Some applications for area, coarea and Rice’s formulas: Real roots of random multidimensional polynomials, chaos expansion and central limit theorem.

Abstract:

The now classical change of variable formula for non bijective functions $ F : {\Bbb R}^d \to {\Bbb R}^j$ with $d\geq j$, which are well known as area and coarea formulas after the work of Federer, are the basis for obtaining the Rice’s formulas for the expectation and the order superior moments of the number of roots or of the volume of the level set for a Gaussian random field. In this talk, we use these Rice’s formulas for the first and second moment of the level functionals and also an expansion into the chaos that we can get using the area and coarea formulas, to obtain the asymptotic variance and a central limit theorem for multidimensional level functionals. This result is then applied to several multidimensional polynomial models and random fields.