Recurrent Solutions For Stochastic Differential Equations With Lévy Noise

Recurrence is one of the central topics of both dynamical systems and probability theory,which describes the asymptotic behavior and complexity of dynamical systems and Markov processes.By Poincaré recurrence theorem and Birkhoff recurrence theorem,recurrence exists extensively in dynamical systems.On the other hand,(positive)recurrence is essentially equivalent to the existence of invariant(probability)measures for Markov processes,and it is known that the existence of invariant measures play a crucial role in the study of Markov processes.In this thesis we attempt to investigate the following two questions.1.The existence,uniqueness and stability of recurrent solutions for stochastic differential equations with Levy noise.Consider the following stochastic differential equation driven by Levy noise:dY(t)=(A Y(t)+f(t,Y(t)))dt+g(t,Y(t))dW(t)+?|x|U<1F(t,Y(t-),x)N(dt,dx)+?|x|U?1 G(t,Y(t-),x)N(dt,dx),where the operator A is exponentially stable and linear(generally unbounded),coefficients f,g,F,G are recurrent in t.We use a unified framework to study recurrent(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for the above stochastic differential equation driven by infinite dimensional Levy noise with large jumps.We prove that under suitable conditions on coefficients,the equation admits a unique bounded solution which inherits the recurrence of coefficients in distribution sense.Moreover we show that these solutions are globally asymptotically stable in square-mean sense.Further we illustrate the above theoretical results by some examples.2.The averaging principle on the infinite intervals for stochastic ordinary differential equations with Levy noise.Consider the following stochastic differential equation driven by Levy noise:dY(t)=s(A(t)y(t)+f{t,Y(t)))dt+(?)g(t,Y(t))d W(t)+(?)?|x|U<1 F(t,Y(t-),x)N(dt,dx)+?|x|U?1 G(t,Y(t-),x)N(dt,dx),with the time scale 0<?<<1 a small parameter,operator A bounded,coefficients f,g,F,G recurrent in t.In contrast to existing works on stochastic averaging on finite intervals,we establish an averaging principle on the infinite intervals,i.e.the second Bogolyubov theorem,for the above stochastic ordinary differential equation driven by Levy noise with recurrence coefficients.We prove that if the coefficients are recurrent(including periodic,quasi-periodic,almost periodic,almost automorphic etc),then there exists a unique bounded solution of the original equation which has the same character of recurrence as the coefficients,and the recurrent solution uniformly converges to the stationary solution of the averaged equation on the whole real axis in distribution sense as the time scale approaches zero.Finally we give two examples.