{
  "abstract": "We analyze the mean square error of stochastic Runge-Kutta integrators for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge-Kutta integrators proposed by Yang & Wang (2026), both of strong order $\\frac32$ and weak order $2$, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over $N$ steps with step size $h$ is $\\mathcal O ( \\frac{1}{Nh} + h^4 )$, which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in $\\mathbb R^2$, and compare the integrators on a diffusion model of CIFAR-10.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11528",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuda Ye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1q88x09mgfx5r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11528",
  "fulltext_url": null,
  "id": "1q88x09mgfx5r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11528",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Numerical methods for differential equations"
    ]
  ],
  "openalex": "W7212227919",
  "pdf_url": "https://arxiv.org/pdf/2609.11528",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1q88x09mgfx5r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.11528",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11528"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Mean Square Error Analysis of Stochastic Runge-Kutta Integrators",
  "title_text": "Mean Square Error Analysis of Stochastic Runge-Kutta Integrators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/1q88x09mgfx5r",
  "venue": "arXiv",
  "versions": []
}