Stochastic Partial Differential Equations

We study regularizing effects of noise for nonlinear SPDE. In this regard we are interested in phenomena where the inclusion of stochastic perturbations leads to increased regularity of solutions as compared to the unperturbed, deterministic cases. Closely related, we study effects of production of uniqueness of solutions by noise, i.e. instances of nonlinear SPDE having a unique solution, while non-uniqueness holds for the deterministic counterparts.

We study degenerate, quasilinear SPDE perturbed by rough noise. For such SPDE "classical" methods of proving well-posedness (e.g. the variational approach) do not apply anymore. The development of new methods applicable in these cases is one of the aims of this project.

We investigate stochastic dynamics induced by SPDE. In this regard we are interested in the generation of random dynamical systems by SPDE and their long-time behavior. In particular, stabilizing effects of noise are analyzed. For example, the inclusion of noise in a stochastic (partial) differential equation can lead to a globally stable long-time behavior, in the sense that the associated random attractor is a single point, while the corresponding deterministic dynamics are not globally stable.