Recurrence Of Two Classes Of Stochastic Differential Equations With Regime-switching

In the study of random dynamical systems,a crucial subject is about recurrence.For stochastic systems with regime-switching,recurrence remains a challenging problem.This thesis investigates recurrence for two classes of stochastic systems with regime-switching,including periodic solutions in distribution,stationary solutions and invariant measures.Commencing with finite-dimensional stochastic systems,we discuss the recurrence,periodic solutions in distribution,stationary solutions and invariant measures for stochastic differential equations driven by Levy noise with regimeswitching.Subsequently,we investigate the existence of periodic solutions in distribution for stochastic lattice differential equations with regime-switching.This thesis consists of five chapters,with the main content outlined as follows:In Chapter 1,we introduce the research background of this thesis,including the historical and current perspectives of regime-switching,stochastic differential equations with regime-switching,stochastic lattice differential equations with regime-switching,and recurrence.In addition,we introduce the main work of this paper and the full text arrangement.In Chapter 2,we present the basic knowledge used in this paper,including basic concepts and related properties,basic inequalities and formulas,as well as classical theorems.In Chapter 3,we consider regime-switching jump diffusion processes generated by non-autonomous stochastic differential equations driven by Levy noises.Utilizing Mmatrix theory and Lyapunov functions,we establish sufficient conditions for recurrence,positive recurrence and transience of a state-dependent regime-switching jump diffusion process in a finite state space.The case of infinite countable state spaces is certainly difficult.We construct a finite partition based on the transition probability matrix of regime-switching to transform the original system into a finite state space.Then we establish the recurrence and positive recurrence for state-dependent regime-switching jump diffusion processes in infinite countable state spaces.Furthermore,we study a specific recurrence,namely periodic solutions in distribution,and derive a periodic law of large numbers.In addition,we present some illustrative examples to demonstrate the main theorems.In Chapter 4,we focus on two specific recurrence for stochastic differential equations driven by Levy noise,namely,stationary solutions and invariant measures.Under the background of random dynamical systems,we construct a random differential equation with Markovian switching,which is conjugated to the original equation.And we prove that the random differential equation with Markovian switching admits a unique stationary solution.Consequently,by the conjugate relationship among stationary solutions,we establish the existence and uniqueness of stationary solution for the original equation.Additionally,we demonstrate that the Markov transition semigroup forms a Feller semigroup.We establish the existence of invariant measures through the solutions of a class of Fokker-Planck equations and the Markov transition semigroup.In Chapter 5,we consider stochastic lattice differential equations with regimeswitching.First,we discuss the well-posedness of solutions for stochastic lattice differential equations with regime-switching.To investigate the existence of periodic solutions in distribution,we first provide a prior estimate.Subsequently,based on the infinite-dimensional version of the Skorohod theorem,we establish the existence of periodic solutions in distribution and a law of large numbers for the periodicity.In addition,we illustrate our main results with an example of the Hooke’s law equation of motion.