{
  "abstract": "We study probability measures on the space of holomorphic functions on the unit disk that are invariant under pre-composition by conformal automorphisms and give no mass to constant functions. We determine the slowest possible growth of functions supported by such measures. Let $M_f(R)$ denote the maximum modulus of $f$ on the hyperbolic ball of radius $R$ centered at the origin. We show that given a non-decreasing function $G:[1,\\infty)\\to(0,\\infty)$, there exists an $\\textrm{Aut}(\\mathbb{D})$-invariant probability measure $μ$ on $\\textrm{Hol}(\\mathbb{D}, \\mathbb{C}) \\setminus \\{\\textrm{constant functions} \\}$ such that $\\log^+ M_f(R)=O_f(G(R))$ for $μ$ almost every $f$ if and only if $\\int_1^\\infty (R/G(R)) \\, dR <\\infty$. For the lower bound, we use a mass transport argument to show that almost every trajectory of the gradient flow of $\\log |f|$ has finite energy. To establish sharpness, we construct holomorphic functions of slow growth on infinite Riemann surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40057",
  "author_total": 2,
  "authors": [
    {
      "name": "Oleg Ivrii"
    },
    {
      "name": "Uri Kreitner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21mmy0wnx6dte.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40057",
  "fulltext_url": null,
  "id": "21mmy0wnx6dte",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40057",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21mmy0wnx6dte.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Invariant probability measures on holomorphic functions on the unit disk",
  "title_text": "Invariant probability measures on holomorphic functions on the unit disk",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/21mmy0wnx6dte",
  "venue": "arXiv",
  "versions": []
}