Analysis of the archetypalfunctional equation in the non-critical case

Leonid V.  Bogachev; Gregory  Derfel; Stanislav A.  Molchanov

doi:10.3934/proc.2015.0132

Article Contents

2015, Volume 2015: 132-141. Doi: 10.3934/proc.2015.0132 open access

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Analysis of the archetypal functional equation in the non-critical case

1.
Department of Statistics, School of Mathematics, University of Leeds, Leeds, LS2 9JT
2.
Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva 84105
3.
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223

Received: August 31, 2014

Revised: November 30, 2014

Published: October 2015

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Abstract

We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(da,db)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(da,db) =\mathbb{E}\{\ln|\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(\alpha<0)>0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x ⇝ \alpha(x-\beta)$.

Keywords:

Functional & functional-differential equations,
pantograph equation,
Markov chain,
harmonic function,
martingale.

Mathematics Subject Classification: Primary: 39B05; Secondary: 34K06, 39A22, 60G42, 60J05.

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References

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