Random Fields Related to the Symmetry Classes of Second-Order Symmetric Tensors

  • Anatoliy Malyarenko
  • Martin Ostoja-Starzewski
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)


Under the change of basis in the three-dimensional space by means of an orthogonal matrix g, a matrix A of a linear operator is transformed as \(A\mapsto gAg^{-1}\). Mathematically, the stationary subgroup of a symmetric matrix under the above action can be either \(D_2\times Z^c_2\), when all three eigenvalues of A are different, or \(\mathrm {O}(2)\times Z^c_2\), when two of them are equal, or \(\mathrm {O}(3)\), when all three eigenvalues are equal. Physically, one typical application relates to dependent quantities like a second-order symmetric stress (or strain) tensor. Another physical setting is that of dependent fields, such as conductivity with such three cases is the conductivity (or, similarly, permittivity, or anti-plane elasticity) second-rank tensor, which can be either orthotropic, transversely isotropic, or isotropic. For each of the above symmetry classes, we consider a homogeneous random field taking values in the fixed point set of the class that is invariant with respect to the natural representation of a certain closed subgroup of the orthogonal group. Such fields may model stochastic heat conduction, electric permittivity, etc. We find the spectral expansions of the introduced random fields.


Random field Symmetry class Spectral expansion 


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Authors and Affiliations

  1. 1.Division of Applied Mathematics, School of Education Culture and CommunicationMälardalen UniversityVästeråsSweden
  2. 2.University of Illinois at Urbana-ChampaignUrbanaUSA

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