The lectures are planned for the Ph.D. in mathematics and can be attended by Ph.D. students of other programs. It is open to students of the Master degree in Mathematics, keeping in mind that prerequisits are some knowledge of stochastic calculus in finite dimensional spaces - stochastic processes, Brownian motion and its proprieties, elements of martingale theory, stochastic integration and Ito formula, elements on Girsanov formula and stochastic differential equations in finite dimensions, with the possibility however fror the latter to be shortly recalled in the first part of the lectures - and some experience with partial differential equations especially of parabolic type, possibly only elements of the theory of heat equation. No knowledge is required on equations of fluid mechanics.
The 40 hours of lecture will be divided approximatively in three parts, as follows.
The first part, about 15 hours, will deal with foundations of stochastic calculus in infinite dimensional spaces - the concept of cylindrical white noise, for instance - and first elements of theory of stochastic partial differential equations, for instance the stochastic convolution. In this part it is possible, to some extent, to cover rapidly some of the prerequisites, if asked by students.
The second part, about 15 hours, will deal with more advanced examples like stochastic Euler and Navier-Stokes equations, and more advanced questions like the link with the associated Kolmogorov and Fokker-Planck equations in infinite dimensions.
The third part, about 10 hours, will discuss applications, mainy to climatology, dealing both with modelling questions - like the role of the noise for modelling of subgrid processes - and the difficult problem of computing probabilities and expected values of solutions.
The goal of the lectures is to introduce students to some of the methodologies of investigation of SPDEs, giving more space to questions and tools of general interest and wide applicability. Students will also see the interest in application to climate change studies.
Notes given by the lecturer.