{
  "abstract": "We establish mathematical foundations for optimal spatiotemporal vector fields governing continuous score-guided probability flows over bounded submanifolds $M \\subset SE(3)$ on the Special Euclidean group $SE(3) \\cong \\mathbb{R}^3 \\times SO(3)$. Modeling dynamic density evolution via a score-perturbed Lie-algebraic continuity equation under zero-flux boundary conditions, we resolve the non-local score drift singularity $s(g,t) = \\nabla_{SE(3)} \\log ρ(g,t)$ as $ρ\\rightarrow 0^+$ via weighted parabolic flux dissipation bounds. Using left-invariant exponential group mollifiers, we derive boundary-preserving DiPerna--Lions commutator estimates tracking structure constants $\\{c_{ij}^k\\}$. Second, interpreting the optimality system as a 7D space-time elliptic operator $L_{7D} = -Δ_{SE(3)} - \\partial_t^2$ with zero-order corner compatibility on $\\partial M \\times \\{0,T\\}$, we establish Agmon--Douglis--Nirenberg $W_p^2(Q_T)$ shift estimates tracking Lie algebra constants, boundary geometry $\\|\\partial M\\|_{C^2}$, and Ricci curvature $\\|Ric_{SE(3)}\\|_{L^\\infty}$, yielding a full $C^\\infty(\\overline{Q}_T)$ Schauder bootstrap expansion. Third, we extend this system to non-cooperative Markov Mean-Field Games (MFGs) on $SE(3)$, establishing score-guided Nash equilibria via Schauder--Tychonoff fixed points. Crucially, the constant diagonal Ricci curvature $Ric_{SE(3)} = \\mathrm{diag}(0,0,0,1/2,1/2,1/2)$ from $SO(3)$ induces a sharp Bochner--Weitzenböck spectral gap boost $λ_1(-Δ_{SE(3)}) \\ge λ_1(-Δ_{\\mathbb{R}^3}) + 1/2 > 0$. This positive rotational curvature enhances the Lie-algebraic Poincaré coercivity constant, demonstrating that intrinsic group curvature stabilizes score-guided flows against dispersion and expands the critical Nash uniqueness horizon $T_{SE(3)}^*$ beyond flat Euclidean limits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03278",
  "author_total": 1,
  "authors": [
    {
      "name": "Pilhwa Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/acztrt5qx7ydv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03278",
  "fulltext_url": null,
  "id": "acztrt5qx7ydv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03278",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03278",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/acztrt5qx7ydv.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03278",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03278"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on $SE(3)$: Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games",
  "title_text": "On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on SE(3): Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/acztrt5qx7ydv",
  "venue": "arXiv",
  "versions": []
}