{
  "id": "2207.09946",
  "version": "v1",
  "published": "2022-07-20T14:43:03.000Z",
  "updated": "2022-07-20T14:43:03.000Z",
  "title": "Convergence of space-discretised gKPZ via Regularity Structures",
  "authors": [
    "Yvain Bruned",
    "Usama Nadeem"
  ],
  "comment": "58 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we propose a semi-general extension to the convergence result found in (Erhard & Hairer, 2021). In that paper, the authors showed that a discrete form of the Parabolic Anderson model driven by a (rescaled) symmetric simple exclusion process on the torus of dimension $3$, can be renormalised in such a manner that the sequence of solutions converges in law to a continuous process that solves a (renormalised) version of the PAM driven by a generalised Ornstein-Uhlenbeck process. We will showcase our extension by proving a similar convergence result for the discrete formulation of the generalised KPZ equation $\\partial_t u = (\\Delta u) + g(u)(\\nabla u)^2 + k(\\nabla u) + h(u) + f(u)\\xi_t(x)$, where the $\\xi$ is a real-valued random field, $\\Delta$ is the discrete Laplacian, and $\\nabla$ is a discrete gradient, without fixing the spatial dimension. Our major contribution lies in extending the very general bounds on labelled rooted binary trees presented in (Hairer & Quastel, 2018), and further proposing a new renormalisation procedure that involves local transformations on the level of the trees that makes it possible to treat some divergences that were not tractable under the framework in (Erhard & Hairer, 2021). While this procedure falls short of complete generality as provided by (Bruned et al., 2019) for singular SPDEs in the continuum, we believe this is a possible intermediate step in that direction for the discrete case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-20T14:43:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60L30",
      "60L90",
      "60H17"
    ],
    "keywords": [
      "regularity structures",
      "space-discretised gkpz",
      "parabolic anderson model driven",
      "symmetric simple exclusion process",
      "similar convergence result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}