Continuous square-integrable martingale noise with positive quadratic variation density induces pathwise exponential decay of the L2 norm in stochastic nonlinear Schrödinger equations via a rescaling that produces a damping potential.
Tsutsumi,Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schr¨odinger equations, SIAM, J., Math., Anal.,15(2) (1984)
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Exponential Decay of $L^2$-Solutions to Stochastic Nonlinear Schr\"odinger Equations Driven by Continuous Martingales
Continuous square-integrable martingale noise with positive quadratic variation density induces pathwise exponential decay of the L2 norm in stochastic nonlinear Schrödinger equations via a rescaling that produces a damping potential.