Nonlinear random ergodic theorems for affine operators

Let (Ω, ß,μ) be a finite measure space and let (S,F, ν) be another probability measure space on which a measure preserving transformation ϕ 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 Fmeasurable family {Ts +ξ(s, ·): s ∈ S} of affine operators, where B is a reflexive Banach space, {Ts : s ∈ S} is a strongly F-measurable family of linear contractions on L1(Ω,B) as well as on L∞(Ω,B) and ξ is a function in (I − T )Lp(S × Ω,B) (1 p <∞) with the operator T defined by Tf (s,ω) = [Tsfϕs](ω) which denotes the F ⊗ ß-measurable version of Tsfϕs(ω). 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. © 2009 Elsevier Inc. All rights reserved.