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      On bounded continuous solutions of the archetypal equation with rescaling

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          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 \(y(X_n)\).

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          Author and article information

          Journal
          2014-09-19
          2015-05-11
          10.1098/rspa.2015.0351
          1409.5648

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

          Custom metadata
          Primary: 39B05, Secondary: 34K06, 39A22, 60G42, 60J05
          Substantially revised. The title is modified
          math.PR math.CA

          Probability

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