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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 $$y(X_n)$$.

### Author and article information

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