Local properties of some Gaussian processes

The purpose of this paper is twofold. First we obtain exact formulae for the Hausdorff dimensions of the level sets and graph, and of the image of a fixed time set, for a Gaussian process with stationary increments and monotone incremental variance. Inequalities for these dimensions have been obtained by Kahane in the case where the process is the sum of a trigonometric series with random coefficients. Secondly we obtain some precise results for the brownian motion process. We consider when an image set has positive Lebesgue measure and when the zero set has positive capacity with respect to a given kernel. We show that certain conditions, which Kahane has shown to be sufficient to ensure these properties, are also necessary.