A Gaussian Kinematic Formula

In this paper, we consider smooth, real-valued random fields built up from i.i.d. copies of centered, unit variance smooth Gaussian fields on a manifold M . Specifically, we consider random fields of the form fp = F (y1(p), . . . , yk(p)) for F ∈ C(R;R) and (y1, . . . , yk) a vector of C centered, unit-variance Gaussian fields. For fields of this type, we compute the expected Euler characteristic, χ, of the excursion sets f−1[u,+∞) in terms of geometric quantities related to the Riemannian structure induced by the covariance function of any one of the yi’s and of F−1[u,+∞) ⊂ R. The motivation for studying these expectations comes from the heuristic approximation P[supp∈M f(p) ≥ u] ' E[χ(f−1[u,+∞))]. The form of this expected Euler characteristic is reminiscent of the classical Chern-Federer Kinematic Fundamental Formula (KFF), which, for two embedded submanifolds M1, M2 in R and g ∈ G = R × O(k) (the group of rigid motions on R), relates the integral of χ(M1 ∩ gM2) against a Haar measure on G to the Lipschitz-Killing curvatures ofM1 andM2. Another classical integral geometric formula which involves the Lipschitz-Killing curvatures is Weyl’s Tube Formula, where they appear in an expression for the volume of a tube of radius r around an embedded submanifold M of R. Our main results can be thought of as Gaussian analogues of the classical formulae.