Distribution results for the occupation measures of continuous Gaussian fields

For a d-dimensional random field X(t) define the occupation measure corresponding to the level α by the Lebesgue measure of that portion of the unit cube over which X(t)⩾α. Denoting this by M[X, α], it is shown that for sample continuous Gaussian fields P{M[X,α]>β}=exp{−12α2kβ(1+0(1))}as α→∞, for a particular functional kβ. This result is applied to a variety of fields related to the planar Brownian motion, and for each such field we obtain bounds for kβ.