Some Central Limit Theorems for Markov Paths and Some Properties of Gaussian Random Fields

摘要

The Authors' primary aim is to build versions of generalised Gaussian processes from simple, elementary components in such a way that as many as possible of the esoteric properties of these elusive objects become intuitive. For generalised Gaussian processes, or fields, indexed by smooth functions or measures on IR sub d, our building blocks will be simple Markov processes whose state space is IR sub d. Roughly speaking, by summing functions of the local times of the Markov processes the Gaussian field shall be obtained via a central limit theorem type of result. This central limit result, together with related results indicating how additive functionals of the Markov processes generate additive functionals of the fields, yield considerable insight into properties of generalised Gaussian processes such as Markovianess, self-similarity, locality, of functionals, etc. Although the paper is comprised primarily of new results, and despite the fact that the subject matter is somewhat esoteric, our aims are primarily didactic and expository - we want to try to initiate the uninitiated into some of the mysteries of generalised processes via an easily understood model.