A Gaussian kinematic formula

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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 P[ₚ∈ ₘf(p)≥ u][χ(f⁻¹[u,+∞))].