On the stochastic Strichartz estimates and the stochastic nonlinear Schr\"odinger equation on a compact riemannian manifold

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities

Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

We prove the pathwise uniqueness of solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov, to the stochastic setting. The proof is based on the deterministic and new stochastic spectrally localized Strichartz estimates and the Littlewood-Paley decomposition

Almost Sure Scattering at Mass Regularity for Radial Schr\"odinger Equations

We consider the radial nonlinear Schr\"odinger equation $i\partial_tu +\Delta u = |u|^{p-1}u$ in dimension $d\geqslant 2$ for $p\in \left[1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure $\mu$ which support is almost $L^2_{\text{rad}}$ and such that $\mu$ - almost every initial data gives rise to a unique global solution. Furthermore, for $p>1+\frac{2}{d}$ the solutions constructed scatter in a space which is almost $L^2$. This paper can be viewed as the higher dimensional counterpart of the work of Burq and Thomann, in the radial case.Comment: Typos correcte