Global well-posedness of the energy-critical nonlinear Schr\"odinger equation on the three-dimensional torus with additive L\'evy noise

Abstract

This paper establishes global well-posedness for the defocusing energy-critical nonlinear Schr\"odinger equation on the three-dimensional torus with a class of additive L\'evy noises. For arbitrary deterministic initial data in the energy space, we construct probabilistically strong solutions that are pathwise unique in the critical solution class. Continuous dependence on initial data, convergence of finite-activity approximations, and a vanishing-noise limit hold on finite time intervals. When the effective noise is nonzero, quantitative bounds for mass exit times imply almost-sure unboundedness of the mass and exclude free scattering. The expected energy then admits a cubic lower bound in time. The analysis couples pathwise critical perturbation theory with energy estimates adapted to Poisson compensation and stopping-time arguments based on the exact mass balance.