{
  "abstract": "Brownian motion and the Brownian bridge have covariance kernels that are Green's functions of the negative second derivative on the unit interval, with mixed Dirichlet-Neumann and Dirichlet boundary conditions, respectively. We characterize the Green's functions of all strictly positive real self-adjoint realizations of this operator. The kernels are in one-to-one correspondence with positive-semidefinite $2\\times2$ endpoint covariance matrices. We call the associated centered Gaussian processes Brownian processes. Each is a standard Brownian bridge plus the affine interpolation of an independent Gaussian endpoint vector. Singular endpoint covariance matrices give hard endpoint constraints, such as the pinned start $X(0)=0$ of Brownian motion. This parametrization gives explicit operator domains and energy spaces. On every mesh it gives an exact sparse precision matrix, so the likelihood of $n$ observations can be evaluated in $O(n)$ operations. Nonnegative realizations with a null space are treated separately through orthogonally normalized pseudoinverses, which are proper Gaussian covariances.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36762",
  "author_total": 1,
  "authors": [
    {
      "name": "Lars Lau Raket"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4v4v7ems7c46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36762",
  "fulltext_url": null,
  "id": "c4v4v7ems7c46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36762",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7224088009",
  "pdf_url": "https://arxiv.org/pdf/2610.36762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4v4v7ems7c46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Brownian covariances on an interval: a complete endpoint classification",
  "title_text": "Brownian covariances on an interval: a complete endpoint classification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/c4v4v7ems7c46",
  "venue": "arXiv",
  "versions": []
}