RANDOM FUNCTIONS OF POISSON TYPE

Let (X, {A figure is presented}, μ) be a σ-finite nonatomic measure space. We think of the customary analysis based upon (X, {A figure is presented}, μ) as continuum analysis. By contrast discrete analysis is based upon an arbitrary countable subset of X, rather than upon X itself, and all countable subsets are treated alike with a Poisson process used to distinguish among them probabilisticly. The sort of functions appropriate for discrete analysis are the Campbell functions, or, as they are called in the present paper, the random functions of Poisson type. The paper presents an account of the ideas underlying discrete analysis and treats briefly the specifies of representation, stochastic integrals, and duality theory for random functions of Poisson type. It is chiefly concerned, however, with those random functions which occur in connection with the discrete analysis of Brownian motion, (for example, with Gaussian noise). In particular it shows that there is a completely positive map which carries such discrete processes onto an algebraic version of Wiener's Brownian motion process, and that under this map, random functions of Poisson type go over to the appropriate random functions of Wiener type. It also shows that the map carries random variables into noncommuting operators characteristic of quantum theory. © 1979.