{
  "id": "1310.4600",
  "version": "v3",
  "published": "2013-10-17T07:33:00.000Z",
  "updated": "2014-09-30T06:54:04.000Z",
  "title": "Hölder continuity and bounds for fundamental solutions to non-divergence form parabolic equations",
  "authors": [
    "Seiichiro Kusuoka"
  ],
  "comment": "The previous versions included serious mistakes. And, the title was changed",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to the equations exists uniquely, has the Gaussian bounds and is locally H\\\"older continuous. In the present paper, we concern the Gaussian bounds and the lower bound of the index of the H\\\"older continuity with respect to the initial point. We use the pinned diffusion processes for the probabilistic representation of the fundamental solutions and apply the coupling method to obtain the regularity of them. Under some assumptions weaker than the H\\\"older continuity of the coefficients, we obtain the Gaussian bounds and the $(1-\\varepsilon)$-H\\\"older continuity of the fundamental solution in the initial point.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-02T08:15:08.000Z",
      "title": "Hölder continuity of the fundamental solutions to parabolic equations with irregular coefficients",
      "abstract": "We consider non-degenerate second-order parabolic partial differential equations with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to the equations exists uniquely and is locally H\\\"older continuous. In the present paper, we concern the lower bound of the index for the H\\\"older continuity with respect to initial point. We use the pinned diffusion processes for the probabilistic representation of the fundamental solutions and the coupling method to obtain the regularity of them. Under some assumptions weaker than H\\\"older continuity of the coefficients, we obtain $(1-\\varepsilon)$-H\\\"older continuity in initial point.",
      "comment": "The previous version included a serious mistake",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-30T06:54:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35K10",
      "60H10",
      "60H30",
      "60J60"
    ],
    "keywords": [
      "fundamental solution",
      "irregular coefficients",
      "hölder continuity",
      "parabolic equations",
      "second-order parabolic partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.4600K"
    }
  }
}