{
  "id": "2309.05546",
  "version": "v1",
  "published": "2023-09-11T15:32:14.000Z",
  "updated": "2023-09-11T15:32:14.000Z",
  "title": "Metastability and time scales for parabolic equations with drift 1: the first time scale",
  "authors": [
    "Claudio Landim",
    "Jungkyoung Lee",
    "Insuk Seo"
  ],
  "comment": "60 pages, 3 figures",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math.AP"
  ],
  "abstract": "Consider the elliptic operator given by $$ \\mathscr{L}_{\\epsilon}f= {b} \\cdot \\nabla f + \\epsilon \\Delta f $$ for some smooth vector field $ b\\colon \\mathbb R^d \\to\\mathbb R^d$ and a small parameter $\\epsilon>0$. Consider the initial-valued problem $$ \\left\\{ \\begin{aligned} &\\partial_ t u_\\epsilon = \\mathscr L_\\epsilon u_\\epsilon,\\\\ &u_\\epsilon (0, \\cdot) = u_0(\\cdot), \\end{aligned} \\right. $$ for some bounded continuous function $u_0$. Denote by $\\mathcal M_0$ the set of critical points of $b$, $\\mathcal M_0 =\\{x\\in \\mathbb R^d : b(x)=0\\}$, assumed to be finite. Under the hypothesis that $ b = -(\\nabla U + \\ell)$, where $ \\ell$ is a divergence-free field orthogonal to $\\nabla U$, the main result of this article states that there exist a time-scale $\\theta^{(1)}_\\epsilon$, $\\theta^{(1)}_\\epsilon \\to \\infty$ as $\\epsilon \\rightarrow 0$, and a Markov semigroup $\\{p_t : t\\ge 0\\}$ defined on $\\mathcal M_0$ such that $$ \\lim_{\\epsilon\\to 0} u_\\epsilon (t\\theta^{(1)}_\\epsilon, x) =\\sum_{m'\\in \\mathcal M_0} p_t(m, m')\\, u_0( m'), $$ for all $t>0$ and $ x$ in the domain of attraction of $m$ for the ODE $\\dot{x}(t)= b( x(t))$. The time scale $\\theta^{(1)}$ is critical in the sense that, for all time scale $\\varrho_\\epsilon$ such that $\\varrho_\\epsilon \\to \\infty$, $\\varrho_\\epsilon/\\theta^{(1)}_\\epsilon \\to 0$, $$ \\lim_{\\epsilon\\to 0} u_\\epsilon (\\varrho_\\epsilon, x)=u_0(m) $$ for all $x \\in \\mathcal D(m)$. Namely, $\\theta_\\epsilon^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_\\epsilon$ evolves smoothly in time and we show that the solution $u_\\epsilon$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-11T15:32:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K15",
      "60J60"
    ],
    "keywords": [
      "first time scale",
      "parabolic equations",
      "metastability",
      "divergence-free field orthogonal",
      "critical points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}