带谱底的Bures - Wasserstein协方差动力学的哈密顿提升
Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor
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中文总结 AI 辅助
研究正定锥上协方差动力学，构造其在Sym$^+_n$上带Bures - Wasserstein度量的有限维哈密顿提升，通过自然力学拉格朗日量等得出相关动量和哈密顿量，添加瑞利耗散恢复梯度流，确定保守母系统及谱底诱导的局部刚度尺度。
中文摘要 AI 辅助
正定锥上的协方差动力学通常由梯度流描述，它编码耗散弛豫但掩盖了潜在的相空间结构。我们在配备Bures - Wasserstein度量的Sym$^+_n$上构造了协方差动力学的有限维哈密顿提升。自然力学拉格朗日量产生正则动量$\Pi=\tfrac12 L_\Sigma[\dot\Sigma]$，其中$L_\Sigma$是李雅普诺夫算子，还有显式哈密顿量$\mathcal{H}(\Sigma,\Pi) = 2{\rm tr}(\Pi\Sigma\Pi)+V(\Sigma)$。添加瑞利耗散在过阻尼极限下恢复Bures - Wasserstein梯度流。对于谱底和迹控制势，各向同性平衡附近的二次涨落哈密顿量分离了迹和无迹模式；当平衡协方差接近谱底时，基线刚度以$(s - \nu)^{-2}$发散。该构造为受约束的Bures - Wasserstein协方差弛豫确定了一个保守母系统，并确定了由谱底引起的局部刚度尺度。
英文摘要
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 $Π=\tfrac12 L_Σ[\dotΣ]$, where $L_Σ$ is the Lyapunov operator, and explicit Hamiltonian $\mathcal{H}(Σ,Π) = 2{\rm tr}(ΠΣΠ)+V(Σ)$. 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.
