{
  "id": "2408.09116",
  "version": "v1",
  "published": "2024-08-17T06:54:39.000Z",
  "updated": "2024-08-17T06:54:39.000Z",
  "title": "Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes",
  "authors": [
    "Feng-Yu Wang",
    "Bingyao Wu",
    "Jie-Xiang Zhu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\\mathbb E} [{\\mathbb W}_p^q(\\mu_T,\\mu)])^{\\frac{1}{q}} (T \\to \\infty)$ uniformly in $(p,q)\\in [1,\\infty) \\times (0,\\infty)$, where $\\mu_T$ is the empirical measure of the diffusion process, $\\mu$ is the unique invariant probability measure, and ${\\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\\mathbb E} |T {\\mathbb W}_2^2(\\mu_T,\\mu)-\\Xi(T)|^q \\to 0$ as $T\\to\\infty$ for any $q\\ge 1$, where $\\Xi(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-17T06:54:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusion process",
      "wasserstein distance",
      "empirical measure",
      "unique invariant probability measure",
      "exact convergence rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}