Stochastic Perturbations in the Fractional Nonlinear Schr\"odinger Equation: Well-posedness and Blow-up

Abstract

This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the energy-subcritical regime for the stochastic fractional nonlinear Schr\"odinger equation. Global existence is subsequently demonstrated through stochastic evolution of mass and energy. In focusing supercritical settings, we derive blow-up criteria via localized virial inequality, revealing how multiplicative noise measurably suppresses blow-up formation compared to deterministic dynamics.