Ergodic Properties of generalized Ornstein { Uhlenbeck Processes

Let an Rd-valued random process ∫ ξ be the solution of an equation of the kind ξ(t) = ξ(0) + t 0 A(u)ξ(u)dι(u) + S(t), where ξ(0) is a random variable measurable w. r. t. some σ-algebra F(0), S is a random process with F(0)-conditionally independent increments, ι is a continuous numeral random process of locally bounded variation, and A is a matrix-valued random process such that for any t > 0 ∫ t 0 ∥A(s)∥ |dι(s)| < ∞. Conditions guaranteing the existence of the limiting, as t → ∞, distribution of ξ(t) are found. The characteristic function of this distribution is written explicitly. An ergodic theorem for generalized Ornstein – Uhlenbeck processes is proved.