{
  "id": "1803.05646",
  "version": "v1",
  "published": "2018-03-15T09:05:23.000Z",
  "updated": "2018-03-15T09:05:23.000Z",
  "title": "Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators",
  "authors": [
    "Franziska Kühn"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $A$ be a pseudo-differential operator with symbol $q(x,\\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\\infty}(\\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends continuously on the space variable $x$, then the existence of solutions is well understood, and therefore the focus lies on martingale problems for pseudo-differential operators with discontinuous coefficients. We prove an existence result which allows us, in particular, to obtain new insights on the existence of weak solutions to a class of L\\'evy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Moreover, we establish a Markovian selection theorem which shows that - under mild assumptions - the $(A,C_c^{\\infty}(\\mathbb{R}^d))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to L\\'evy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-15T09:05:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35"
    ],
    "keywords": [
      "martingale problem",
      "lévy-type operators",
      "markovian selection theorem",
      "non-local operators",
      "pseudo-differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}