Stochastic Mappings and Random Distribution Fields II. Stationarity

TL;DR

This paper extends the theory of multivariate stochastic mappings and random distribution fields by exploring stationarity, providing new spectral representations and isomorphism theorems for stationary multivariate fields.

Contribution

It introduces a specific form of operator cross covariance distribution for multivariate random distribution fields and derives a Kolmogorov type isomorphism theorem and spectral representation.

Findings

Derived a particular form for the operator cross covariance distribution.

Established a Kolmogorov type isomorphism theorem for stationary fields.

Provided a spectral representation for stationary multivariate random distribution fields.

Abstract

As a continuation of [GasparPopa] this paper treats the stationary and stationarily cross-correlated multivariate stochastic mappings. Moreover for the case of multivariate random distribution fields, a particular form for the operator cross covariance distribution is given, from which a Kolmogorov type isomorphism theorem and a spectral representation of a stationary multivariate random distribution field are derived.