Inhomogeneous refinement equations with random affine maps

TL;DR

This paper characterizes solutions to inhomogeneous refinement equations involving random affine transformations, expanding understanding of their structure in the context of probability spaces and random variables.

Contribution

It provides new characterizations of solutions to inhomogeneous refinement equations with random affine maps, a novel approach in the analysis of such equations.

Findings

Characterizations of solutions to the refinement equation with random affine maps

Conditions under which solutions exist in L^1(R)

Insights into the structure of solutions involving random variables L and M

Abstract

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