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

Abstract

Given a probability space (Omega ,mathcal {A},mathbb {P}), a complete separable Banach space X with the sigma -algebra mathcal B(X) of all its Borel subsets, an operator Lambda :Omega rightarrow L(X,X) and xi :Omega rightarrow X we consider the mathcal {B}(X)otimes mathcal A-measurable function f:Xtimes Omega rightarrow X given by f(x,omega )=Lambda (omega )x+xi (omega ) and investigate the continuous dependence of a weak limit pi ^f of the sequence of iterates (f^n(x,cdot ))_{nin mathbb {N}} of f, defined by f^0(x,omega )=x, f^{n+1}(x,omega )=f(f^n(x,omega ),omega _{n+1}) for xin X and omega =(omega _1,omega _2,dots ). Moreover for X taken as a Hilbert space we characterize pi ^f via the functional equation φf(u)=∫Ωφf(Λ(ω)u)φξ(u)P(dω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\varphi ^f(u)=\\int _{\\Omega }\\varphi ^f(\\Lambda (\\omega )u)\\varphi ^{\\xi }(u)\\mathbb {P}(d\\omega ) \\end{aligned}$$\\end{document}with the aid of its characteristic function varphi ^f. We also indicate the continuous dependence of a solution of that equation.