Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits
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(StN)q, with StN the empirical variance of the system, κ>0 and q≥ 0. With the interaction kernel given by K(z)=|z|p z, for p≥ 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.
Related
- A law of large numbers for kinetic interacting diffusions
- Formation of clusters and coarsening in weakly interacting diffusions
- Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle
- Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
- Jump limits of continuous multiscale diffusions: Homogenization under heavy-tailed activity bursts
BibTeX
@misc{hambly2026variancefeedback,
title = {{Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits}},
author = {Ben Hambly and Vinh Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20771},
eprint = {2610.20771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20771},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits
AU - Ben Hambly
AU - Vinh Nguyen
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20771
UR - https://arxiv.org/abs/2610.20771
AB - 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≥ 0. With the interaction kernel given by K(z)=|z|^p z, for p≥ 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.
ER -