Stochastic dispersive PDEs with additive space-time white noise

Abstract

In this thesis, we will discuss the Cauchy problem for some nonlinear dispersive PDEs with additive space-time white noise forcing. We will focus on two different models: the stochastic nonlinear beam equation (SNLB) with power nonlinearity posed on the three dimensional torus, and the stochastic nonlinear wave equation with cubic nonlinearity in two dimensions, posed both on the torus and on the Euclidean space (SNLW). For (SNLB), we will present a joint work with R. Mosincat, O. Pocovnicu and Y. Wang, which settles local well-posedness for every nonlinearity of the type |u|p-1u, and global welllposedness for p < 11=3. In the case p = 3, we also consider a damped version of the equation, for which we can show invariance of the Gibbs measure. Moreover, we describe the long time-behaviour of the flow, by showing unique ergodicity of the Gibbs measure, and convergence to equilibrium for smooth initial data. In the case of (SNLW) with cubic nonlinearity, we consider a renormalised version of the equation, which was introduced by M. Gubinelli, H. Koch and T. Oh. In their work, they established local well-posedness on the two-dimensional torus. We show global existence for these solutions (joint with M. Gubinelli, H. Koch and T. Oh, and local and global wellposedeness for the same equation posed on the two-dimensional Euclidean space.