Open Access
September 2020 On the absolute continuity of random nodal volumes
Jürgen Angst, Guillaume Poly
Ann. Probab. 48(5): 2145-2175 (September 2020). DOI: 10.1214/19-AOP1418


We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, nondegenerate and stationary Gaussian field $(f(x),{x\in\mathbb{R}^{d}})$. Under mild conditions, we prove that in dimension $d\geq3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable bearing in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac–Rice type formulas allowing one to express the volume of the set $\{f=0\}$ as integrals of explicit functionals of $(f,\nabla f,\operatorname{Hess}(f))$ and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau–Hirsch criterion then gives conditions ensuring the absolute continuity.


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Jürgen Angst. Guillaume Poly. "On the absolute continuity of random nodal volumes." Ann. Probab. 48 (5) 2145 - 2175, September 2020.


Received: 1 November 2018; Revised: 1 September 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152638
Digital Object Identifier: 10.1214/19-AOP1418

Primary: 26C10 , 60H07
Secondary: 30C15 , 42A05

Keywords: Absolute continuity , Kac–Rice formula , Nodal volume

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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