{
  "id": "2206.10069",
  "version": "v1",
  "published": "2022-06-21T01:41:12.000Z",
  "updated": "2022-06-21T01:41:12.000Z",
  "title": "Moments and asymptotics for a class of SPDEs with space-time white noise",
  "authors": [
    "Le Chen",
    "Yuhui Guo",
    "Jian Song"
  ],
  "comment": "37 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \\begin{equation*} \\left(\\partial^{\\beta}_t+\\dfrac{\\nu}{2}\\left(-\\Delta\\right)^{\\alpha / 2}\\right) u(t, x)= ~ I_{t}^{\\gamma}\\left[\\lambda u(t, x) \\dot{W}(t, x)\\right] \\quad t>0,~ x\\in\\mathbb R^d, \\end{equation*} where $\\dot{W}$ is space-time white noise, $\\alpha>0$, $\\beta\\in(0,2]$, $\\gamma \\ge 0$, $\\lambda\\neq0$ and $\\nu>0$. The existence and uniqueness of solution in the It\\^o-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the $p$-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the $p$-th moment Lyapunov exponents. In particular, by letting $\\beta=2$, $\\alpha=2$, $\\gamma=0$, and $d=1$, we confirm the following standing conjecture for the stochastic wave equation: \\begin{align*} t^{-1}\\log\\mathbb E[u(t,x)^p] \\asymp p^{3/2}, \\quad \\text{for $p\\ge 2$ as $t\\to \\infty$.} \\end{align*} The method for the lower bounds is inspired by a recent work by Hu and Wang [HW21], where the authors focus on the space-time colored Gaussian noise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-21T01:41:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "space-time white noise",
      "nonlinear stochastic partial differential equation",
      "asymptotics",
      "th moment lyapunov exponents",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}