PROPERTIES OF THE SET OF POSITIVITY FOR THE DENSITY OF A REGULAR WIENER FUNCTIONAL

Let f be an R-d-valued Wiener functional, which is smooth and non-dege nerate in the sense of the Malliavin calculus. Let p be the density, w ith respect to the Lebesgue measure on R-d, of its law. We are interes ted in the set U = {p > 0}. We prove that U is connected. As a consequ ence, the intrinsic distance d(f) associated with f on U is a true dis tance (in particular, it is finite). We give in the end an answer to a conjecture of Malliavin about d(f). (C) Elsevier, Paris.