{
  "abstract": "This paper concerns the global solutions of the kinetic Fokker-Planck (KFP) equation in $\\mathbb R^3$ driven by a given external force $E=E(t,x)$ and an inhomogeneous source term $S_0=S_0(t,x,v)$. First, when $E$ and $S_0$ are sufficiently small in a suitable hybrid homogeneous Besov-Sobolev framework without any time-decay assumption, we establish the global well-posedness of the Cauchy problem in the hybrid homogeneous space $L_v^2(\\dot B^{ {1}/{2}}_{2,\\infty}(\\mathbb R^3_x) \\cap\\dot H^N(\\mathbb R^3_x))$ with $N\\geq 3$. We further obtain the uniform stability of the Cauchy problem via semi-group theory and a refined energy method. Second, when both $E$ and $S_0$ are time-periodic of the period $T$, we show the existence of time-periodic solutions with the same period $T$ and their asymptotic stability as $t\\to \\infty$. Moreover, we derive the time-decay estimate of the perturbation around time-periodic solutions for initial data in $L^p$-norm with $ {3}/{2}< p \\leq 2$. Finally, when both $E$ and $S_0$ are independent of $t$, we establish the existence of stationary solutions and their asymptotic stability. Our results provide a unified treatment of time-periodic dynamics and stationary states for the KFP equation with small force and source terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23191",
  "author_total": 4,
  "authors": [
    {
      "name": "Renjun Duan"
    },
    {
      "name": "Fucai Li"
    },
    {
      "name": "Jinkai Ni"
    },
    {
      "name": "Zhu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29pg1frzq9jjq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23191",
  "fulltext_url": null,
  "id": "29pg1frzq9jjq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23191",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Gas Dynamics and Kinetic Theory"
    ]
  ],
  "openalex": "W7222958393",
  "pdf_url": "https://arxiv.org/pdf/2610.23191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29pg1frzq9jjq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Solutions to the Fokker-Planck equation with time-dependent force and source in $\\mathbb R^3$",
  "title_text": "Solutions to the Fokker-Planck equation with time-dependent force and source in ℝ^3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/29pg1frzq9jjq",
  "venue": "arXiv",
  "versions": []
}