Inhomogeneous refinement equations with random affine maps

Abstract

Given a probability space $(\Omega ,\mathcal{A},P)$, random variables $L,M:\Omega \to \mathbb{R}$ 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 $f(x)={\int }_{\Omega }|L(\omega )|f(L(\omega )x-M(\omega ))P(d\omega )+g(x)$.

the final version will be published in Journal of Difference Equations and Applications