{
  "abstract": "This paper establishes subordination representations from multi-term time-fractional differential equations to single-term ones, a direction complementary to the previously known subordination from single-term to multi-term equations; combining the two complementary directions also yields subordination representations between two multi-term equations. The crucial step is to prove that certain implicitly defined functions are Bernstein functions. We show that the quotient $\\big(\\sum_{j=1}^m a_jλ^{α_j}\\big)/\\big(\\sum_{j=1}^m a_jα_jλ^{α_j}\\big)$ is a Stieltjes function whenever $0\\leqα_1-α_m\\leq 1$, and consequently that the inverse function of $λ=\\big(\\sum_{j=1}^m a_jφ^{α_j}\\big)^{1/β}$ is a Bernstein function for every $0<β\\leqα_m$. As applications, we obtain subordination representations with a clear probabilistic meaning: from multi-term to single-term time-fractional diffusion-wave equations, between abstract nonlocal-in-time equations with kernels of type $(\\mathcal{PC})$ (including equations of tempered stable type), and between nonlocal-in-time telegraph equations. A necessary and sufficient condition for $[(λ+b)^β-b^β]^γ$ to be a Bernstein function also yields a negative result: in general, no such subordination exists from an $α$-stable type dynamics to a tempered $β$-stable one when $α<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25278",
  "author_total": 2,
  "authors": [
    {
      "name": "Calvin Khor"
    },
    {
      "name": "Cheng-Gang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e43wwc01ctm0r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25278",
  "fulltext_url": null,
  "id": "e43wwc01ctm0r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25278",
  "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.AP"
    ],
    [
      "openalex",
      "2611|Fractional Differential Equations Solutions"
    ]
  ],
  "openalex": "W7223010986",
  "pdf_url": "https://arxiv.org/pdf/2610.25278",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e43wwc01ctm0r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25278",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25278"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "New Bernstein functions and subordination representations for time-nonlocal evolution equations",
  "title_text": "New Bernstein functions and subordination representations for time-nonlocal evolution equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/e43wwc01ctm0r",
  "venue": "arXiv",
  "versions": []
}