Existence of invariant measures for the stochastic damped Schrödinger equation

Existence of invariant measures for the stochastic damped Schrödinger equation In this paper, we address the long time behavior of solutions of the stochastic Schrödinger equation $$du+(\lambda u+i\Delta u +i\alpha |u|^{2\sigma }u)dt=\Phi dW_t$$ d u + ( λ u + i Δ u + i α | u | 2 σ u ) d t = Φ d W t in $${{\mathbb {R}}}^{d}$$ R d . We prove the existence of an invariant measure in $$H^{1}$$ H 1 for $$\sigma <2/(2-d)$$ σ < 2 / ( 2 - d ) in the defocusing case and for $$\sigma <2/d$$ σ < 2 / d in the focusing case. We also establish asymptotic compactness of invariant measures, implying in particular the existence of an ergodic measure. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Stochastical Partial Differential Equations Springer Journals

Existence of invariant measures for the stochastic damped Schrödinger equation

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Springer US
Copyright © 2017 by Springer Science+Business Media New York
Mathematics; Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis
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