The main goal of the course is to develop a first knowledge of the variational approach to stochastic PDEs.

The first part of the course is devoted to the development of a general theory of stochastic analysis in infinite dimensions: this will cover, in particular, Gaussian measures in Hilbert spaces, stochastic processes and martingales in Hilbert spaces, quadratic variation and tensor quadratic variation, measurability of stochastic processes, stochastic compactness methods, stochastic integration with respect to a square-integrable continuous martingale, Wiener processes and cylindrical Wiener processes, maximal inequalities for square integrable martingales, Burkholder-Davis-Gundy inequality, Itô formula.

The second part of the course covers applications of such tools to the variational theory of stochastic PDEs: existence/uniqueness results for SPDEs with Lipschitz nonlinearities, existence/uniqueness results for SPDEs of monotone type in Hilbert triplets.

Eventually, selected examples of stochastic partial differential equations will be discussed, with focus on the state of the art and possible open problems: these will mainly cover phase-field-type SPDEs such as stochastic Allen-Cahn and stochastic Cahn-Hilliard equations.

Prerequistes: Real and Functional Analysis, Probability.