Spectral Representation of Vector Random Field

Abstract

A homogeneous multi-dimensional random field is represented as the moving average by means of the multi-dimensional Wiener integral, from which the Fourier spectral representation of the random field is derived. The spectral representation of a homogeneous and isotropic vector random field in 3-dimensional space is achieved using the result of the previous work on the generalized spherical Bessel function and vector harmonics. The correlation tensor and the spectral density tensor are expressed as the vector Hankel transforms of each other. The spectral representation of the random field is obtained in terms of the solid vector harmonics and the random spectral measures labelled by three quantum numbers. One of quantum numbers indicates a longitudinal and two transverse parts as well as their mutual independence. The expression is simplified considerably by introduction of a single random phase. Lastly, various vector random fields, including the potential, the solenoidal and the curl fields, are discussed using the spectral representation.