Continuous dependence of the weak limit of iterates of some random-valued vector functions

Abstract Given a probability space $$(\Omega ,\mathcal {A},\mathbb {P})$$ ( Ω , A P ) , complete separable Banach X with the $$\sigma $$ σ -algebra $$\mathcal B(X)$$ B X of all its Borel subsets, an operator $$\Lambda :\Omega \rightarrow L(X,X)$$ Λ : → L and $$\xi X$$ ξ we consider {B}(X)\otimes \mathcal A$$ ⊗ -measurable function $$f:X\times \Omega f × given by $$f(x,\omega )=\Lambda (\omega )x+\xi )$$ x ω = + investigate continuous dependence weak limit $$\pi ^f$$ π sequence iterates $$(f^n(x,\cdot ))_{n\in \mathbb {N}}$$ n · ∈ N f defined $$f^0(x,\omega )=x, f^{n+1}(x,\omega )=f(f^n(x,\omega ),\omega _{n+1})$$ 0 1 for $$x\in $$\omega =(\omega _1,\omega _2,\dots 2 ⋯ . Moreover taken as Hilbert characterize via functional equation $$\begin{aligned} \varphi ^f(u)=\int _{\Omega }\varphi ^f(\Lambda )u)\varphi ^{\xi }(u)\mathbb {P}(d\omega ) \end{aligned}$$ φ u ∫ d aid characteristic $$\varphi We also indicate solution that equation.