Integration by Parts Formulas and Regularity of Probability Laws

Abstract

We present an abstract setting for integration by parts inspired by the Malliavin calculus. In this framework we give the so-called Malliavin–Thalmaier formula which allows us to represent the density of the law of a multi-dimensional random variable using just one integration by parts, and we investigate the properties of the density using this formula. Finally we present a new argument, based on an interpolation method, which permits us to obtain regularity properties for the density under quite weak regularity assumptions.