Level Crossings for Random Fields

Abstract

For an $n$-dimensional random field $X(\mathbf{t})$ we define the excursion set $A$ of $X(\mathbf{t})$ by $A = \{\mathbf{t} \in \mathbf{I}_0: X(\mathbf{t}) \geqq u\}$, where $I_0$ is the unit cube in $R^n.$ 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 $X(\mathbf{t})$ is a homogeneous Gaussian field satisfying certain regularity conditions.