Properties of the set of positivity for the density of a regular Wiener functional
type: Item
instance of: scholarly article
title: Properties of the set of positivity for the density of a regular Wiener functional
zbMATH Open document ID: 0897.60060
DOI: 10.1016/S0007-4497(98)80074-2
author: Francis Hirsch
author: Shiqi Song
published in: Bulletin des Sciences Mathématiques
publication date: 1998-10-11T00:00:00Z
review text: Consider an \(\mathbb{R}^n\)-valued Wiener functional, which is in \(\mathbb{D}^\infty\) and nondegenerate, denote by \(p_f\) its density on \(\mathbb{R}^n\), and set \(U_f= \{p_f>0\}\). The authors first show that \(\{f\notin U_f\}\) is slim, and deduce that \(U_f\) is connected, thereby improving a result of Fang. Then they show that the intrinsic distance \(d_f\) on \(U_f^2\) induces the Euclidean topology on \(U_f\), and they finally built a positive function \(f\) (on the Gaussian space \((\mathbb{R}^2, \gamma))\) such that \(d_f(0,1) <\infty\) (and \(\lim_0 p_f=0)\), thereby negatively answering a conjecture of Malliavin.
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1138708
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: regular Wiener functional
zbMATH Keywords: set of positivity for the density
zbMATH Keywords: intrinsic distance
reviewed by: Jacques Franchi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1138708