Analysis of the archetypal functional equation in the non-critical case

TL;DR

This paper investigates the existence and properties of solutions to a fundamental functional equation involving random affine transformations, revealing conditions under which non-trivial solutions exist or must be constant, based on the expected logarithm of the scaling factor.

Contribution

It provides a comprehensive analysis of the non-critical case of the archetypal functional equation, including existence, uniqueness, and behavior of solutions, especially distinguishing between positive and negative scaling scenarios.

Findings

Non-trivial solutions exist if the expected log of the scale factor is positive.

No non-constant bounded solutions exist if the expected log of the scale factor is negative.

Solutions with limits at infinity are constant when the scale factor can be negative.

Abstract

We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln|\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(\alpha<0)>0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha(x-\beta)$.