无穷维随机微分方程的渐近概周期解

Asymptotically Almost Periodic Solutions for Stochastic Differential Equations in Infinite Dimensions

该文引入了渐近θ-概周期随机过程的概念,并在算子半群理论框架下研究了一类带有渐近概周期系数的无穷维随机微分方程,利用随机分析理论建立了此类随机微分方程渐近θ-概周期解的存在性.此外该文还引入了依路径分布渐近概周期过程的概念,并证明了上述渐近θ-概周期解还是依路径分布渐近概周期的.值得注意的是,在早期的研究结果中,建立的均是更弱的一维分布渐近概周期解的存在性.

In this paper,we introduce the notion of asymptotically θ-almost periodic stochastic process and study a class of stochastic differential equations in infinite dimensions with asymptotically almost periodic coeffcients under the framework of operator semigroup theory.Using stochastic analysis theory,the existence of asymptotically θ-almost periodic solutions of these equations is established.In addition,the concept of asymptotically almost periodic process in path distribution is introduced,and we prove that the above solutions are also asymptotically almost periodic in path distribution.It is noteworthy that all the earlier related results only give the existence of asymptotically almost periodic solutions in one-dimensional distribution,which are weaker than asymptotically almost periodic solutions in path distribution.