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Asymptotic Dynamics of Stochastic Lattice Differential Equations: A Review

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Book cover Continuous and Distributed Systems II

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 30))

Abstract

This is an expository article on asymptotic dynamics of stochastic lattice differential equations. In particular, we investigate the long-term behavior of stochastic lattice differential equations, by using the concept of global random pullback attractor in the framework of random dynamical systems. General results on the existence of global compact random attractors are first provided for general random dynamical systems in weighted spaces of infinite sequences. They are then used to study the existence of global pullback random attractors for various types of stochastic lattice dynamical systems with white noise.

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Notes

  1. 1.

    A random set \(D(\omega )\) is said to be tempered with respect to \((\theta _t)_{t \in \mathbb {R}}\) if for a.e. \(\omega \in {\varOmega }\), \(\lim _{t \rightarrow \infty }e^{-\gamma t} \sup _{x \in D(\theta _{-t} \omega )} \Vert x\Vert _X = 0\) for all \(\gamma >0\). A random variable \(\omega \mapsto r(\omega )\in \mathbb {R}\) is said to be tempered with respect to \((\theta _t)_{t\in \mathbb {R}}\) if for a.e. \(\omega \in {\varOmega }\), \( \lim \limits _{t\rightarrow +\infty }e^{-\gamma t}\sup \limits _{t\in \mathbb {R}}|r(\theta _{-t}\omega )|=0\) for all \(\gamma >0\).

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Acknowledgments

This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2011-22411.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL, 36849, USA

    Xiaoying Han

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  1. Xiaoying Han
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Correspondence to Xiaoying Han .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Viktor A. Sadovnichiy

  2. Kiev Polytechnic Institute, National Technical University Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

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Han, X. (2015). Asymptotic Dynamics of Stochastic Lattice Differential Equations: A Review. In: Sadovnichiy, V., Zgurovsky, M. (eds) Continuous and Distributed Systems II. Studies in Systems, Decision and Control, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-19075-4_7

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  • DOI: https://doi.org/10.1007/978-3-319-19075-4_7

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