{
  "abstract": "We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus ($T^2$) with a potential $V(x)\\in H^{2^+}(T^2)$. The initial data, $u_0$,is given by a centered gaussian variable with covariance $(-Δ+V)^{-1}$, and thus is almost surely in $H^{0^-}(T^2)$. We show that it is almost surely well-possed for all time in the weak sense, and that $u-e^{it(Δ-V)}u_0$ is in $H^{s}(T^2)$ for some $s>0$, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schr{ö}dinger} Equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21969",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Lindstrom"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egytzndg9gd22.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21969",
  "fulltext_url": null,
  "id": "egytzndg9gd22",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21969",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7213898931",
  "pdf_url": "https://arxiv.org/pdf/2609.21969",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egytzndg9gd22.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.21969",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21969"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "2D Torus defocusing NLSE with a potential and random initial data",
  "title_text": "2D Torus defocusing NLSE with a potential and random initial data",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/egytzndg9gd22",
  "venue": "arXiv",
  "versions": []
}