Abstract
This chapter is mainly developed to consider stochastic NLSEs possessing the stochastic multi-symplectic conservation law and the charge conservation law (see e.g. [64, 119]). For this kind of conservative equations, it then suffices to consider its dynamic behavior on the unit sphere without loss of generality. We show in Sect. 6.1 that the finite dimensional approximation (FDA) based on the midpoint scheme is ergodic with a unique invariant measure. Also, the ergodic limit of this FDA can be approximated via the temporal average of an ergodic fully discrete scheme(FDS), see Sect. 6.3.
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Hong, J., Wang, X. (2019). Approximation of Ergodic Limit for Conservative Stochastic Nonlinear Schrödinger Equations. In: Invariant Measures for Stochastic Nonlinear Schrödinger Equations. Lecture Notes in Mathematics, vol 2251. Springer, Singapore. https://doi.org/10.1007/978-981-32-9069-3_6
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DOI: https://doi.org/10.1007/978-981-32-9069-3_6
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