Asymptotic Properties Of Stochastic Functional Differential Equations With Infinite Delay

Many scholars have researched into stochastic functional differential equa-tions(SFDEs) and plenty of literature on SFDEs is generated. However, most of these works are confined to SFDEs with finite delay. This paper discusses the asymptotic behavior of SFDEs with infinite delay, including moment boundedness, stability and asymptotic path-wise estimation.The existence and uniqueness of the global solution is the basis of investigating the asymptotic properties of the equations. Under local Lipschitz condition and giving two con-crete Lyapunov functions U(x) and V(x), this paper establishes some general controllable conditions on (?)U(t, x,φ) and (?)V(t, x,φ) by introducing a class ofΨ-type functions which have rich connotations. These general conditions replace the linear growth condition and guarantee the existence and uniqueness of the global solution.This paper investigates the moment boundedness firstly, including pth moment ultimate boundedness and qth moment boundedness average in time. By making full use of the properties ofΨ-type function, a strong control on (?)V(t, x,φ) is imposed to establish the general conditions of pth moment unltimate boundedness and qth moment boundedness average in time. These general conditions are stronger than those for the existence and uniqueness of the global solution.Stability is the most important asymptotic property. This paper studies pth moment general decay stability and almost sure pathwise general decay stability. The general decay rate is described by theΨ-type function. Adopting the same idea as the general conditions for moment boundedness is established, we impose some control on (?)V(t, x,φ), which guarantees that pth moment general decay stability and almost sure pathwise general decay stability are obtained simultaneously.As another approach to study pth moment stability and pathwise stability with general decay rate, Razumikhin-type method been exploited. Under the assumption of existence and uniqueness of the global solution, this paper establishes a Razumikhin-type theorem of pth moment general decay stability firstly. Then, on the basis of this theorem, an improved one is obtained by imposing some condition on LV(x(t)) to guarantee pth moment general decay stability. This condition are different from afore mentioned ones. Moreover, by adding some controllable conditions on the coefficient g, we can further get almost sure pathwise general decay stability.Pathwise estimation has its special value. This paper examines the pathwise growth with no more than polynomial rate. Assume that the equation admits a unique global so-lution on R_×~n and let V(x)=|x|2 be the Lyapunov function, some general aconditions on pathwise growth with no more than polynomial rate are established with the help of expo-nential martingale inequality and Borel-Cantelli Lemma.It is not easy to apply all the above general conditions on asymptotic properties since they do not relate to the coefficients f arid g directly. The main work of this paper is to establish some conditions which are explicitly described by the coefficients f and g to satisfy the general conditions of corresponding asymptotic property. Assume that the local Lipschitz condition is satisfied, for each kind of asymptotic property, by imposing several polynomial growth conditions which control f and g and establishing some inequalities on the parameters of the polynomials, the general conditions established for the corresponding asymptotic property are realized. It is worth mentioning that, these inequalities can also ensure the existence and uniqueness of the global solution simultaneously.