算子分式布朗运动的整表示与性质
Integral representations and properties of operator fractional Brownian motions
摘要：
算子分数布朗运动（OFBM）是（i）高斯过程，（ii）算子自相似过程和（iii）平稳增量过程.它们是分数布朗运动的自然多元推广。由于可能缺乏时间可逆性，定义性质（i）-（iii）一般不表征OFBM的协方差结构。为了避免这个问题，类OFBM的特点是在这里通过他们的整表示在频谱和时域。对于谱域表示，这涉及到如何运营商的自相似形状的平稳增量过程的一般表示的谱密度。时域表示是通过使用主矩阵函数并对确定性谱域核进行傅里叶变换而导出的。根据OFBM的谱表示和时域表示，建立了OFBM时间可逆的充要条件。它还表明，固定增量的OFBM的谱密度有一个刚性的结构，这里称为二分法原理。算子布朗运动的概念也进行了探讨。
英文摘要：
Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)--(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.
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