Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor

Abstract

Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Sym$^+_n$ equipped with the Bures--Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum $\Pi=\tfrac12 L_\Sigma[\dot\Sigma]$, where $L_\Sigma$ is the Lyapunov operator, and explicit Hamiltonian $\mathcal{H}(\Sigma,\Pi) = 2{\rm tr}(\Pi\Sigma\Pi)+V(\Sigma)$. Adding Rayleigh dissipation recovers the Bures--Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as $(s-\nu)^{-2}$ as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures--Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.