Global solutions of the nonlinear Schrödinger equation with multiplicative noise

Abstract

In this thesis, we investigate existence and uniqueness of solutions to the stochastic nonlinear Schrodinger equation (NLS), i.e. the NLS perturbed by a multiplicative noise. First, we present a fixed point argument based on deterministic and stochastic Strichartz estimates. In this way, we prove local existence and uniqueness of stochastically strong solutions of the stochastic NLS with nonlinear Gaussian noise for initial values in $L^2(\mathbb{R}^d)$ and $H^1(\mathbb{R}^d),$ respectively. Using a stochastic generalization of mass conservation, we show that the $L^2$-solution exists globally under an additional restriction of the noise. In the second part, we develop a general existence theory for global martingale solutions of the stochastic NLS with a saturated Gaussian multiplicative noise. The proof is based on a modified Galerkin approximation and a limit process due to the tightness of the approximated solutions. As an application, we get existence results for the stochastic defocusing and focusing NLS and fractional NLS on various geometries like bounded domains with Dirichlet or Neumann boundary conditions as well as compact Riemannian manifolds. The martingale solution constructed by the Galerkin method is not necessarily unique. For this reason, we independently show pathwise uniqueness of solutions to the stochastic NLS with linear conservative Gaussian noise. The proof works in special cases as 2D and 3D Riemannian manifolds and is based on spectrally localized Strichartz estimates. In the last chapter, we replace the Gaussian noise by a Poisson noise in the Marcus form and transfer the proof of existence of martingale solutions to this case.