Properties of the set of positivity for the density of a regular Wiener functional
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.
scientific article; zbMATH DE number 1138708