Some Local Properties of Gaussian Vector Fields

Abstract

Formulas for the Hausdorff dimension of the graph, image, and level sets of Gaussian vector fields are given under general conditions which allow for different local behavior of the components and for dependence among them. Conditions for the field to have a local time, to hit any fixed point, and for the image to have positive Lebesgue measure are given, and relations between these properties are discussed. Applications of the results are given and include a discussion of when differentiable planar fields have critical points at fixed levels.