In Section 7.4 we develop and exploit some special properties of Gaussian processes that are associated with Borel right processes. In this chapter we consider the question of characterizing associated Gaussian processes. In order to present these results in their proper generality, we must leave the familiar framework of Borel right processes and consider local Borel right process, which are introduced in the final sections of Chapter 4. The reader should note that this is the first place in this book, after Chapter 4, that we mention local Borel right processes. We remind the reader that Borel right processes are local Borel right processes, and for compact state spaces, there is no difference between local Borel right processes and Borel right processes.

Let S be a locally compact space with a countable base. A Gaussian process {Gx ; x ∈ S} is said to be associated with a strongly symmetric transient local Borel right process X on S, with reference measure m, if the covariance Γ = Γ(x, y) = E(GxGy) is the 0-potential density of X for all x, y ∈ S. Not all Gaussian processes are associated. It is remarkable that some very elementary observations about the 0-potential density of a strongly symmetric transient Borel right process show what is special about associated Gaussian processes.

One obvious condition is that Γ(x, y) ≥ 0 for all x, y ∈ S, since the 0-potential density of a strongly symmetric transient Borel right process is nonnegative (see Remark (3.3.5)).