{
  "id": "2103.11482",
  "version": "v1",
  "published": "2021-03-21T20:46:01.000Z",
  "updated": "2021-03-21T20:46:01.000Z",
  "title": "Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields",
  "authors": [
    "D. Kinzebulatov",
    "Yu. A. Semenov"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "We consider Kolmogorov operator $-\\nabla \\cdot a \\cdot \\nabla + b \\cdot \\nabla$ with measurable uniformly elliptic matrix $a$ and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field $b$ and divergence ${\\rm div\\,}b$. More precisely, we prove: (1) Gaussian lower bound, provided that ${\\rm div\\,}b \\geq 0$, and $b$ is in the class of form-bounded vector fields (containing e.g.\\,the class $L^d$, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\\rm loc}^{2+\\varepsilon}$, $\\varepsilon>0$); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper and lower bounds, provided that $b$ is form-bounded, ${\\rm div\\,}b$ is in the Kato class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-21T20:46:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector field",
      "heat kernel bounds",
      "parabolic equations",
      "gaussian lower bound",
      "gaussian upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}