Abstract
In this paper, we address the long time behavior of solutions of the stochastic Schrödinger equation \(du+(\lambda u+i\Delta u +i\alpha |u|^{2\sigma }u)dt=\Phi dW_t\) in \({{\mathbb {R}}}^{d}\). We prove the existence of an invariant measure in \(H^{1}\) for \(\sigma <2/(2-d)\) in the defocusing case and for \(\sigma <2/d\) in the focusing case. We also establish asymptotic compactness of invariant measures, implying in particular the existence of an ergodic measure.
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Acknowledgements
I.E. was partly supported by the Swiss National Foundation Grant SNF 200021-153555, I.K. was supported in part by the NSF Grants DMS-1311943 and DMS-1615239, while M.Z. was supported in part by the NSF Grant DMS-1109562.
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Ekren, I., Kukavica, I. & Ziane, M. Existence of invariant measures for the stochastic damped Schrödinger equation. Stoch PDE: Anal Comp 5, 343–367 (2017). https://doi.org/10.1007/s40072-016-0090-1
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DOI: https://doi.org/10.1007/s40072-016-0090-1