Asymptotic Behavior For Several Classes Of Stochastic Differential Equations

Dynamical system is the mathematical science which study on theevolution of systems. Inreality, many systems will inevitably be affectedby random factors. Considering the random factors in the dynamicalsystem will produce random dynamical system. Random dynamicalsystems appear in many important practical applications, such asphysics, mechanics, oceanography, meteorology, biology,communications engineering, other science and engineeringtechnology.One of the most basic and important research topics of dynamicalsystems is to study the asymptotic behavior of dynamical systems.Random attractor is an effective tool to describe the asymptoticbehavior of random dynamical systems. This paper is devoted to studythe asymptotic behavior of random dynamical systems generated byseveral classes of stochastic differential equations driven by Brownmovement, as well as the asymptotic behavior of stochastic retardedlattice dynamical systems, consider the existence of random attractorsof these systems. This is important not only in theory, but also inapplication.This paper is divided into three parts:The first part is the first two chapters. First of all, chapter1introducesthe background and status of this study, as well as the content andsignificance of this study. Then, chapter2describes some basicknowledge related to this paper. The second part is the core content ofthis paper, including the chapters3,4,5and6. Firstly, chapter3isdevoted to study the asymptotic behavior of the stochastic stronglydamped wave equation with homogeneous Neumann boundarycondition. We investigate the existence of a random attractor for therandom dynamical system associated with the equation. Secondly, chapter4and chapter5study the asymptotic behavior of stochasticdamped wave equation and the stochastic reaction–diffusionequation with multiplicative noise on unbounded domain, and considerthe existence of random attractors for the random dynamical systemsgenerated by the two equations, respectively. Finally, chapter6considers the asymptotic behavior of stochastic retarded latticedynamical systems. We first present some sufficient conditions for theexistence of a global random attractor for random dynamical systemsdefined on the separable Banach space C ([ν,0],lpρ). Then we applythe obtained abstract results to the first-order stochastic retarded latticesystem with random coupled coefficients and additive white noises,and prove that this system possesses a random attractor.The third part is chapter7, this chapter is to summarize the mainresults obtained in this paper, and propose some problems for futureresearch.