Research On Random Attractors Of Two Classes Of Stochastic Equations Based On Wong-Zakai Approximation

Stochastic(partial)differential equation(S(P)DE)can effectively describe random phenom-ena or uncertain phenomena in the real world,and the stochastic excitation sometimes even plays a decisive role,which can lead to fundamental changes in the system.At present,several non-linear S(P)DE models have been proposed and applied in many fields such as engineering tech-nology,life sciences and financial management.These models have profoundly explained some realistic phenomena that are difficult to describe in deterministic systems.This fully shows that it is necessary to carry out in-depth research on S(P)DEs and provide accurate mathematical ba-sis for practical applications.As we all know,the research on the qualitative nature of S(P)DEs has always been an important and basic direction in the theoretical system and applications of differential equations and their dynamic systems.In particular,as a compact invariant subset of the solution space,the random(pullback)attractor can characterize the long-term dynamic behavior of the generated random dynamic system,so the research on the random attractor has important scientific significance and application value.In this dissertation,the stochastic chemostat model driven by Brownian motion is studied.We introduce the Mond-Haldane type non-monotonic response function and obtain the exis-tence and explicit expression of the random attractor of the model.It has the internal structure composed of singleton sets.Secondly,by introducing the Wong-Zakai stationary approxima-tion of Brownian motion,we prove the existence,internal structure,and upper semi-continuity of the random attractor of such equation.Next,we focus on the stochastic evolution equation driven by infinite- dimensional fractional Brownian motion,propose a reasonable definition for Wong-Zakai stationary approximation of infinite-dimensional fractional Brownian motion,and introduce an appropriate stopping time sequence to obtain the existence of random attractor of such approximation equation.Finally,the convergence of the infinite-dimensional fractional Gauss noise,stopping time sequence,mild solution,and upper semi-continuity of the corre-sponding random attractor in the sense of Wong-Zakai stationary approximation are proved.The details are as follows:Chapter 1 is the introduction,which gives a survey to the background of this dissertation,the preliminary of stochastic dynamical systems and the basic theory of random attractors,the main inequality and related lemmas,the stochastic integrals for H?lder continuous functions and related properties.Chapter 2 considers a stochastic chemostat model with a Monod--Haldane type non-monotonic response function.By conjugate we prove the existence of the global solution of the model,and further obtain the existence and internal structure of its random attractor.Chapter 3 studies the stochastic chemostat model driven by Wong-Zakai stationary ap-proximation of Brownian motion.Finally we prove the existence,internal structure and upper semi-continuity of the random attractor of such approximation equation.Chapter 4 first proposes a reasonable definition for the Wong-Zakai stationary approxima-tion of infinite-dimensional fractional Brownian motions.By constructing a suitable stopping time sequence,we deal with the difficulty of obtaining the compactness of random absorbing set due to the unboundedness of noise,then we obtain the compactness in a sufficiently small random time interval.Combining the separability of the stochastic integral,thus we obtain the existence of the random attractor of approximation equation.Chapter 5 focuses on the convergence of infinite-dimensional fractional Gauss noise,stop-ping time sequences,mild solution,and the upper semi-continuity of corresponding random attractors in the sense of Wong-Zakai stationary approximation.