Representations of the Random Fields on a Sphere

Abstract

With practical applications in mind, a study is made of the representations of the random fields on a sphere based on the theory of representations of the rotation group. The representation of the rotation group is defined in this paper by a class of rotational shift transformations of the random fields generated by a homogeneous random field on the sphere, ‘homogeneous’ with respect to the rotational motions. First, the spectral decomposition of a homogeneous scalar random field on a sphere is given a simple interpratation : it is a sum of the invariant vectors in the irreducible representation spaces of the rotational shift transformations. Some representations of the random fields are given in terms of the stochastic integrals with respect to a homogeneous random measure on the sphere. A homogeneous l-vector random field with 2 l+ 1 components in an irreducible space of weight l representation is defined as an invariant tensor under rotational shift transformations. A ‘stochastic’ spherical harmonics is defined as one of such random fields, which is a stochastic version of the l-vector spherical harmonic. Similarly, a ‘stochastic’ solid harmonic is defined in terms of stochasic spherical harmonics and generalized spherical Bessel functions. It is expressible as a l-vector Fourier integral over a sphere as well as a tensorial Fourier integral, and satisfies the l-vector Helmholtz equation. Such ‘stochastic’ harmonic functions can be used effectively in dealing with the scattering problem associated with a random sphere.