ON THE ABSOLUTE CONTINUITY OF RANDOM NODAL VOLUMES

• 成果类型: Article
• 署名作者: Angst, Jurgen; Poly, Guillaume
• 署名单位: Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1418
• 发表日期: 2020
• 页码: 2145-2175
• 关键词: length

摘要

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, nondegencrate and stationary Gaussian field (f (x), x is an element of R-d). Under mild conditions, we prove that in dimension d >= 3, 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, del f, 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.