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 Π=1/2 LΣ[Σ̇], where LΣ is the Lyapunov operator, and explicit Hamiltonian ℋ(Σ,Π) = 2tr(ΠΣΠ)+V(Σ). 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−ν)−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.
Versions
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BibTeX
@misc{kerskens2026hamiltonian,
title = {{Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor}},
author = {Christian Kerskens},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.16380},
eprint = {2607.16380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16380},
}
RIS
TY - UNPB
TI - Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor
AU - Christian Kerskens
PY - 2026
DA - 2026/07/17
PB - arXiv
DO - 10.48550/arxiv.2607.16380
UR - https://arxiv.org/abs/2607.16380
AB - 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 Π=1/2 L_Σ[Σ̇], where L_Σ is the Lyapunov operator, and explicit Hamiltonian ℋ(Σ,Π) = 2tr(ΠΣΠ)+V(Σ). 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−ν)^−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.
ER -