54

views

0

recommends

- Record: found
- Abstract: found
- Article: found

Preprint

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 \(y(X_n)\).

http://arxiv.org/licenses/nonexclusive-distrib/1.0/