Stability of Infinite Dimensional Stochastic Differential Equations with Applications

Liu, Kai

doi:10.1201/9781420034820

Cited by 97 publications

(126 citation statements)

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“…The local asymptotic behavior in the neighborhood of equilibria has been extensively studied in the literature (see e.g., [57,60,61]), but will not be commented here. We are interested in analyzing the global behavior of SDEs by the theory of random dynamical systems.…”

Section: Global Asymptotic Behavior Of Sdes: Conjugation Of Rdsmentioning

confidence: 99%

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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PrefaceDuring the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering.The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing.Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering.

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Section: Global Asymptotic Behavior Of Sdes: Conjugation Of Rdsmentioning

confidence: 99%

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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“…Recently, the studies of stochastic differential equations have attracted considerable attention among scholars. Many interesting results have been obtained over the last few years (see, for example, [7,9,10,23] and the references therein). Stability analysis is very important for stochastic delay systems, as we like to know the impact of memory as well as noise.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation

Abou-El-Ela¹,

Sadek²,

Mahmoud³

et al. 2017

Turk J Math

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“…Ichikawa 1982;Da Prato & Zabczyk 1992;Caraballo & Liu 1999;Govindan 2003;Liu 2004;Luo & Liu 2008 and references therein). In particular, many scholars have paid much attention to stochastic partial differential equations with delays.…”

Section: Introductionmentioning

confidence: 99%

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

2009

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The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al. 118, and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

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Stability of Infinite Dimensional Stochastic Differential Equations with Applications

Cited by 97 publications

References 0 publications

Applied Nonautonomous and Random Dynamical Systems

Applied Nonautonomous and Random Dynamical Systems

Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

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