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Measurability of Random Attractors for Quasi Strong-to-Weak Continuous Random Dynamical Systems

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Abstract

In order to obtain the measurability of a random attractor, the RDS is usually required to be continuous which, however, is hard to verify in many applications. In this paper, we introduce a quasi strong-to-weak (abbrev. quasi-S2W) continuity and establish a new existence theorem for random attractors. It is shown that such continuity is equivalent to the closed-graph property for mappings taking values in weakly compact spaces. Moreover, it is inheritable: if a mapping is quasi-S2W continuous in some space, then so it is automatically in more regular subspaces. Also, a mapping with such continuity must be measurable. These results enable us to study random attractors in regularity spaces without further proving the system’s continuity. In addition, applying the core idea to bi-spatial random attractor theory we establish new existence theorems ensuring that the bi-spatial attractors are measurable in regularity spaces. As an application, for a stochastic reaction–diffusion equation with general conditions we study briefly the random attractor in \(H^1(\mathbb {R}^d)\), the \((L^2(\mathbb {R}^d), H^1(\mathbb {R}^d) )\)-random attractor and the \((L^2(\mathbb {R}^d),L^p(\mathbb {R}^d))\)-random attractor, \(p>2\), \(d\in \mathbb {N}\).

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Acknowledgements

The authors would like to express their sincere thanks to the referees for valuable comments and suggestions which led to a good improvement of this paper. Cui and Li was supported by National Natural Science Foundation of China Grant 11571283. Cui was partially funded by China Postdoctoral Science Foundation 2017M612430 and State Scholarship Fund 201506990049. Langa was partially supported by Junta de Andalucí́a under Proyecto de Excelencia FQM-1492, Brazilian-European partnership in Dynamical Systems (BREUDS) from the FP7-IRSES Grant of the European Union and FEDER Ministerio de Economiá y Competitividad Grant MTM2015-63723-P.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Hongyong Cui

  2. Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Seville, Spain

    José A. Langa

  3. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Yangrong Li

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  1. Hongyong Cui
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  2. José A. Langa
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Correspondence to Yangrong Li.

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Cui, H., Langa, J.A. & Li, Y. Measurability of Random Attractors for Quasi Strong-to-Weak Continuous Random Dynamical Systems. J Dyn Diff Equat 30, 1873–1898 (2018). https://doi.org/10.1007/s10884-017-9617-z

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  • DOI: https://doi.org/10.1007/s10884-017-9617-z

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