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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 |
Contents
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.
References
[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