Operator valued distributions and random fields
by
Pastorel Gaspar
West University of Timisoara
Coauthors: Dumitru Gaspar, Nicoale Grindeanu, Lorena Popa

Stochastic distributions were introduced in the specialized literature by K. Itô and I.M. Gelfand, as generalization of stochastic processes with one continuous time parameter. The extension of prediction theory and extrapolation to this distributions frame is due to Yu. A. Rozanov, K. Balagangadharan and refers, as in the case of stochastic processes, to the stationary case. As it is known in the case of finitely many continuous time parameters stochastic processes are called random fields.
In analogy with the "semigroup distribution" introduced by J. L. Lions for the generalization to operatorial distributions of C₀ - continuous operator semigroups, we shall use the appellation random field distribution, for the generalization to the frame of distributions of random fields.
In the present talk, we aim to approach the study of random fields distributions with Fourier analysis methods. The first section contains generalities on random fields distributions and the Fourier representations of stationary random fields distributions.
The third section treats the harmonizability of random fields distributions, which were considered only for continuous fields by M. Loève, H. Cram\' er, Yu. A. Rozanov, M. Rao a. o. It is known that the harmonizable stochastic processes (and random fields) are defined using a bi-measure of finite semivariation. It is therefore to be expected that more general bi-measures play an important role in a similar study of random fields distributions.
Therefore we grant the second section to the treatment of this kind of bi-measures with the aid of distributions. A classification of bi-measure on the pattern of the L. Schwartz classification of measures, which correspond to the continuity of measures (bi-measures) - as functionals - on the basic spaces considered, is evidenced. We are also concerned with the identification of a sufficiently wide class of bi-measures to which a (modified) Fourier transform can be applied.
The results are then used in defining the tempered random fields distributions and to establish the integral representation and other basic properties, but mainly a dilation theorem of harmonizable random fields distributions to stationary random fields distributions.
In the last sections, by using suitable results about the Hilbert C^* - module valued distributions the corresponding extension to the multivariate case is given

Date received: June 14, 2008