Fourier Transforms of Distributions of Homogeneous Random Fields with Independent Increments and Complex Markov--Maslov Chains

Abstract

We prove formulas that describe, under certain sufficiently general assumptions, the Fourier transforms of distributions of processes and fields mentioned in the title. A similar method applied to complex Markov chains on a finite segment can be used to compute the Fourier transform of the complex Poisson measure from [1, Chap. IX] (where it is found by series expansion of the measure) given its finite-dimensional distributions. We assume that the value space of a random field is a Polish (i.e., complete, separable, and metrizable) locally convex space; however, the results seem to be new for real processes as well. An additional assumption on the processes is that their trajectories have no discontinuities of the second kind in the (weakened) topology of the value space; in particular, such are the Poisson jump processes. Earlier [2] such a formula was proved only for processes with strongly continuous trajectories. Also discussed in [2] are applications of a similar construction for continuous fields to the particular case of Wiener–Levy–Chentsov fields [3].