The Research Of Pseudo Almost Automorphic Type Stochastic Processes And Its Appplications In Stochastic Differential Equations

Compared with the periodic phenomena,the almost periodic phenomena is more accurate to depict many regular changes in the nature.As a generalization of almost pe-riodic functions,the range of almost automorphic functions is more extensive.Since Bohr and Bochner introduced the almost periodic functions and almost automorphic functions respectively,there has attracted more and more attention in the investigation of this two concepts and there series of generalizations.Especially,after the concept of square-mean pseudo almost automorphic stochastic processes was proposed in 2011,this notion has been promoted to square-mean weighted pseudo almost automorphic stochastic processes,square-mean?-pseudo almost automorphic stochastic processes,square-mean Stepanov weighted pseudo almost automorphic stochastic processes and so on.In recent years,these pseudo almost automorphic type stochastic processes have been widely applied into the research of qualitative theory in abstract equations,stochastic differential equations,par-tial differential equations et al,which greatly promote the development of the differential equations and dynamical systems.Therefore,it has important theoretical and practical significance of this paper to further study the properties of pseudo almost automorphic type stochastic processes and its applications,the specific contents are as follows:Chapter 1 briefly introduces the concepts of almost automorphic functions,asymp-totically almost automorphic functions and pseudo almost automorphic type functions,presents some elementary knowledge,such as stochastic differential equations and stochas-tic analysis,C₀-semigroup,also provides the research background and main contents of this paper.Chapter 2 mainly investigates the p-mean doubly-weighted pseudo almost automor-phic stochastic processes and its applications.Firstly,based on the concept of weighted pseudo almost automorphic stochastic processes,we give the concept of doubly-weighted pseudo almost automorphic stochastic processes by introducing a class of nonequivalent weight functions.Secondly,we discuss the equivalence,convolution invariance and trans-lation invariance of the space of doubly-weighted pseudo almost automorphic stochastic processes.Further,for a class of nonlinear stochastic differential equations driven by G-Brown motion,we obtain the existence,uniqueness and exponential stability of p-mean doubly-weighted pseudo almost automorphic mild solutions.Finally,we present an example to illustrate the effectiveness of our results.Chapter 3 mainly researches the p-mean Poisson?-pseudo almost automorphic s-tochastic processes and its applications.Firstly,we give the concept of p-mean Poisson?-pseudo almost automorphic stochastic processes,and prove the equivalence between p-mean Poisson?-pseudo almost automorphic stochastic processes and p-mean Poisson asymptotically almost automorphic stochastic processes in a appropriate open set.Fur-ther,we establish a composition theorem for p-mean Poisson?-pseudo almost automor-phic stochastic processes.Moreover,under some assumptions,we obtain the existence,uniqueness and exponential stability of p-mean?-pseudo almost automorphic mild so-lutions for a class of nonlinear stochastic differential equations driven by L?evy process.Finally,we analysis a specific stochastic differential equation to illustrate the effectiveness of our results.Chapter 4 mainly studies the p-mean Stepanov doubly-weighted pseudo almost au-tomorphic stochastic processes and its applications.We firstly introduce the concept of p-mean Stepanov doubly-weighted pseudo almost automorphic stochastic processes.Then,under some conditions,for any p-mean Stepanov doubly-weighted pseudo almost automorphic stochastic process h,we give the Stepanov almost automorphic component f and the disturbance component h₀of h,and obtain the size relation between h and f in the sense of norm.Especially,when h satisfies the Lipschitz assumption,then f also sat-isfies the Lipschitz condition with the same Lipschitz constant as h.Further,we establish the composition theorem of p-mean Stepanov doubly-weighted pseudo almost automor-phic stochastic processes and present the completeness of the space of p-mean Stepanov doubly-weighted pseudo almost automorphic stochastic processes.Moreover,we inves-tigate a class of nonlinear stochastic differential equations driven by Brown motion and obtain the corresponding discriminant theorem to show the existence and uniqueness of p-mean Stepanov doubly-weighted pseudo almost automorphic mild solution.Finally,we discuss an example of stochastic differential equations to illustrate the effectiveness of our theoretical results.