{
  "id": "1409.5648",
  "version": "v1",
  "published": "2014-09-19T13:24:06.000Z",
  "updated": "2014-09-19T13:24:06.000Z",
  "title": "On bounded continuous solutions of the archetypical functional equation with rescaling",
  "authors": [
    "Leonid V. Bogachev",
    "Gregory Derfel",
    "Stanislav A. Molchanov"
  ],
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "abstract": "We study the \"archetypical\" functional equation $y(x)=\\iint_{\\mathbb{R}^2} y(a(x-b))\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)$ ($x\\in\\mathbb{R}$), where $\\mu$ is a probability measure; equivalently, $y(x)=\\mathbb{E}\\{y(\\alpha(x-\\beta))\\}$, where $\\mathbb{E}$ denotes expectation and $(\\alpha,\\beta)$ is random with distribution $\\mu$. Particular cases include: (i) $y(x)=\\sum_{i} p_{i}\\, y(a_i(x-b_i))$ and (ii) $y'(x)+y(x) =\\sum_{i} p_{i}\\,y(a_i(x-c_i))$ (pantograph equation), both subject to the balance condition $\\sum_{i} p_{i}=1$ (${p_{i}>0}$). Solutions $y(x)$ admit interpretation as harmonic functions of an associated Markov chain $(X_n)$ with jumps of the form $x\\rightsquigarrow\\alpha(x-\\beta)$. The paper concerns Liouville-type results asserting that any bounded continuous harmonic function is constant. The problem is essentially governed by the value $K:=\\iint_{\\mathbb{R}^2}\\ln|a|\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)=\\mathbb{E}\\{\\ln |\\alpha|\\}$. In the critical case $K=0$, we prove a Liouville theorem subject to the uniform continuity of $y(x)$. The latter is guaranteed under a mild regularity assumption on the distribution of $\\beta$, which is satisfied for a large class of examples including the pantograph equation (ii). Functional equation (i) is considered with $a_i=q^{m_i}$ ($q>1$, $m_i\\in\\mathbb{Z}$), whereby a Liouville theorem for $K=0$ can be established without the uniform continuity assumption. Our results also include a generalization of the classical Choquet--Deny theorem to the case $|\\alpha|\\equiv1$, and a surprising Liouville theorem in the resonance case $\\alpha(c-\\beta)\\equiv c$. The proofs systematically employ Doob's Optional Stopping Theorem (with suitably chosen stopping times) applied to the martingale $y(X_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-19T13:24:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B05",
      "34K06",
      "39A22",
      "60G42",
      "60J05"
    ],
    "keywords": [
      "archetypical functional equation",
      "bounded continuous solutions",
      "systematically employ doobs optional",
      "concerns liouville-type results asserting",
      "liouville theorem"
    ],
    "publication": {
      "doi": "10.1098/rspa.2015.0351",
      "journal": "Proceedings of the Royal Society of London Series A",
      "year": 2015,
      "month": "Aug",
      "volume": 471,
      "number": 2180,
      "pages": 20150351
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015RSPSA.47150351B"
    }
  }
}