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

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 $x⇝\alpha (x-\beta )$.

Minor editorial corrections, including a small change in the title. To be published in the AIMS Proceedings