On bounded continuous solutions of the archetypical functional equation with rescaling

Leonid V. Bogachev, Gregory Derfel, Stanislav A. Molchanov

Published 2014-09-19Version 1

We study the "archetypical" functional equation $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ denotes expectation and $(\alpha,\beta)$ is random with distribution $\mu$. Particular cases include: (i) $y(x)=\sum_{i} p_{i}\, y(a_i(x-b_i))$ and (ii) $y'(x)+y(x) =\sum_{i} p_{i}\,y(a_i(x-c_i))$ (pantograph equation), both subject to the balance condition $\sum_{i} p_{i}=1$ (${p_{i}>0}$). Solutions $y(x)$ admit interpretation as harmonic functions of an associated Markov chain $(X_n)$ with jumps of the form $x\rightsquigarrow\alpha(x-\beta)$. The paper concerns Liouville-type results asserting that any bounded continuous harmonic function is constant. The problem is essentially governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln |\alpha|\}$. In the critical case $K=0$, we prove a Liouville theorem subject to the uniform continuity of $y(x)$. The latter is guaranteed under a mild regularity assumption on the distribution of $\beta$, which is satisfied for a large class of examples including the pantograph equation (ii). Functional equation (i) is considered with $a_i=q^{m_i}$ ($q>1$, $m_i\in\mathbb{Z}$), whereby a Liouville theorem for $K=0$ can be established without the uniform continuity assumption. Our results also include a generalization of the classical Choquet--Deny theorem to the case $|\alpha|\equiv1$, and a surprising Liouville theorem in the resonance case $\alpha(c-\beta)\equiv c$. The proofs systematically employ Doob's Optional Stopping Theorem (with suitably chosen stopping times) applied to the martingale $y(X_n)$.