Two-dimensional Kac-Rice formula. Application to shot noise processes excursions

TL;DR

This paper extends the Kac-Rice formula to two dimensions, providing a new way to compute the expected Euler characteristic of excursion sets for deterministic and random fields, with applications to shot noise processes.

Contribution

It introduces a 2D Kac-Rice formula for excursion sets, enabling analysis of Euler characteristics and moments for random fields like shot noise processes.

Findings

Derived a 2D Kac-Rice formula for excursion sets.

Proved continuity of the Euler characteristic integral with respect to the function.

Applied the formula to shot noise processes, establishing mean Euler characteristic and moment finiteness.

Abstract

Given a deterministic function f:R^2->R atisfying suitable assump- tions, we show that for h smooth with compact support, the integral of the Euler characteristic of the excursion set of f above some level u against a test function h corresponds to the Lebesgue integral on R^2 of a bounded quantity depending on grad(f)(x),h(f(x)),h'(f(x)) and \partial_{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 C2 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.