Bogachev, LV, Derfel, G and Molchanov, SA (2015) Analysis of the archetypal functional equation in the non-critical case. In: Proceedings of the 10th AIMS Conference. 10TH AIMS International Conference, 07-11 Jul 2014, Madrid, Spain. American Institute of Mathematical Sciences , Wilmington, North Carolina, U.S.A. , pp. 132-141.
Abstract
We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), 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 non-trivial (i.e. non-constant) 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 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 $P(\alpha<0)>0$ 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 $x\rightsquigarrow\alpha (x-\beta)$.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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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 & functional-differential equations, pantograph equation, Markov chain, harmonic function, martingale. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & 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: | 10.3934/proc.2015.0132 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:84145 |