Nonlinear random ergodic theorems for affine operators (Q1011416)

The article deals with nonlinear random ergodic theorems for affine operators in the spaces \(L_p(S \times \Omega,{\mathbb B})\), where \((\Omega,{\mathcal B},\mu)\) is a finite measure space, \((S,{\mathcal F},\nu)\) is a probabilistic measure space, \({\mathbb B}\) is a reflexive Banach space, \(1 \leq p < \infty\). Let \(\{T_s: s \in S\}\) be a strongly \({\mathcal F}\)-measurable family of linear contractions on \(L_1(\Omega,{\mathbb B})\) as well as \(L_\infty(\Omega,{\mathbb B})\), \(Tf(s,\omega) = T_sf_{\varphi s}(\omega)\), \(U_s = T_s + \xi_s\), \(s \in S\), \(\xi \in (I - T)g\) for some \(g \in L_p(S \times \Omega,{\mathbb B})\). The following problem is studied: under which conditions, for every \(f \in L_p(S \times \Omega,{\mathbb B})\), there exist a function \(\eta \in L_p(S \times \Omega,{\mathbb B})\) and a set \(E(f,g) \in {\mathcal F}\) of \(\nu\)-measure zero such that, for each \(s \in S \setminus E(f,g)\), \[ \lim_{n\to\infty} \bigg\|\frac1n \sum_{k=1}^n U_s U_{\varphi s} \cdots U_{\varphi^{k-1}s} f_{\varphi^ks}(\omega)- \eta_s(\omega)\bigg\|_{\mathbb B}=0 \quad \mu\text{-a.e.} \] Moreover, there are described the conditions under which \[ \lim_{n\to\infty} \Bigg(\int_\Omega \bigg\|\frac1n \sum_{k=1}^n U_s U_{\varphi s} \cdots U_{\varphi^{k-1}s} f_{\varphi^ks}(\omega)- \eta_s(\omega)\bigg\|_{\mathbb B}^p \, d\mu\Bigg)^\frac1p = 0 \] holds (the cases \(p > 1\) and \(p = 1\) are considered separately). Several special cases and examples are also treated. At the end of the article, some version of the nonlinear random ergodic theorem is also obtained; moreover, there are presented some examples of affine systems for which the nonlinear ergodic theorem fails to hold.