Periodicity in distribution. I. Discrete systems

We consider the existence of periodic in distribution solutions to the difference equations in a Banach space. A random process is called periodic in distribution if all its finite-dimensional distributions are periodic with respect to shift of time with one period. Only averaged characteristics of a periodic process are periodic functions. The notion of the periodic in distribution process gave adequate description for many dynamic stochastic models in applications, in which dynamics of a system is obviously nonstationary. For example, the processes describing seasonal fluctuations, rotation under impact of daily changes, and so forth belong to this type. By now, a considerable number of mathematical papers has been devoted to periodic and almost periodic in distribution stochastic processes. We give a survey of the theory for certain classes of the linear difference equations in a Banach space. A feature of our treatment is the analysis of the solutions on the whole of axis. Such an analysis gives simple answers to the questions about solution stability of the Cauchy problem on + ∞ , solution stability of analogous problem on − ∞ , or of existence solution for boundary value problem and other questions about global behaviour of solutions. Examples are considered, and references to applications are given in remarks to appropriate theorems.