ABSTRACT
In the previous chapter, we have discussed the existence and uniqueness questions for stochastic evolution equations. Now some properties of the solutions will be studied. In particular, we shall deal with some problems concerning the asymptotic properties of solutions, such as boundedness, asymptotic stability, invariant measures and small random perturbations. For stochastic equations in finite dimension, the asymptotic problems have been studied extensively for many years by numerous authors. For the stability and related questions, a comprehensive treatment of the subject is given in the classic book by Khasminskii [47]. In particular, his systematic development of the stability analysis based on the method of Lyapunov functions has a natural generalization to an infinite-dimensional setting. As will be seen, the method of Lyapunov functionals will play an important role in the subsequent asymptotic analysis. For stochastic processes in finite dimensions, the small random perturbation and the related large deviations problems are treated in the well-known books by Freidlin and Wentzell [32], Deuschel and Stroock [26], Varadhan [91], and Dembo and Zeitouni [25]. For stochastic partial differential equations, there are relatively fewer papers in asymptotic results. A comprehensive discussion of some stability results for stochastic evolution equations in Hilbert spaces can be found in a book by Liu [63], and the subject of invariant measures is treated by Da Prato and Zabczyk [22] in detail.
