{
  "abstract": "We study occupation measures associated with stochastic approximation recursions within the ordinary differential equation (ODE) method. Under Lyapunov conditions ensuring that the limiting ODE has a globally exponentially stable equilibrium, we establish that the continuous-time occupation measures associated with the piecewise linear interpolation of the iterates converge almost surely in the $2$-Wasserstein topology to the Dirac mass at the equilibrium, the unique invariant probability measure of the limiting semiflow. The choice of the $2$-Wasserstein distance is naturally aligned with our continuous- and discrete-time Lyapunov drift analysis. We also consider two discrete-time sequences of occupation measures, induced by the algorithmic time scale and the iteration count, respectively, and we establish their almost sure convergence to the same invariant measure in the $2$-Wasserstein distance. We then derive corresponding convergence rates in expectation, providing a nonasymptotic complement to the almost sure results. These expectation bounds also imply some tail probability estimates. Finally, we verify that our results apply to a canonical class of linear recursions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05921",
  "author_total": 1,
  "authors": [
    {
      "name": "Dimitrios Katselis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f8re16kn5hez8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05921",
  "fulltext_url": null,
  "id": "f8re16kn5hez8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05921",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05921",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f8re16kn5hez8.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05921",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05921"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Occupation Measures in the ODE Method: Wasserstein Convergence and Rates",
  "title_text": "Occupation Measures in the ODE Method: Wasserstein Convergence and Rates",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/f8re16kn5hez8",
  "venue": "arXiv",
  "versions": []
}