Research On Almost Periodicity And Almost Automorphy Of Stochastic Functional Differential Equations

The asymptotic behaviors such as periodic solutions, almost periodic solutions andalmost automorphic solutions of diferential equations have attracted much attention theseyears since they manifest the regular change of the system. Comparing with periodicphenomenon, the almost periodic phenomenon is easier to be observed in the real worldand the almost automorphic phenomenon is the most common of the three. Since H. Bohrproposed the theory of almost periodic function and S. Bochner introduced the almostautomorphic function, they had been found with successful and interesting applicationsin functional diferential equations, impulsive diferential equations, partial diferentialequations and other areas. This fact consumedly enlarges and develops the theory ofalmost periodic functions and almost automorphic functions. So the researchs on almostperiodicity and almost automorphy of diferential equations are interested in the theoryof mathematics and have broad prospects in practical applications.As we all know both stochastic diferential equations and deterministic functionaldiferential equations are two important tools to describe the variation of the changingprocesses. Stochastic functional diferential equations can be viewed as the generaliza-tions of both stochastic diferential equations and deterministic functional diferentialequations. Since the environmental noises and the retarded factor are considered, theycan truthfully simulate the problems in the real world and have been widely appliedto model the corresponding systems in many fields such as chemistry, physics, ecology,medicine, finance, neural networks and control science etc.In this doctorial dissertation, we study the almost periodicity and almost automorphyof stochastic functional diferential equations by using the theory of stochastic analysisand technique of inequality including Ito formula, mean inequality of stochastic integral,H¨older inequality, and combining the methods to research the asymptotic behaviors whichare transplanted from deterministic diferential equations. We introduce two concepts ofp-mean asymptotic almost periodic stochastic process and p-mean almost automorphicstochastic process, and propose some sufcient conditions to ensure the existence andstability of square mean almost periodic mild solutions, square mean asymptotic almostperiodic mild solutions, square mean almost automorphic mild solutions and square meanpseudo almost automorphic mild solutions for stochastic functional diferential equations.Moreover we give some examples to illustrate the efectiveness of these conditions. Thedetailed work of this dissertation are as follows. Chapter1simply introduces the advance and some basic properties of almost peri-odic functions, asymptotic almost periodic functions, pseudo almost periodic functions,almost automorphic functions, asymptotic almost automorphic functions, pseudo almostautomorphic functions and strongly continuous semigroup of linear operators, the theoryof stochastic functional diferential equations and stochastic analysis, the background,the significance and main work of this dissertation.In chapter2, we study the existence and uniform stability of square mean almostperiodic mild solutions for stochastic functional diferential equations. We give somesufcient conditions to ensure the existence of square mean almost periodic mild solutionsfor stochastic functional diferential equations. Based on the existence, we further givesome sufcient conditions to ensure the uniform stability of square mean almost periodicmild solutions. Finally, we study the square mean almost periodic mild solution for astochastic model to illustrate the efectiveness of our abstract results.In chapter3and4, based on the concept of p-mean almost periodic stochastic process,we introduce two new concepts, p-mean asymptotic almost periodic stochastic process andp-mean almost automorphic stochastic process, and study some basic properties of them.Based on these properties, we apply these concepts to study the square mean asymptoticalmost periodic mild solutions and square mean almost automorphic mild solutions forstochastic functional diferential equations, and give some sufcient conditions to ensurethe existence and exponential stability of square mean asymptotic almost periodic mildsolutions and square mean almost automorphic mild solutions for stochastic functionaldiferential equations. Finally, we study the square mean asymptotic almost periodicmild solutions and square mean almost automorphic mild solutions for some stochasticmodels, which illustrate the efectiveness of our abstract results.In chapter5, we study the existence and exponential stability of square mean pseu-do almost automorphic mild solutions for stochastic functional diferential equations. Wegive some sufcient conditions to ensure the existence of square mean pseudo almost au-tomorphic mild solutions for stochastic functional diferential equations. Based on theexistence, we further give some sufcient conditions to ensure the exponential stabilityof square mean pseudo almost automorphic mild solutions. Finally, we study the squaremean pseudo almost automorphic mild solutions for two stochastic models, which illus-trate the efectiveness of our abstract results.