Syllabus: This is an introductory course about analysis in separable Hilbert spaces endowed with non-degenerate Gaussian measures. Sobolev spaces and spaces of continuous functions from X to R will be considered. The basic differential operators (gradient and divergence) will be studied, as well as Ornstein-Uhlenbeck operators and Ornstein-Uhlenbeck semigroups, that are the Gaussian analogues of the Laplacian and the heat semigroup. The most important functional inequalities in this context, such as Poincaré and logarithmic Sobolev inequalities, will be proved. Hermite polynomials and the Wiener chaos decomposition  will be described. 

The reference books are "Gaussian Measures" by V. Bogachev (Mathematical Surveys and Monographs 62, AMS 1998); "Second Order Partial Differential Equations in Hilbert Spaces" by G. Da Prato and J. Zabczyk (Cambridge Univ. Press 2002); “An introduction to Infinite Dimensional Analysis” by G. Da Prato (Springer 2006).

Il corso consiste di 12 incontri da due ore e sarà possibile assistere da remoto. Per essere aggiunti al Team di Teams scrivere all'indirizzo paolo.defazio@unipr.it.