In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise.
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October 2024
Research Article|
October 03 2024
Dynamics of stochastic differential equations with memory driven by colored noise
Special Collection:
Nonautonomous Dynamical Systems: Theory, Methods, and Applications
Ruonan Liu
;
Ruonan Liu
a)
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing – original draft, Writing – review & editing)
1
School of Mathematics and Statistics, Xuzhou University of Technology
, Jiangsu 221008, People’s Republic of China
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Tomás Caraballo
Tomás Caraballo
b)
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing – original draft, Writing – review & editing)
2
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla
, 41012 Sevilla, Spain
b)Author to whom correspondence should be addressed: caraball@us.es
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b)Author to whom correspondence should be addressed: caraball@us.es
a)
Electronic mail: liu794046243@163.com
Chaos 34, 103110 (2024)
Article history
Received:
June 17 2024
Accepted:
September 09 2024
Citation
Ruonan Liu, Tomás Caraballo; Dynamics of stochastic differential equations with memory driven by colored noise. Chaos 1 October 2024; 34 (10): 103110. https://doi.org/10.1063/5.0223756
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