Inhomogeneous refinement equations with random affine maps

Abstract

Given a probability space $(Ω,{\mathcal A},P)$, random variables $L,M\colonΩ\to\mathbb R$ and $g\in L^1(\mathbb R)$ we obtain two characterizations of these $f\in L^1(\mathbb R)$ which are solutions of the inhomogeneous refinement equation with a random affine map of the form $f(x)=\int_Ω|L(ω)|f(L(ω)x-M(ω))P(dω)+g(x)$.