{
  "id": "2408.10326",
  "version": "v1",
  "published": "2024-08-19T18:05:49.000Z",
  "updated": "2024-08-19T18:05:49.000Z",
  "title": "On weak convergence of stochastic wave equation with colored noise on $\\mathbb{R}$",
  "authors": [
    "Wenxuan Tao"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the following stochastic wave equation on the real line $\\partial_t^2 u_{\\alpha}=\\partial_x^2 u_{\\alpha}+b\\left(u_\\alpha\\right)+\\sigma\\left(u_\\alpha\\right)\\eta_{\\alpha}$. The noise $\\eta_\\alpha$ is white in time and colored in space with covariance structure $\\mathbb{E}[\\eta_\\alpha(t,x)\\eta_\\alpha(s,y)]=\\delta(t-s)f_\\alpha(x-y)$ where $f_\\alpha(x)\\propto |x|^{-\\alpha}$ is the Riesz kernel. We state the continuity of the solution $u_\\alpha$ in terms of $\\alpha$ for $\\alpha\\in (0,1)$ with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also prove that the probability measure induced by $u_\\alpha$ converges weakly to the one induced by the solution in the space-time white noise case as $\\alpha$ goes to $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T18:05:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic wave equation",
      "weak convergence",
      "colored noise",
      "space-time white noise case",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}