We give an overview of some recent results for doubly nonlinear stochastic evolution equations in Hilbert spaces, with applications to stocastic Stefan problems. In the first part of the talk we introduce the prototypes of the problems in consideration and we discuss the main existence results. Secondly, we focus on the direction of uniqueness by noise for doubly nonlinear evolutions, highlighting some recent contributions and open problems.
The works presented in the talk are based on joint collaborations with Prof. Ulisse Stefanelli (University of Vienna, Austria) and Prof. Carlo Orrieri (University of Pavia, Italy).