{
  "id": "2211.08929",
  "version": "v1",
  "published": "2022-11-16T14:21:18.000Z",
  "updated": "2022-11-16T14:21:18.000Z",
  "title": "The Liouville theorem for a class of Fourier multipliers and its connection to coupling",
  "authors": [
    "David Berger",
    "René L. Schilling",
    "Eugene Shargorodsky"
  ],
  "comment": "Work in Progress",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "The classical Liouville property says that all bounded harmonic functions in $\\mathbb{R}^n$, i.e.\\ all bounded functions satisfying $\\Delta f = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of L\\'evy processes. For generators of L\\'evy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time L\\'evy processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-16T14:21:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connection",
      "sufficient conditions",
      "space-time levy processes",
      "strong liouville theorem",
      "fourier multiplier operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}