Quaderni di Matematica

The aim of these lecture notes is to present some basic facts and ideas of the theory of Gaussian measures on infinite dimensional Hilbert spaces and to show to the reader how this theory can be applied to solve the infinite dimensional heat equation and, more generally, its perturbations by a linear drift term.

In particular, the Cameron-Martin theorem will be useful to obtain regularity properties of the semigroup generated by the Gross-Laplacian and the Ornstein-Uhlenbeck semigroup.

These notes originated from a course given by the second author at the University of Lecce in May 2002 and at the University of Halle-Wittenberg in May 2003.