Abstract Malliavin calculus          
     and regularization properties     

Vlad Bally
Université Paris-Est, Marne-la-Vallée

A course for the PhD School in Mathematics at Tor Vergata
Spring semester 2019

The first outstanding application of Malliavin calculus, introduced in 1975 by Paul Malliavin, was a probabilistic proof of the Hörmander regularity criterion. But in the 40 last years it gave rise to a huge amount of applications, and in particular it has been developed as a branch of stochastic analysis on the Wiener space (see the classical book of Nualart [1], as well as the more recent books of Nourdin and Peccati [2] and of Nualart and Nualart [3]). The present course does not go in this direction, but focuses on the initial application of this calculus: regularity of probability laws. Our aim is to present an abstract framework (close to Dirichlet forms) in which such properties may be obtained by using some integration by parts techniques which are strongly inspired by the ones initiated by Malliavin. But there are two specific differences with respect to the classical calculus. First of all, it does not work only in the Gaussian framework but extends to a large class of random variables by means of a splitting technique. Moreover, the classical Malliavin calculus is an infinite dimensional differential calculus: the idea is that one defines the differential operators on the space of some “smooth functionals” (which essentially are finite dimensional) and then extends (closes) the operators - in this way one obtains an infinite dimensional calculus. In contrast, we restrict ourselves to a finite dimensional calculus (which follows the lines of Malliavin calculus), and then we use a “balance argument” (which essentially fits in the framework of the interpolation theory) in order to obtain results for infinite dimensional functionals which are not in the domain of the differential operators. For these two reasons we are far from the point of view of the analysis on Wiener space.
As we mentioned before, the central tool in our approach is an integration by parts formula. We give two main applications. First, we consider the regularity of the law of some random variables and estimates of the density. And a second application concerns the use of the regularization effect in order to obtain the convergence in total variation distance. All over we try to remain in an elementary framework and to avoid technicalities - so we do not aim to obtain the maximal generality or the weaker possible hypothesis.

[1] D. Nualart, The Malliavin calculus and related topics. Springer, 2008.
[2] I. Nourdain, G. Peccati, Normal approximations with Malliavin calculus. From Stein's method to universality. Cambridge Tracts in Mathematics, 2012.
[3] D. Nualart, E. Nualart, Introduction to Malliavin Calculus. IMS Textbooks, Cambridge University Press, 2018.

Schedule: from May 21st 2019 to June 20th 2019