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.
Hornung,The stochastic nonlinear Schr¨odinger equation in unbounded domains and non-compact manifolds, Nonlinear Differ
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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.