Dipartimento di Matematica, Sala Seminari.<\/p>\n
Consider the nodal volume of a non-degenerate (in a sense to specify) 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 Fr\u00e9chet differentiability. Moreover, we give an esplicit formula for the derivative (the mean curvature).<\/p>\n
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. We prove that the nodal volume admits a L2 Malliavin derivative, for d>2 and that in the case d=2, this is false, but the Malliavin derivative still exists in L1.<\/p>\n
A fundamental ingredient is to understand the Sobolev regularity of the function f(t) that expresses the volume of the level t of a “typical” Morse function.<\/p>\n
(A joint work with Giovanni Peccati.)<\/p>\n
Further information is available on the event page<\/a> on the Indico platform.<\/p>\n","protected":false},"excerpt":{"rendered":" Consider the nodal volume of a non-degenerate (in a sense to specify) 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…<\/p>\n