Nodal Volumes as Differentiable Functionals of Gaussian fields

2024/03/29 by Giovanni Peccati, Peccati, Giovanni, Michele Stecconi +1 · 2 citations
Earth and Planetary Sciences · #28A75 #58K05 #60G15 #60G60 #60H07 #Differential Geometry (math.DG) #FOS: Mathematics #Geophysics and Gravity Measurements #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2403.20243

openalex publication_date 2024/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C² 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.

Cited by

• Tightness of Stationary Nodal Measures
• Malliavin differentiability for the excursion measure of a Gaussian field

Related

• On the absolute continuity of random nodal volumes
• Invariant Gaussian fields on homogeneous spaces: explicit constructions and mean nodal volume
• New chaos decomposition of Gaussian nodal volumes
• Differential Topology of Gaussian Random Fields
• Tightness of Stationary Nodal Measures