31 mar 2021 -- 11:00 [open in google calendar]
University of Ferrara (on-line)
Abstract.
In 1976, at an international Symposium on stochastic differential equations held at Kyoto University, Paul Malliavin mentioned a new theory of calculus, the stochastic calculus of variations. This theory made a big impact, and it turned out to have applications not only to PDEs but also many other fields; due to his huge contribution, the theory is usually called Malliavin Calculus. In the first part of the talk I’ll present some historical background on the theory of integration and differentiation in infinite dimension developed in the 20th century, in particular on classical and abstract Wiener space, which inspired Malliavin to create his theory. The second part of the talk will be devoted to discuss the basic framework and tools of Malliavin Calculus: Gaussian subspaces, isonormal processes, Hermite polynomials, Wiener chaos decomposition, derivative operator, Sobolev spaces, divergence operator and Ornstein-Uhlenbeck semigroup. Sometimes it will be useful to consider at first an easiest case (dimension 1) and then to generalize to infinite dimension. Finally, I will state a recent result about equivalence of norms in Sobolev spaces. This is a joint work with M. Muratori (PoliMi) and M. Rossi (Milano Bicocca).