{
  "abstract": "We study gradient-type dynamics for weak stochastic control through the laws of controlled diffusions on path space. Although the objective is generally nonconvex in the drift, a uniformly convex running cost induces a strictly convex functional of the corresponding path law and hence a natural Bregman geometry. We construct the associated implicit proximal scheme and prove its well-posedness and convergence to a continuous-time mirror flow. When the mirror cost coincides with the running cost and the law-dependent potential is linear, the flow becomes affine in centered entropy coordinates. This yields a variational construction of the resulting Newton flow, uniform stability estimates, and exponential convergence to the optimizer. For a general uniformly convex mirror cost and a nonlinear convex potential of the law, we construct the natural-gradient flow globally in learning time using BMO estimates for quadratic BSDEs, and establish quantitative convergence and a first-order discretization error. We further analyze interacting finite-particle approximations, proving exponential convergence to the finite-particle optimum and an O(1/N) normalized discrepancy from the mean-field optimum. In the Markovian setting, reflection coupling yields uniform spatial regularity and exponential convergence of the feedback controls on Rd, including for nonquadratic state-dependent Hamiltonians.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03487",
  "author_total": 4,
  "authors": [
    {
      "name": "Yupeng Bai"
    },
    {
      "name": "Louis-Pierre Chaintron"
    },
    {
      "name": "Zhenjie Ren"
    },
    {
      "name": "Songbo Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24qj6dd5kx1zb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03487",
  "fulltext_url": null,
  "id": "24qj6dd5kx1zb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03487",
  "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.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7220421789",
  "pdf_url": "https://arxiv.org/pdf/2610.03487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24qj6dd5kx1zb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convex geometry of diffusion laws and gradient flows for stochastic control",
  "title_text": "Convex geometry of diffusion laws and gradient flows for stochastic control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/24qj6dd5kx1zb",
  "venue": "arXiv",
  "versions": []
}