Nodal volumes as differentiable functionals of Gaussian fields

Peccati, Giovanni; Stecconi, Michele

doi:10.1090/tran/9440

Nodal volumes as differentiable functionals of Gaussian fields
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by Giovanni Peccati and Michele Stecconi;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9440

Published electronically: May 1, 2025

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Abstract:

We characterize the absolute continuity of the law and the Malliavin-Sobolev regularity of random nodal volumes associated with smooth Gaussian fields on generic $\mathcal {C}^2$ manifolds with arbitrary dimension. Our results extend and generalize the seminal contribution by Angst and Poly (2020) about stationary fields on Euclidean spaces and cover, in particular, the case of two-dimensional manifolds, possibly with boundary and corners. The main tools exploited in the proofs include the use of Gaussian measures on Banach spaces, Morse theory, and the characterization of Malliavin-Sobolev spaces in terms of ray absolute continuity. Several examples are analyzed in detail.

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Nodal volumes as differentiable functionals of Gaussian fields
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Abstract: