  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 $y(X_n)$.
