Dear Colleagues,

We would like to invite you to the following SPASS seminar, jointly organized by UniPi, SNS, UniFi and UniSi:

Sobolev-Malliavin regularity of the nodal volume

by Michele Stecconi (University of Luxemburg)

The seminar will take place on TUE, 21.11.2023 at 14:00 CET in Aula Seminari, Dipartimento di Matematica, UNIPI and streamed online at the link below.

The organizers,

A. Agazzi, G. Bet, A. Caraceni, F. Grotto, G. Zanco
https://sites.google.com/unipi.it/spass

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Abstract: 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 Fréchet differentiability. Moreover, we give an explicit formula for the derivative (the mean curvature).

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.
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.
(A joint work with Giovanni Peccati.)