{
  "abstract": "We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple \\[ V\\hookrightarrow H\\simeq H^*\\hookrightarrow V^*. \\] The evolution problem is formulated with the posterior state, \\[ D_g u(t)+A(t) S_g u(t)=f(t),\\qquad u(a)=u_0. \\] This choice is decisive at atoms of the Stieltjes measure: if $δ=Δ_g(r)>0$, the equation yields the implicit variational resolvent \\[ (I+δA(r))u(r^+)=u(r)+δf(r). \\] We prove a Banach-valued lateral product formula, a weak--strong reconstruction theorem, and a Stieltjes analogue of the Lions--Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution \\[ \\frac12\\sum_r\\|u(r^+)-u(r)\\|_H^2. \\] Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin--Lions theorem for anterior Stieltjes--Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00171",
  "author_total": 1,
  "authors": [
    {
      "name": "Francisco J. Fernández"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8ym567dh1g8g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00171",
  "fulltext_url": null,
  "id": "e8ym567dh1g8g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00171",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Differential Equations and Boundary Problems"
    ]
  ],
  "openalex": "W7215967658",
  "pdf_url": "https://arxiv.org/pdf/2610.00171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8ym567dh1g8g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States",
  "title_text": "Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/e8ym567dh1g8g",
  "venue": "arXiv",
  "versions": []
}