{
  "abstract": "We study a system of $N$ attractive interacting diffusions where the variance feeds back into the noise. That is the common and idiosyncratic Brownian forcing has amplitude $κ(S_t^N)^q$, with $S_t^N$ the empirical variance of the system, $κ>0$ and $q\\geq 0$. With the interaction kernel given by $K(z)=|z|^p z$, for $p\\geq 0$, feedback through the empirical variance gives the system critical homogeneity at $q=(p+2)/4$. For this exponent an intrinsic time change separates the variance from the normalized centered configuration: the latter evolves autonomously on a unit sphere and, when it is not purely driven by common noise and the sphere has positive dimension, is a reversible elliptic diffusion with an explicit invariant Gibbs measure. In this case, for $p>0$, the stationary averaged dissipation gives an exact finite-particle trichotomy between consensus, recurrent excursions through arbitrarily small and large variance scales, and finite-time explosion. At critical homogeneity we also derive an autonomous unit-variance mean-field equation and its constrained free-energy dissipation identity. Across the subcritical, critical, and supercritical regimes, we establish well-posedness and boundary classifications and prove conditional propagation of chaos on intervals where the limiting solution is unique and nonexplosive. We connect the sharp finite-particle coercivity constants to a mean-field variational problem. Finite-size Itô corrections enlarge the universal coercive consensus region. We solve the linear model, where $p=0$, completely, including invariant laws, explosion probabilities, noncommuting limits, and an $N^{-1/2}$ transition window.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20771",
  "author_total": 2,
  "authors": [
    {
      "name": "Ben Hambly"
    },
    {
      "name": "Vinh Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6bg8n7yqyscfg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20771",
  "fulltext_url": null,
  "id": "6bg8n7yqyscfg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20771",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.20771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6bg8n7yqyscfg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits",
  "title_text": "Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6bg8n7yqyscfg",
  "venue": "arXiv",
  "versions": []
}