# 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

, Volume 5 (3) – Jan 3, 2017
25 pages

/lp/springer_journal/existence-of-invariant-measures-for-the-stochastic-damped-schr-dinger-HEqoUbTgNR
Publisher
Springer US
Subject
Mathematics; Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis
ISSN
2194-0401
eISSN
2194-041X
D.O.I.
10.1007/s40072-016-0090-1
Publisher site
See Article on Publisher Site

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