On closure methods in nonlinear stochastic dynamics

Abstract

Please cite this article as: Bobryk, R.V., On closure methods in nonlinear stochastic dynamics. Statistics and Probability Letters (2010Letters ( ), doi:10.1016Letters ( /j.spl.2010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTOn closure methods in nonlinear stochastic dynamics Roman V. BobrykInstitute of Mathematics, Jan Kochanowski University, 25406 Kielce, Poland AbstractThe paper discusses some problems in applications of closure techniques to moment hierarchies by considering equation of overdamped oscillator with cubic nonlinearity and additive Gaussian white noise. It is rigorously shown that some closure schemes can be inconsistent in spite of they good convergence properties.

Closure schemes in stochastic nonlinear dynamics: A validation case study

“…Nevertheless, it is showed rigorously that this closure scheme is convergent to (6) if α < 0 and to the inconsistent solution (8) if α > 0 [20]. A more sophisticated closure scheme consists in applying to the nth equation the relation [19] y 2n+2 (α,β,σ ) = γ y 2n (α,β,σ ), γ > 0, (11) but it gives the same result [20].…”