Prerequisites: The course should be accessible to students with a background either in probability or in analysis. The material covered in Stochastic Analysis is not assumed, although this course may be taken as a follow-up to V4F1.
Contents: The course gives an introduction to analysis on function spaces endowed with a probability measure (e.g. the Wiener space), and to stochastic partial differential equations.
Gaussian measures on function spaces: Existence, regularity, Wiener processes, Gaussian free field, perturbations of Gaussian measures.
Semigroups and generators, Dirichlet forms.
Functional inequalities on Gaussian measure spaces: Logarithmic Sobolev Inequality, Isoperimetric Inequality, Concentration Inequalities.
Linear SPDE: Stochastic integral, stochastic convolutions, Ornstein-Uhlenbeck processes on Hilbert spaces, time and space regularity, long-time behaviour.
Semilinear SPDE: Reaction-diffusion equations, Navier-Stokes.