CRITERIA OF POSITIVITY FOR THE DENSITY OF THE LAW OF A WIENER FUNCTIONAL

We consider a function f on an abstract Wiener space, with values in R d which is regular and non-degenerate in the sense of the Malliavin ca lculus. Its law admits a density p which is C-infinity on R-d. We give criteria ensuring that p(xi) > 0 for a fixed xi in R-d. The first one is stated in terms of capacities c(r,p) on the Wiener space. The seco nd one is related to some ''restrictions'' of f to subspaces of dimens ion d of the Cameron-Martin space. Finally, we apply this criterion to obtain another one, previously given by AIDA-KUSUOKA-STROOCK, in the case where function f admits a ''skeleton''. The criteria are actually proved in the more general situation where the basic Wiener measure i s replaced by a measure with a regular density.