Bogachev, LV, Derfel, G and Molchanov, SA (2015) Analysis of the archetypal functional equation in the noncritical case. In: Proceedings of the 10th AIMS Conference. 10TH AIMS International Conference, 0711 Jul 2014, Madrid, Spain. American Institute of Mathematical Sciences , Wilmington, North Carolina, U.S.A. , pp. 132141.
Abstract
We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(xb))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of nontrivial (i.e. nonconstant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln a \mu(da,db) =E \{\ln \alpha\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a nontrivial solution constructed as the distribution function of a certain random series representing a selfsimilar 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 $P(\alpha<0)>0$ 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 $x\rightsquigarrow\alpha (x\beta)$.
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Copyright, Publisher and Additional Information:  (c) 2015, American Institute of Mathematical Sciences. This is an author produced version of a paper accepted for publication in AIMS Proceedings. Uploaded with permission from the publisher. 
Keywords:  Functional & functionaldifferential equations, pantograph equation, Markov chain, harmonic function, martingale. 
Dates: 

Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  17 Mar 2015 09:43 
Last Modified:  08 Mar 2016 18:16 
Published Version:  http://dx.doi.org/10.3934/proc.2015.0132 
Status:  Published 
Publisher:  American Institute of Mathematical Sciences 
Identification Number:  https://doi.org/10.3934/proc.2015.0132 