermite polynomials; Malliavin calculus; integration by parts formulas

Domaines: Mathématiques [math]; Probabilités [math.PR]

Vlad Bally, Lucia Caramellino. Convergence and regularity of probability laws by using an interpolation method. Annals of Probability, 2017, 45 (2). ⟨hal-01109276v2⟩