Variance-Feedback Noise in Interacting Diffusions: Shape Dynamics and Mean-Field Limits

Source 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.