Introduced by Paul Malliavin in 1978, Malliavin calculus can be roughly described as an infinite-dimensional differential calculus whose operators act on sets of random objects associated with Gaussian or more general noises. Originally introduced and exploited for studying the regularity of the laws of Wiener functionals (such as the solutions of stochastic differential equations), it has found many other applications during the last two decades, including concentration inequalities, anticipated stochastic calculus or computation of Greeks in mathematical finance.
Stein’s method gathers a collection of probabilistic techniques for assessing the distance between probability distributions by means of differential operators. This approach was originally developed by Charles Stein in the seventies. In recent years, Stein’s method has become one of the most popular and powerful tools for computing explicit bounds in probabilistic limit theorems, with applications to fields as diverse as random matrices, random graphs, probability on groups and spin glasses. In 2009, it was discovered by Giovanni Peccati and the speaker that one can fruitfully combine Stein’s method with Malliavin calculus, in order to obtain probability approximations for random systems that are driven by some underlying Gaussian noise. The introduction of such a method has been the starting point of many applications and generalizations by dozens of authors in a number of areas, ranging from Gaussian analysis to stochastic geometry, and from combinatorics to non-commutative probability, information theory, concentration inequalities and universality results. The goal of this series of lectures is to introduce the audience to this combination of Malliavin calculus and Stein’s method, nowadays called the Malliavin-Stein approach. We will also review some recent applications. All the needed material will be introduced progressively; no specific knowledge is required, beyond the definitions and very basic properties of Gaussian processes (mostly Brownian motion).