{
  "id": "2008.01335",
  "version": "v1",
  "published": "2020-08-04T05:26:35.000Z",
  "updated": "2020-08-04T05:26:35.000Z",
  "title": "Harnack Inequalities, Ergodicity, and Contractivity of Stochastic Reaction-Diffusion Equation in $L^p$",
  "authors": [
    "Liu",
    "Zhihui"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive rough noise in the $L^p(\\OOO)$-space, for any $p \\ge 2$, where $\\OOO$ is a bounded, open subset of $\\rr^d$. These inequalities are used to study the ergodicity and contractivity of the corresponding Markov semigroup $(P_t)_{t \\ge 0}$. The main ingredients of our method are a coupling by the change of measure and a uniform exponential moments' estimation $\\sup_{t \\ge 0} \\ee \\exp(\\epsilon \\|\\cdot\\|_p^p)$ with some positive constant $\\epsilon$ for the solution. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)_{t \\ge 0}$ possesses a unique and thus ergodic invariant measure and is supercontractive in $L^p$, which is independent of the Lipschitz term.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T05:26:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H10",
      "37H05"
    ],
    "keywords": [
      "stochastic reaction-diffusion equation",
      "contractivity",
      "ergodicity",
      "lipschitz term",
      "ergodic invariant measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}