Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Spectral expansions of homogeneous and isotropic tensor-valued random fields. (English) Zbl 1348.60077 Z. Angew. Math. Phys. 67, No. 3, Article ID 59, 20 p. (2016). Summary: We establish spectral expansions of tensor-valued homogeneous and isotropic random fields in terms of stochastic integrals with respect to orthogonal scattered random measures previously known only for the case of tensor rank 0. The fields under consideration take values in the 3-dimensional Euclidean space \(E^3\) and in the space \(\mathrm{S}^2(E^3)\) of symmetric rank 2 tensors over \(E^3\). We find a link between the theory of random fields and the theory of finite-dimensional convex compact sets. These random fields furnish a stepping-stone for models of rank 1 and rank 2 tensor-valued fields in continuum physics, such as displacement, velocity, stress and strain, providing appropriate conditions (such as the governing equation or positive-definiteness) are imposed. Cited in 6 Documents MSC: 60G60 Random fields 60H05 Stochastic integrals 60G57 Random measures 74B05 Classical linear elasticity Keywords:tensor-valued random fields; spectral expansion; stochastic integrals; random measures PDFBibTeX XMLCite \textit{A. Malyarenko} and \textit{M. Ostoja-Starzewski}, Z. Angew. Math. Phys. 67, No. 3, Article ID 59, 20 p. (2016; Zbl 1348.60077) Full Text: DOI arXiv References: [1] Arad I., L’vov V.S., Procaccia I.: Correlation functions in isotropic and anisotropic turbulence: the role of the symmetry group. Phys. Rev. E. 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