Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases

Abstract

We study the stochastic nonlinear Schrödinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global well-posedness of solutions in both mass-critical and energy-critical cases. We also prove the rescaled scattering behavior of global solutions in the spaces ${\mathit{L}}^2$, ${\mathit{H}}^1$ as well as the pseudo-conformal space for dimensions $\mathit{d}\ge 3$ in the case of finite global quadratic variation of noise. Furthermore, the Stroock–Varadhan type theorem is also obtained for the topological support of the probability distribution induced by global solutions in the Strichartz and local smoothing spaces. Our proof is based on the construction of a new family of rescaling transformations indexed by stopping times and on the stability analysis adapted to the multiplicative noise.