SPASS - Probability, Stochastic Analysis and Statistics Seminar
# Sobolev-Malliavin regularity of the nodal volume

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Sala Seminari (Dipartimento di Matematica)
### Sala Seminari

#### Dipartimento di Matematica

Description

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 esplicit 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.)