{
  "id": "1906.03422",
  "version": "v1",
  "published": "2019-06-08T08:53:02.000Z",
  "updated": "2019-06-08T08:53:02.000Z",
  "title": "Asymptotic Formulas for Empirical Measures of (Reflecting) Diffusion Processes on Riemannian Manifolds",
  "authors": [
    "Feng-Yu Wang"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\\in C^2(M)$ such that $\\mu(d x):=e^{V(x)}d x$ is a probability measure, and let $\\{\\lambda_i\\}_{i\\ge 1} $ be all non-trivial eigenvalues of $-L$ with Neumann boundary condition if the boundary exists. Then the empirical measures $\\{\\mu_t\\}_{t>0}$ of the diffusion process generated by $L$ (with reflecting boundary if the boundary exists) satisfy $$ \\lim_{t\\to \\infty} \\big\\{t \\mathbb E^x [W_2(\\mu_{t},\\mu)^2]\\big\\}= \\sum_{i=1}^\\infty\\frac 2 {\\lambda_i^2}\\ \\text{ uniformly\\ in\\ } x\\in M,$$ where $\\mathbb E^x$ denotes the expectation taken for the diffusion process starting at point $x$, and $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\\le 3$, and when $d\\ge 4$ two-sided estimates are presented for the convergence rate of $\\mathbb E[W_2(\\mu_t,\\mu)^2].$ The same type asymptotic formula is also derived for the modified empirical measures of the (reflecting) diffusion process on a class of non-compact Riemannian manifolds with or without boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-08T08:53:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusion process",
      "empirical measures",
      "connected riemannian manifold possibly",
      "reflecting",
      "type asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}