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

Abstract

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