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Analysis of the archetypical functional equation in the non-critical case | | Leonid V. Bogachev
; Gregory Derfel
; Stanislav A. Molchanov
; | | Date: |
22 Sep 2014 | | Abstract: | We study the archetypical functional equation of the form
$y(x)=iint_{mathbb{R}^2} y(a(x-b)),mu(mathrm{d}a,mathrm{d}b)$
($xinmathbb{R}$), where $mu$ is a probability measure on $mathbb{R}^2$;
equivalently, $y(x)=mathbb{E}{y(alpha(x-eta))}$, where $mathbb{E}$ is
expectation with respect to the distribution $mu$ of random coefficients
$(alpha,eta)$. Existence of non-trivial (i.e., non-constant) bounded
continuous solutions is governed by the value
$K:=iint_{mathbb{R}^2}ln|a|,mu(mathrm{d}a,mathrm{d}b)=mathbb{E}{ln|alpha|}$;
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 $(alpha,eta)$. 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 $pminfty$ 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
$x
ightsquigarrowalpha(x-eta)$. | | Source: | arXiv, 1409.6126 | | Services: | Forum | Review | PDF | Favorites | |
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