{
  "abstract": "Coherent sets-regions of phase space that resist mixing-organize transport in stochastic dynamical systems and are encoded in the leading singular functions of the stochastic Koopman operator. These are typically computed numerically and globally; a local, analytic theory near a fixed point has been missing, although fixed points and the transitions between them organize the flow. We supply the local theory for a saddle, the transition state through which probability flows between basins. Naively the construction fails-probability escapes along unstable directions, the invariant measure is non-normalizable, the singular functions leave $L^2$, and singular values exceed one. We show that conditioning on non-escape (a Doob's $h$-transform) repairs this; that the required $h$ is the unique stabilizing solution of an algebraic Riccati equation; and that the resulting slow modes decay at the escape rate along the unstable manifold. We obtain these modes in closed form-products of Hermite polynomials in canonical coordinates times the $h$-function-and use their unitary structure to derive a data-driven estimator: canonical correlation analysis on conditioned trajectories. When a single unstable direction dominates, the leading mode is the reaction coordinate, locally recovering the stable manifold and committor of transition-path theory. The construction connects coherent sets to transition-path theory, to the stabilizing Riccati solution of linear control, and to the time-reversed dynamics of generative diffusion models.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35627",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuehao Ding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3j9s9ygg46m8q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35627",
  "fulltext_url": null,
  "id": "3j9s9ygg46m8q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35627",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "nlin.CD"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223819772",
  "pdf_url": "https://arxiv.org/pdf/2610.35627",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3j9s9ygg46m8q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.35627",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35627"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator",
  "title_text": "Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3j9s9ygg46m8q",
  "venue": "arXiv",
  "versions": []
}