Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion |
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Authors: | María J Garrido-Atienza Peter E Kloeden Andreas Neuenkirch |
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Institution: | 1.Dpto. de Ecuaciones Diferenciales y Análisis Numérico,Universidad de Sevilla, Apdo. de Correos 1160,Sevilla,Spain;2.Institut für Mathematik,Johann Wolfgang Goethe-Universit?t,Frankfurt am Main,Germany |
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Abstract: | In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under
a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary
solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit
Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges
to the unique stationary solution of the original system as the stepsize of the discretization decreases.
Partially supported by the DAAD, Ministerio de Educación y Ciencia (Spain) and FEDER (European Community) under grants MTM2005-01412
and HA2005-0082, by Junta de Andalucía under the Proyecto de Excelencia P07-FQM-02468, and the DFG-project “Pathwise numerics
and dynamics of stochastic evolution equations”. |
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Keywords: | Fractional Brownian motion Random dynamical system Random attractor One-sided dissipative Lipschitz condition Implicit Euler scheme |
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