Nonlinear random ergodic theorems for affine operators

摘要

Let (Ω,β, _) be a finite measure space and let (S, F, v) be another probability measure space on which a measure preserving transformation (p is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly .measurable family {T_s +ξ(s, _): s∈S} of affine operators, where B is a reflexive Banach space, {T_s: s∈S} is a strongly F-measurable family of linear contractions on L_1(Ω, B) as well as on L_∞(Ω, B) and ξ is a function in (I - T)L_p(S x Ω, B) (1≤ p < ∞) with the operator Т defined by T f (s, ω) = [T_s f_φs](ω) which denotes the F_ β-measurable version of T_s f_φs (w). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.