{
  "abstract": "We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against \"noncommutative $L^p$-integrators\" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23948",
  "author_total": 3,
  "authors": [
    {
      "name": "David A. Jekel"
    },
    {
      "name": "Todd A. Kemp"
    },
    {
      "name": "Evangelos A. Nikitopoulos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9wvxy4xn4d59s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23948",
  "fulltext_url": null,
  "id": "9wvxy4xn4d59s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23948",
  "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.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7214049053",
  "pdf_url": "https://arxiv.org/pdf/2609.23948",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9wvxy4xn4d59s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23948",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23948"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Noncommutative $L^p$-integrators and stochastic differential equations in $L^p$",
  "title_text": "Noncommutative L^p-integrators and stochastic differential equations in L^p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/9wvxy4xn4d59s",
  "venue": "arXiv",
  "versions": []
}