Malliavin calculus and normal approximations

This is an introductory course on Malliavin calculus and its applications. The Malliavin calculus is a stochastic calculus of variations with respect to the Brownian motion, that was introduced by Paul Malliavin in the 70's to provide a probabilistic proof of Hörmander's hypoellipticity theorem. The course will cover the main applications of Malliavin calculus including the existence and regularity of probability densities and the rate of convergence in normal approximations via Stein's method.  We will also discuss some recent applications to ergodicity and asymptotic behavior of spatial averages of stochastic partial differential equations.

The following topics will be treated:

1. Brownian motion and the Wiener space.
2. An introduction to the stochastic calculus with respect to the Brownian motion and change of variables formulas.
3. Derivative and divergence operators. The divergence operator as a stochastic integral.
4. The Wiener chaos and the Orsntein-Uhlenbeck semigroup. Meyer inequalities.
5. Stochastic integral representations. Clark-Ocone formula.
6. Proving tightness using Malliavin calculus.
7. Density formulas. Criteria for existence and regularity of probability densities and its application to diffusion processes.
8. Stein's method for normal approximations. Rate of convergence in total variation distance.
9. Central limit theorems for stationary sequences.
10. Ergodicity and asymptotic behavior of spatial averages of stochastic partial differential equations.