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

Abstract

We study the stochastic nonlinear Schroedinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global existence and uniqueness of solutions in both cases. When the quadratic variation of noise is globally bounded, we also obtain the rescaled scattering behavior of solutions in the spaces $L^2$, $H^1$ as well as the pseudo-conformal space. Moreover, the Stroock-Varadhan type theorem for the topological support of solutions are also obtained in the Strichartz and local smoothing spaces.

References in corpus (3)

• Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$
• On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact riemannian manifold
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions

Cited by in corpus (6)

• On long time behavior for stochastic nonlinear Schrödinger equations with a multiplicative noise
• A note on log-log blow up solutions for stochastic nonlinear Schrödinger equations
• Decay of the stochastic linear Schrödinger equation in $d\ge 3$ with small multiplicative noise
• Minimal mass blow-up solutions to rough nonlinear Schroedinger equations
• Long time behavior of stochastic NLS with a small multiplicative noise
• A Wong-Zakai theorem for the stochastic mass critical NLS