Some Classes of Random Fields inn-Dimensional Space, Related to Stationary Random Processes

The purpose of this paper is to establish a spectral theory for certain types of random fields and random generalized fields (multidimensional random distributions) in the Euclidean n-space Rₙ similar to the well-known spectral theory for stationary random processes. Let D denote the Schwartz space of all complex-valued C_ functions (x) defined on Rₙ whose carrier is compact. Following Ito 6 and Gelfand 7 we shall call the random linear functional () on D satisfying (1. 4) the random generalized field. We can identify a continuous random field (x) on Rₙ with a random generalized field (1. 5) and therefore we can consider the ordinary random fields (x) as special cases of random generalized fields. We shall only deal with the first moment m () and the second moment B (₁, ₂) of the random generalized field () and shall call them a mean value functional and a covariance functional of this field. For an ordinary random field the mean value and covariance function defined by (1. 1) play a similar role. The random generalized field () is called homogeneous if its mean value functional m () and covariance functional B (₁, ₂) are invariant under all shift transformations in the space D, i. e. if the relations (2. 2) and (2. 3) hold true. The theory of this homogeneous random generalized field, which is very similar to Ito’s theory of a stationary random distribution 6, is treated in Section 2. The main results of the section deal with the spectral representation of the covariance functional of such fields and of random generalized fields themselves. Let D₁ denote the subspace of the space D consisting of all functions (x) satisfying (3. 1). The continuous linear functional () on D₁ is called a locally homogeneous random generalized field if its mean value functional m () and its covariance functional B (₁, ₂) are invariant under all shift transformations in D₁. The locally homogeneous random fields mentioned above are treated in Section 3. In this section we obtain the spectral representation of a covariance functional of the locally homogeneous random generalized field and the spectral representation of this field itself; these results generalize the spectral theory for random processes with stationary increments. The homogeneous random generalized field is called homogeneous and isotropic if its mean value functional m () and covariance functional B (₁, ₂) are invariant under 320 Some Classes of Random Fields all transformations in D induced by orthogonal transformations (motions and reflections) in the space Rₙ. In the case of the n-dimensional random generalized field () = = \ ₁ (), , ₙ () \ its mean value functional m () = \ m₁ (), , mₙ () \ forms a vector in Rₙ and its covariance functionals B₈₉ (₁, ₂) form a tensor in Rₙ. The n-dimensional homogeneous field () is called a homogeneous and isotropic random generalized vector field if the vector and the tensor are invariant under all motions and reflections in Rₙ and the simultaneous transformations in D induced by this motion or reflection Homogeneous and isotropic random generalized fields and random generalized vector fields are treated in Section 4. In this section we obtain the general form of the functionals m () (or m () ) and B (₁, ₂) (or B₈₉ (₁, ₂) ) for these fields. The locally homogeneous random generalized field (), D₁ is called locally homogeneous and locally isotropic if its mean value functional m () and covariance functional B (₁, ₂) are invariant under all transformations in D₁ induced by orthogonal transformations in Rₙ ; it is also clear how we can define the notion of the locally homogeneous and locally isotropic random generalized vector field (), D₁. Locally homogeneous and locally isotropic random generalized vector fields are treated in Section 5. Here we obtain the general form of the mean value functional and of the covariance for these fields; in particular, we obtain the general form of a mean value and of a covariance function of ordinary random fields (scalar and vector) which are locally homogeneous and locally isotropic in the sense of Kolmogorov 17.