On bounded continuous solutions of the archetypal equation with rescaling

TL;DR

This paper investigates bounded continuous solutions of the archetypal rescaling equation, establishing conditions under which such solutions are constant, and applies probabilistic methods to analyze these functional equations and their variants.

Contribution

It provides Liouville-type theorems for bounded solutions of the archetypal equation with rescaling, including critical cases, using martingale techniques and stopping times.

Findings

Bounded continuous solutions are constant under certain conditions.

Liouville-type results hold in the critical case with uniform continuity.

The approach employs martingale and optional stopping theorem methods.

Abstract

The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, with random $\alpha,\beta$ and $\mathbb{E}$ denoting expectation. Examples include: (i) functional equation $y(x)=\sum_{i} p_{i} y(a_i(x-b_i))$; (ii) functional-differential (`pantograph') equation $y'(x)+y(x)=\sum_{i} p_{i} y(a_i(x-c_i))$ ($p_{i}>0$, $\sum_{i} p_{i}=1$). Interpreting solutions $y(x)$ as harmonic functions of the associated Markov chain $(X_n)$, we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the `critical' case $\mathbb{E}\{\ln|\alpha|\}=0$ such a theorem holds subject to uniform continuity of $y(x)$; the latter is guaranteed under mild regularity assumptions on $\beta$, satisfied e.g.\ for the pantograph equation (ii). For equation (i) with $a_i=q^{m_i}$ ($m_i\in\mathbb{Z}$, $\sum_i p_i m_i=0$), the result can be proved without the uniform continuity assumption. The proofs utilize the iterated equation $y(x)=\mathbb{E}\{y(X_\tau)\,|\,X_0=x\}$ (with a suitable stopping time $\tau$) due to Doob's optional stopping theorem applied to the martingale $y(X_n)$.