Stochastic nonlinear Schrödinger equation on half-line with boundary noise

Abstract

We consider the stochastic nonlinear Schrödinger equations on the half-line with Neumann brown-noise boundary conditions. We establish the global existence and uniqueness of solutions to initial-boundary value problem with values in H1. We are also interested in the regularity behavior of the first spatial derivative of the solutions near the origin, where the boundary data are highly irregular. To obtain optimal estimate of the stochastic boundary response we propose new method based on Laplace transform and Cauchy theory of complex analysis. Also we adopt Sthriharts estimates and the Gagliardo–Nirenberg interpolation inequalities for the case of stochastic equations on a half-line.