{
  "abstract": "We develop a Dirichlet-form framework for relaxation and steady-state entropy production in preconditioned Langevin stochastic partial differential equations. In infinite dimensions, the usual Fokker--Planck calculations based on ambient Lebesgue densities and the corresponding probability-current densities are generally unavailable. For the detailed-balance class, a quasi-regular symmetric Dirichlet form on the Gibbs state space yields an exact de Bruijn entropy-dissipation formula for regular densities and an integrated inequality for arbitrary finite-entropy initial laws. Self-adjointness gives detailed balance and stationary path reversal, while a coordinate-martingale criterion identifies the associated process with the prescribed SPDE. For the one-dimensional $Φ^4_1$ and convex Allen--Cahn-type Gibbs dynamics, we combine the established strong well-posedness theory with direct verification of the logarithmic derivatives, form closure, quasi-regularity and form--SPDE correspondence, and obtain relative-entropy decay bounds with exponents $2$ and $2(1-λ/m)$, respectively, with $m>λ$ in the latter case. Away from detailed balance, bounded skew-adjoint linear forcing preserves a Gaussian invariant law without requiring commutation between the forcing and covariance. We identify the antisymmetric action on cylinder observables and its Cameron--Martin current and prove that the squared current energy equals the steady-state entropy-production rate defined by forward--reverse path-space relative entropy per unit time, as well as the monotone limit of the Galerkin rates. Under commutation, we additionally obtain an explicit transient Onsager decomposition after a mass quench. Exclusion processes and Gaussian rotors provide finite-state and Gaussian benchmarks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18201",
  "author_total": 4,
  "authors": [
    {
      "name": "Shuyuan Fan"
    },
    {
      "name": "Yuanke Chen"
    },
    {
      "name": "Lifei Wang"
    },
    {
      "name": "Jinqiao Duan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5a81mtcmmve70.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18201",
  "fulltext_url": null,
  "id": "5a81mtcmmve70",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18201",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "3109|Advanced Thermodynamics and Statistical Mechanics"
    ]
  ],
  "openalex": "W7213530388",
  "pdf_url": "https://arxiv.org/pdf/2609.18201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5a81mtcmmve70.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2609.18201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18201"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach",
  "title_text": "Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/5a81mtcmmve70",
  "venue": "arXiv",
  "versions": []
}