{
  "abstract": "We prove the local well-posedness of the (renormalized) parabolic $\\Phi^4_3$ model associated with the harmonic oscillator on $\\mathbb{R}^3$, that is, the equation formally written as \\begin{equation*} \\partial_t X + HX= -X^3+\\infty\\cdot X + \\xi, \\quad t>0, \\quad x \\in \\mathbb{R}^3, \\end{equation*} where $H:=-\\Delta_{\\mathbb{R}^3} +|x|^2$ and $\\xi$ denotes a space-time white noise. This model is related to the stochastic Gross-Pitaevskii equation, used in the description of Bose-Einstein condensation. \\smallskip Our overall formulation of the problem, based on the so-called paracontrolled calculus, follows the strategy outlined by Mourrat and Weber for the $\\Phi^4_3(\\Delta)$ model on the three-dimensional torus. To extend the paracontrolled approach to the harmonic oscillator setting, we develop several new analytical tools. These include a microlocal analysis adapted to the paracontrolled framework, resonance-type/high–low frequency operators, and the construction of the stochastic diagrams governing the dynamics. Taken together, these developments constitute the core technical contribution of the present work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Aurélien Deya"
    },
    {
      "name": "Reika Fukuizumi"
    },
    {
      "name": "Laurent Thomann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq1brzvbx7h9t.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "dq1brzvbx7h9t",
  "issns": [],
  "landing_url": "https://hal.science/hal-05016338v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-ap"
    ],
    [
      "hal",
      "1.math.math-pr"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05016338/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq1brzvbx7h9t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:24Z",
      "native_id": "hal-05016338",
      "source": "hal",
      "url": "https://hal.science/hal-05016338v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ap",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-pr",
      "subfield": "probability"
    }
  ],
  "title": "On the parabolic $\\Phi_3^4$ model for the harmonic oscillator: diagrams and local existence",
  "title_text": "On the parabolic Φ_3^4 model for the harmonic oscillator: diagrams and local existence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:24Z",
  "url": "https://research.draconitic.com/paper/dq1brzvbx7h9t",
  "venue": "HAL",
  "versions": []
}