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

Research paper by **Leonid V. Bogachev, Gregory Derfel, Stanislav A. Molchanov**

Indexed on: **15 Jan '15**Published on: **15 Jan '15**Published in: **Mathematics - Probability**

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#### 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
$x\rightsquigarrow\alpha(x-\beta)$.