Sobolev-Malliavin regularity of the nodal length.
Consider the nodal volume of a Gaussian random field defined on a compact Riemannian manifold of dimension d greater or equal to 2. We prove that the law of such random variable has an absolutely continuous component, as a direct consequence of its Fréchet differentiability. Moreover, we give an explicit formula for the derivative (the mean curvature) and precise conditions under which the nodal volume admits a Malliavin derivative. The non-singularity of the law had already been established by Angst and Poly for stationary fields on the d-torus, in dimension d>2, via Malliavin calculus. In this work the two dimensional case remained open, in particular, the Malliavin differentiability of the nodal length was unknown. In this talk, I will focus on the d=2 case. A fundamental ingredient is a deterministic study of the function f(t) that expresses the (d-1)-volume of the level t of a Morse function. (A joint work with Giovanni Peccati.)
Area: IS8 - Random Fields (Maurizia Rossi)
Keywords: Gaussian, random field, manifold, nodal length, Malliavin