A Gaussian kinematic formula (Q2493176)

Given a \(C^2\), centered, unit-variance Gaussian process \(y\) on a manifold \(M\) and \(F \in C^2(R^k;R)\). The author considers the field \(f_p = F(y_1(p),\dots,y_k(p))\) where the fields \(y_i\) are iid copies of \(y\). He studies the expected Euler characteristic \(\chi\) of the excursions \(f^{-1}[u,\infty) \subset M\) motivated by the approximation \(P(\sup_{p \in M} f(p) \geq u)\simeq E\chi (f^{-1}[u,\infty))\). The main result, Theorem 4.1, is a Gaussian analogue of the Chern-Federer kinematic fundamental formula and states a formal series expansion for the RHS.