{
  "abstract": "We introduce an intrinsic notion of p-roughness for continuous paths,for p>1, defined by the uniform convergence of discrete p-energies over all shifted sufficiently fine uniform grids. We prove that this self-averaging property is equivalent to mesoscopic cancellation of the coarse-graining error for discrete p-energy, yielding a characterization that can be verified on a single uniform multiresolution. $p$-roughness refines the finite p-th variation property and implies the invariance of p-th variation across a class of partition sequences. We prove that Brownian motion is almost surely 2-rough and that fractional Brownian motion with Hurst parameter $H$ is almost surely $1/H$-rough. We also derive criteria for p-roughness based on Faber-Schauder coefficients. These results yield partition-robust formulations of higher-order pathwise calculus and energy occupation measures. Finally, we interpret coarse-graining as a renormalization flow for the p-energy and show that the p-roughness class is stable under critical time-amplitude scaling, with linear p-energy profiles as fixed points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18855",
  "author_total": 1,
  "authors": [
    {
      "name": "Rama Cont"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/00f31mj22y4gh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18855",
  "fulltext_url": null,
  "id": "00f31mj22y4gh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18855",
  "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.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/00f31mj22y4gh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18855"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "$p$-roughness of paths and invariance of $p$-th variation",
  "title_text": "p-roughness of paths and invariance of p-th variation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/00f31mj22y4gh",
  "venue": "arXiv",
  "versions": []
}