{
  "id": "1409.6126",
  "version": "v1",
  "published": "2014-09-22T09:20:45.000Z",
  "updated": "2014-09-22T09:20:45.000Z",
  "title": "Analysis of the archetypical functional equation in the non-critical case",
  "authors": [
    "Leonid V. Bogachev",
    "Gregory Derfel",
    "Stanislav A. Molchanov"
  ],
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "abstract": "We study the archetypical functional equation of the form $y(x)=\\iint_{\\mathbb{R}^2} y(a(x-b))\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)$ ($x\\in\\mathbb{R}$), 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 $K:=\\iint_{\\mathbb{R}^2}\\ln|a|\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)=\\mathbb{E}\\{\\ln|\\alpha|\\}$; 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 $(\\alpha,\\beta)$. 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 $x\\rightsquigarrow\\alpha(x-\\beta)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-22T09:20:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B05",
      "34K06",
      "39A22",
      "60G42",
      "60J05"
    ],
    "keywords": [
      "archetypical functional equation",
      "non-critical case",
      "proofs employ martingale techniques",
      "random coefficients",
      "random series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.6126B"
    }
  }
}