分数维纳混沌：第二部分 // 界面谱混沌与集体相关性的衰减
Fractional Wiener chaos: Part 2. Interface spectral chaos
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中文总结 AI 辅助
本文构造了基于抛物柱面函数的乘积谱分解，在变形下得到非高斯分布并恢复经典维纳混沌，利用该分解推导了共享与独立逆稳定时钟下集体阈值相关性的显式展开，发现共享时钟下衰减更慢。
中文摘要 AI 辅助
我们由幂归一化的抛物柱面函数构造了一个乘积谱分解。在非整数实数阶时，这些函数关于相应的全直线高斯测度不是平方可积的。将可容许的半直线分支引入同一高斯空间并在原点处匹配，得到修正的正交归一特征基。通过酉变换到加权概率空间，使对应于最小特征值的特征函数为常数，并允许一致的可数乘积。所得分布在变形下是非高斯的，当变形移除时，该分解恢复经典维纳混沌。我们识别了相关的闭梯度、伴随散度和变形数算子。Rodrigues公式覆盖了谱分支中出现的所有阶，包括负阶。然而，在变形下，将两个分支阶同时改变相同的非零量会破坏它们之间所需的关系。通过谱演算定义的分数演化保持该基。在酉识别后，单坐标分数演化算子在算子范数下收敛到高斯和三维径向Ornstein-Uhlenbeck分数演化，分别在相应的参数端点处。收敛在远离零的紧时间区间上是一致的。我们利用所构造的乘积谱分解推导了在共享和独立逆稳定时钟下集体阈值相关性的显式展开。这些展开确定了代数衰减速率和首项系数，并提供了显式的截断误差界。在平衡初始化下，两种模型具有相同的单坐标过程律和相同的固定时间联合律，然而在共享时钟下集体相关性衰减得更慢。
英文摘要
We construct a product spectral decomposition from power-normalised parabolic-cylinder functions. At noninteger real orders, these functions are not square integrable with respect to the corresponding full-line Gaussian measure. Bringing the admissible half-line branches into the same Gaussian space and matching them at the origin gives a corrected orthonormal eigenbasis. A unitary transformation to a weighted probability space makes the eigenfunction corresponding to the lowest eigenvalue constant and permits consistent countable products. The resulting law is non-Gaussian under deformation, and the decomposition recovers classical Wiener chaos when the deformation is removed. We identify the associated closed gradient, adjoint divergence and deformed number operator. The Rodrigues formula covers all orders occurring in the spectral branches, including negative orders. Under deformation, however, changing both branch orders by the same nonzero amount breaks the required relation between them. Fractional evolution defined through spectral calculus preserves the basis. After unitary identification, the one-coordinate fractional evolution operators converge in operator norm to Gaussian and three-dimensional radial Ornstein--Uhlenbeck fractional evolutions at the respective parameter endpoints. Convergence is uniform on compact time intervals away from zero. An application example with inverse stable clock is provided in the accompanying github repository.
