{
  "id": "2105.03013",
  "version": "v1",
  "published": "2021-05-07T00:19:50.000Z",
  "updated": "2021-05-07T00:19:50.000Z",
  "title": "A Sobolev space theory for the Stochastic Partial Differential Equations with space-time non-local operators",
  "authors": [
    "Kyeong-Hun Kim",
    "Daehan Park",
    "Junhee Ryu"
  ],
  "comment": "49 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $$ \\partial_{t}^{\\alpha}u=\\left( \\phi(\\Delta) u +f(u) \\right) + \\partial_t^\\beta \\sum_{k=1}^\\infty \\int_0^t g^k(u)\\,dw_s^k, \\quad t>0, x\\in \\mathbb{R}^d; \\,\\,\\, u(0,\\cdot)=u_0 $$ as well as the SPDE driven by space-time white noise $$ \\partial^{\\alpha}_{t}u=\\phi(\\Delta)u + f(u) + \\partial^{\\beta-1}_{t}h(u) \\dot{W}, \\quad t>0,x\\in \\mathbb{R}^d; \\quad u(0,\\cdot)=u_{0}. $$ Here, $\\alpha\\in (0,1), \\beta\\in (-\\infty, \\alpha+1/2)$, $\\{w_t^k : k=1,2,\\cdots\\}$ is a family of independent one-dimensional Wiener processes, and $\\dot{W}$ is a space-time white noise defined on $[0,\\infty)\\times \\mathbb{R}^d$. The time non-local operator $\\partial_{t}^{\\gamma}$ denotes the Caputo fractional derivative if $\\gamma>0$ and the Riemann-Liouville fractional integral if $\\gamma\\leq0$. The the spatial non-local operator $\\phi(\\Delta)$ is a type of integro-differential operator whose symbol is $-\\phi(|\\xi|^2)$, where $\\phi$ is a Bernstein function satisfying \\begin{equation*} \\kappa_0\\left(\\frac{R}{r}\\right)^{\\delta_{0}} \\leq \\frac{\\phi(R)}{\\phi(r)}, \\qquad \\forall\\,\\, 0<r<R<\\infty \\end{equation*} with some constants $\\kappa_0>0$ and $\\delta_0\\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-07T00:19:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60",
      "26A33",
      "47G20"
    ],
    "keywords": [
      "stochastic partial differential equation",
      "sobolev space theory",
      "space-time non-local operators",
      "space-time white noise",
      "wiener processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}