arXiv 1101.45632011-01-24math.PR

算子分式布朗运动的指数、对称群和分类

Exponents, symmetry groups and classification of operator fractional Brownian motions

Gustavo Didier, Vladas Pipiras

摘要：

算子分数布朗运动（OFBM）是零均值，算子自相似（o.s.s.），平稳增量高斯过程。他们将单变量分数布朗运动推广到多变量的情形。众所周知，所谓的对称群的一个o.s.s.过程与正交群的子群共轭。此外，根据Hudson和Mason的一个著名结果，算子自相似过程的所有指数的集合可以与它的对称群的切空间相联系.本文基于OFBM的谱域整表示，重新研究了OFBM的对称群和指数集。一般描述的对称群OFBM的频谱域参数的中心化的子集。研究了任意维数下具有极大型和极小型对称群的OFBM。特别是，它表明，OFBM具有最小的对称群（因此，独特的指数）一般来说，在拓扑意义上。更精细的分类结果OFBM的基础上，明确建设他们的对称群，给出了在较低的维度2和3。它还表明，谱域整表示的参数化，在适当的意义上，不受指数的多重性，而同样是不正确的时域整表示。

英文摘要：

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group. In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

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