An analogue of Kac-Rice formula for Euler characteristic

Given a deterministic function f : R 2 → R satisfying suitable assumptions , we show that for h smooth with compact support, R χ({f u})h(u)du = R 2 γ(x, f, h)dx, where χ({f u}) is the Euler characteristic of the excursion set of f above the level u, and γ(x, f, h) is a bounded function depending on ∇f (x), h(f (x)), h ′ (f (x)) and ∂ ii f (x), i = 1, 2. This formula can be seen as a 2-dimensional analogue of Kac-Rice formula. It yields in particular that the left hand member is continuous in the argument f , for an appropriate norm on the space of C 2 functions. If f is a random field, the expectation can be passed under integrals in this identity under minimal requirements, not involving any density assumptions on the marginals of f or his derivatives. We apply these results to give a weak expression of the mean Euler characteristic of a shot noise process, and the finiteness of its moments