{
  "id": "2205.00248",
  "version": "v1",
  "published": "2022-04-30T11:50:52.000Z",
  "updated": "2022-04-30T11:50:52.000Z",
  "title": "Regularity theory for a new class of fractional parabolic stochastic evolution equations",
  "authors": [
    "Kristin Kirchner",
    "Joshua Willems"
  ],
  "comment": "40 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "A new class of fractional-order stochastic evolution equations of the form $(\\partial_t + A)^\\gamma X(t) = \\dot{W}^Q(t)$, $t\\in[0,T]$, $\\gamma \\in (0,\\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert space $H$ and the spatiotemporal driving noise $\\dot{W}^Q$ is an $H$-valued cylindrical $Q$-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process $X$ are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of $A$. In addition, the covariance of $X$ and its long-time behavior are analyzed. These abstract results are applied to the cases when $A := L^\\beta$ and $Q:=\\widetilde{L}^{-\\alpha}$ are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Mat\\'ern fields to space-time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-30T11:50:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60H15",
      "47D06",
      "35R11"
    ],
    "keywords": [
      "fractional parabolic stochastic evolution equations",
      "elliptic second-order differential operators",
      "regularity theory",
      "fractional-order stochastic evolution equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}