Under localized damping and noise, the randomly forced NLS on the 1D torus admits a unique invariant measure and exponentially fast convergence in the dual-Lipschitz metric.
Almost surely smoothed scattering for cubic NLS
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We consider cubic NLS in dimensions 2, 3, 4 and we prove that almost surely solutions with randomized initial data at low regularity scatter. Moreover, we establish some smoothing properties of the associated scattering operator and precise the rate of convergence.
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Exponential mixing for the randomly forced NLS equation
Under localized damping and noise, the randomly forced NLS on the 1D torus admits a unique invariant measure and exponentially fast convergence in the dual-Lipschitz metric.