{
  "id": "2310.01670",
  "version": "v1",
  "published": "2023-10-02T22:02:08.000Z",
  "updated": "2023-10-02T22:02:08.000Z",
  "title": "Asymptotic behavior of Wasserstein distance for weighted empirical measures of diffusion processes on compact Riemannian manifolds",
  "authors": [
    "Jie-Xiang Zhu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_t)_{t \\geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $\\alpha > 0$, let $$ \\mu_t^{(\\alpha)} = \\frac{\\alpha}{t^\\alpha} \\int_{0}^{t} \\delta_{X_s} \\, s^{\\alpha - 1} \\mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\\mathbb{E}^\\nu \\big[ \\mathrm{W}_2^2(\\mu_t^{(\\alpha)}, \\mu) \\big]$ for sufficient large $t$, where $\\mathrm{W}_2$ is the quadratic Wasserstein distance and $\\mu$ is the invariant measure of the process. In the particular case $\\alpha = 1$, our result sharpens the limit theorem achieved in [26]. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-02T22:02:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact riemannian manifold",
      "weighted empirical measure",
      "diffusion process",
      "asymptotic behavior",
      "quadratic wasserstein distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}