Level Crossings for Random Fields

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = ∈ I₀: X(t) u\, where I₀ is the unit cube in Rⁿ. It is shown that the natural generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields is via the characteristic of the set A introduced by Hadwiger (1959). This characteristic is related to the number of connected components of A. A formula is obtained for the mean value of this characteristic when $n = 2, 3$. This mean value is calculated explicitly when X(t) is a homogeneous Gaussian field satisfying certain regularity conditions.