Local Sample Path Properties of Gaussian Fields

Abstract

A zero-one law is derived for a class of Gaussian fields $\{X(t) : t \in R^d\}$ including the generalized multiparameter Brownian motion. Under very general conditions, the joint distribution of the suprema of several Gaussian processes defined over compact metric spaces is shown to be absolutely continuous with a bounded density. Sufficient conditions are given for the existence of proper scaling limits of $\{X(t)\}$. The results are then combined to study local oscillations and local maxima.