论文标题
基于Hilbert-Schmidt操作员，桥接Koopman操作员和时间序列自动相关
Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator
论文作者
论文摘要
给定平稳的连续时间过程$ f（t）$，可以为每个有限的$τ$ \ cite {vautard1989singularsa}定义Hilbert-Schmidt操作员$A_τ$。令$λ_{τ，i} $为$a_τ$带有降序的特征值。在本文中，定义了Hilbert Space $ \ MATHCAL {H} _f $和（时间班）连续的ISOMETRIES的单参数$ \ MATHCAL {K}^S $。令$ \ {v_i，i \ in \ mathbb {n} \} $为所有$ s \ geq 0 $的$ \ mathcal {k}^s $的eigenVectors。令$ f = \ displayStyle \ sum_ {i = 1}^{\ infty} a_iv_i + f^{\ perp} $是正交分解，降低$ | a_i | $。我们证明$ \ displaystyle \ lim_ {τ\ to \ infty}λ_{τ，i} = | a_i |^2 $。连续的单参数semigroup $ \ {\ Mathcal {k}^s：s \ geq 0 \} $几乎可以肯定地与经典的koopman单参数半群在$ l^2（x，n）上定义的经典单参数半群，如果动态系统是eRgodic and ofgarient and ovariant量$ n $ n $ n $ n，则
Given a stationary continuous-time process $f(t)$, the Hilbert-Schmidt operator $A_τ$ can be defined for every finite $τ$\cite{Vautard1989SingularSA}. Let $λ_{τ,i}$ be the eigenvalues of $A_τ$ with descending order. In this article, a Hilbert space $\mathcal{H}_f$ and the (time-shift) continuous one-parameter semigroup of isometries $\mathcal{K}^s$ are defined. Let $\{v_i, i\in\mathbb{N}\}$ be the eigenvectors of $\mathcal{K}^s$ for all $s\geq 0$. Let $f = \displaystyle\sum_{i=1}^{\infty}a_iv_i + f^{\perp}$ be the orthogonal decomposition with descending $|a_i|$. We prove that $\displaystyle\lim_{τ\to\infty}λ_{τ,i} = |a_i|^2$. The continuous one-parameter semigroup $\{\mathcal{K}^s: s\geq 0\}$ is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on $L^2(X,ν)$, if the dynamical system is ergodic and has invariant measure $ν$ on the phase space $X$.
