{
  "id": "2408.15437",
  "version": "v1",
  "published": "2024-08-27T22:46:49.000Z",
  "updated": "2024-08-27T22:46:49.000Z",
  "title": "Mosco convergence of gradient forms with non-convex potentials II",
  "authors": [
    "Martin Grothaus",
    "Simon Wittmann"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let $d\\in\\mathbb N$, $y_1,\\dots,y_M\\in\\mathbb R$ and $f\\in C_b(\\mathbb R)$ be fixed. For each $N\\in\\mathbb N$ we consider a $k_N$-dimensional, skew reflecting distorted Brownian motion $(X^{N,i}_t)_{i=1,\\dots,k_N}$, $t\\geq 0$, and investigate the scaling limits for $N\\to\\infty$. The drift includes skew reflections at height levels $\\tilde y_j:=N^{1-\\frac{d}{2}}y_j$ with intensities $\\beta_j/N^d$ for $j=1,\\dots,M$. The corresponding SDE is given by \\begin{equation} d X^{N,i}_t=-\\big(A_N X^{N}_t\\big)_id t-\\frac{1}{2}N^{-\\tfrac{d}{2}-1}\\,f\\big(N^{\\frac{d}{2}-1}X^{N,i}_t\\big)d t \\\\+\\sum_{j=1}^M\\tfrac{1-e^{-\\beta_j/N^d}}{1+e^{-\\beta_j/N^d}}d l_t^{N,i, \\tilde y_j} +d B_t^{N,i}, \\end{equation} where ${(B_t^{N,i})}_{t\\geq 0}$, $i=1,\\dots, k_N$, are independent Brownian motions and $ l_t^{N,i, \\tilde y_j}$ denotes the local time of ${(X^{N,i}_t)}_{t\\geq 0}$ at $\\tilde y_j$. We prove the weak convergence of the equilibrium laws of \\begin{equation*} u_t^N=\\Lambda_N\\circ X^{N}_{N^2t},\\quad t\\geq 0, \\end{equation*} for $N\\to\\infty$, choosing suitable injective, linear maps $\\Lambda_N:\\mathbb R^{k_N}\\to \\{h\\,|\\,h:D\\to\\mathbb R\\}$. The scaling limit is a distorted Ornstein-Uhlenbeck process whose state space is the Hilbert space $H=L^2(D, dz)$. We characterize a class of height maps, such that the scaling limit of the dynamic is not influenced by the particular choice of ${(\\Lambda_N)}_{N\\in\\mathbb N}$ within that class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T22:46:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J46",
      "47D07",
      "82B31"
    ],
    "keywords": [
      "mosco convergence",
      "gradient forms",
      "non-convex potentials",
      "scaling limit",
      "mesoscopic interface models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}