A Gaussian kinematic formula

Abstract

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are 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∈C2(ℝk;ℝ) and (y1,…,yk) a vector of C2 i.i.d. centered, unit-variance Gaussian fields. The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest. As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets M∩f−1[u,+∞)=M∩y−1(F−1[u,+∞)) of the field f. The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.