Gaussian processes, kinematic formulae and Poincar\'e's limit

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz--Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper. Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the $n$-sphere. The $n\to\infty$ limit, which gives the final result, is handled via recent extensions of the classic Poincar\'e limit theorem.