Ergodicity Of Periodic Measures And Random Quasi-periodic Problems In Stochastic Differential Equations

A dynamic system is the study of the long-term behavior of an evolving system.The modern theory of dynamic systems originated from Poincare’s research on the geometric properties of the solutions of nonlinear equations at the end of 19th century,such as the stability of solutions and the existence of periodic solutions(see[55]).Since then,Birkhoff continued the research on Poincare’s work,discovered many different types of long-term limite behaviors and proposed the term "dynamic system" in[8].The evolution of a specific state in a dynamic system is called an orbit,and the fixed point,periodic orbit and quasi-periodic orbit have always been very important concepts in the orbit research of the dynamic system.When it comes to a dynamic system on a measure space,researchers often focus on the existence and ergodicity of its invariant measures.Ergodic theory studies the statistical properties of dynamic systems relative to invariant measures.The name comes from the "ergodic hypothesis" in classical statistical mechanics,that is,the time average of observations is equal to the state average.However,dynamic systems are often affected by random factors,such as external fluctuations,internal disturbances,initial conditions of fluctuations,and uncertain parameters.In the 1990s,Ludwig Arnold[1]established the basic theoretical framework of random dynamic systems from stochastic equations,and obtained linear theoretical results of finite-dimensional random dynamic systems.Moreover,Crauel and Flandoli[14,15]and Crauel,Debussche and Flandoli[13]obtained a series of important results in infinite-dimensional random dynamical systems.In 2009,in order to characterize the random periodic phenomena in the real world systems,such as the daily maximum temperature,business cycle and the El Nino phenomena,Zhao and Zheng[60]first proposed the concept of random periodic solutions of random dynamic systems.Later,Feng and Zhao[26]studied the ergodicity of periodic measures of random dynamical systems and established the "equivalence" of random periodic processes and periodic measures.For a random dynamical system on a metric dynamical system(Ω,F,P,(θt)t∈T),its randomness is contained in the probability space(Ω,F,P).However,in most of our real-world systems,the probability measure P cannot be determined.At this time,it is necessary to introduce a sublinear expectation theory that can solve problems with probability uncertainties.In 2004,Peng[47,48,49,52,53]directly established the nonlinear expectation theory with dynamic compatibility.Later,Feng and Zhao[25]established the theoretical framework of sublinear expected dynamic system and its ergodicity.Assume the existence of random periodic paths in random dynamical systems,we find the sufficient and necessary conditions for the PS-ergodicity of periodic measures which generated by the random periodic path on product space and state space respectively.Moreover,we have proved that the upper expectations of these periodic measures are ergodic sublinear expectations.In addition to periodic motions,quasi-periodic motions are also very common phenomena in nature.For example,when studying the three-body problem with KAM theory,planetary motion usually has a quasi-periodic horseshoe-shaped orbit.However,many problems in nature are both quasi-periodic and random.For example,the temperature process is random and have both one day and one year periods.But as far as we know,there is no related mathematical theory of random quasi-period.In order to study the random quasi-periodic phenomenon in nature,we first tried to propose the concept of random quasi-periodic path and obtained a series results of random quasi-periodic problems.The dissertation is organized as follows.In Chapter 1,we give the basic concepts of dynamical systems and random dynamical systems,and review the definitions of ergodicity and mixing of dynamical systems.In Chapter 2,first we study the random periodic path of a random dynamical system on state space X and obtain a periodic measure μs,s∈R of skew product Θt,t≥0 on product space(Ω,F):=(Ω×X,F(?)B(X)).Then we prove that for each s∈R,the skew product dynamical system(Ω,F,μs,(Θτn)n≥0)is ergodic if and only if the noise metric dynamical system(Ω,F,P,(Θτn)n≥0)is ergodic.If the random dynamical system is Markovian,the transition probability of the random dynamical system generates a Markov semigroup Pt,t≥0.Then we can construct a periodic measure ρs,s∈R on state space X from the random periodic path and prove that for each s∈R,ρs is an invariant measure of the discrete Markov semigroup Pτn,n∈N,meanwhile,we give a sufficient and necessary condition for the periodic measure ρ3 being PS-ergodic.In addition,we prove that the continuous time canonical dynamical system driven by Brownian motion(Ω,F,P,(θt)t∈R)and their discrete dynamical system(Ω,F,P,(θτn)n≥0)are ergodic.But in general,the ergodicity of a discrete time dynamical system is stronger than that of continuous time canonical dynamical system,and we provide an example that the continuous time dynamical system is ergodic,but its discretization is not.In Chapter 3,first we review the basic concepts of sublinear expectation and sublinear Markov semigroup,then we give the definitions of sublinear expectation dynamical system and its ergodicity.In the previous chapter,we obtain the periodic measure μs on product space(Ω,F)and periodic measure ps on state space(X,B(X)).Then we can construct the sublinear(upper)expectations E and T from periodic measures μs andρs respectively.Under the sufficient and necessary conditions for the periodic measuresμs and ρs being PS-ergodic,we can also conclude that E and T are ergodic sublinear expectations.In addition,we also give examples of an ergodic discrete time sublinear expectation dynamical system and an ergodic sublinear expectation dynamical system on torus(continuous time).In Chapter 4,we try to establish a random quasi-periodic mathematical theory to model the random quasi-periodic phenomena in real world.First,we give the defmitions of random quasi-periodic path and quasi-periodic measure and find a sufficient condition for the semi-flow u of a stochastic differential equation to have a unique random quasi-periodic pathφ(φ)with periods τ1,τ2,meanwhile we prove that the distribution measure p(p)of the random quasi-periodic path is the unique quasiperiodic measure of the stochastic differential equation with periods τ1,τ2.After lifting the semi-flow u to X:=[0,τ1)×[o,τ2)× X,we obtained a perfect cocycle random dynamical system Φ,its corresponding Markov semigroup P*and a quasi-periodic measureμs:=δs mod τ1×δs mod τ2×ρs of P*with periods τ1,τ2.Moreover,there is a unique invariant measure of P*which is the average 1/τ1τ2 ∫0τ1 ∫0τ2 δs1×δs2×ρs1,s2 ds1ds2.At the end of this chapter,we study the existence of the density of the quasi-periodic measure p and give a sufficient condition for its density to satisfy the Fokker-Planck equation.In Chapter 5,We summarize the main results and contributions of this thesis and give some possible research directions for the future work.