{
  "abstract": "The covariance of Brownian motion is the Green kernel of the one-dimensional Laplacian with a pinned left endpoint and a free right endpoint. This correspondence between Green's functions for the Laplacian on an interval and Brownian covariance functions turns out to be a general phenomenon. We use this correspondence to organize Brownian-derived Gaussian processes by three operations: choosing a self-adjoint realization, recentering a path, and conditioning a fixed law. Local Robin and selected nonlocal boundary conditions yield a catalogue of twelve processes with closed-form kernels, Brownian-path constructions, spectra and traces. The periodic and Neumann-Neumann realizations annihilate the constants; projecting the source and imposing zero time average selects the corresponding covariance pseudoinverses. Recentering and conditioning can both give a Brownian path zero time average, but the resulting laws differ: a constraint alone does not specify a process. A rank-one conditioning identity gives a scale-invariant criterion for the covariance effect of a shrinking family of observations. For Brownian finite differences of any fixed positive order, the covariance effect vanishes uniformly as the spacing tends to zero. For $m$-fold integrated Brownian motion, derivative constraints of orders $1,\\ldots,m$ remain detectable at interior times. For simulation, we identify which grid constructions are exact and which are approximate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33747",
  "author_total": 1,
  "authors": [
    {
      "name": "Lars Lau Raket"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdrw6t6rx8qcx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33747",
  "fulltext_url": null,
  "id": "fdrw6t6rx8qcx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33747",
  "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|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7223911989",
  "pdf_url": "https://arxiv.org/pdf/2610.33747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdrw6t6rx8qcx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning",
  "title_text": "A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fdrw6t6rx8qcx",
  "venue": "arXiv",
  "versions": []
}