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Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

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Abstract

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 (and 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 multiplicity of exponents, whereas the same is not true for time domain integral representations.

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Authors and Affiliations

  1. Mathematics Department, Tulane University, 6823 St. Charles Avenue, New Orleans, LA, 70118, USA

    Gustavo Didier

  2. Dept. of Statistics and Operations Research, UNC-Chapel Hill, CB#3260, Smith Bldg., Chapel Hill, NC, 27599, USA

    Vladas Pipiras

  3. CEMAT, Instituto Superior Técnico, Lisboa, Portugal

    Vladas Pipiras

Authors
  1. Gustavo Didier
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  2. Vladas Pipiras
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Corresponding author

Correspondence to Gustavo Didier.

Additional information

The first author was supported in part by the Louisiana Board of Regents award LEQSF(2008-11)-RD-A-23. The second author was supported in part by the NSF grants DMS-0505628 and DMS-0608669.

The authors are thankful to Profs. Eric Renault and Murad Taqqu for their comments on this work.

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Didier, G., Pipiras, V. Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions. J Theor Probab 25, 353–395 (2012). https://doi.org/10.1007/s10959-011-0348-5

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  • DOI: https://doi.org/10.1007/s10959-011-0348-5

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