Spectral Expansions of Homogeneous and Isotropic Tensor-Valued Random Fields

We establish spectral expansions of homogeneous and isotropic random fields taking values in the $3$-dimensional Euclidean space $E^3$ and in the space $\mathsf{S}^2(E^3)$ of symmetric rank $2$ tensors over $E^3$. The former is a model of turbulent fluid velocity, while the latter is a model for the random stress tensor or the random conductivity tensor. We found a link between the theory of random fields and the theory of finite-dimensional convex compacta.