{'fraction_non_alphanumeric': 0.026022304832713755, 'fraction_numerical': 0.0012391573729863693, 'mean_word_length': 3.9876543209876543, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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 calculus. Its law admits a density p which is C∞ on R d . We give criteria ensuring that p(ζ) > 0 for a fixed ζ in R d . The first one is stated in terms of capacities c r,p on the Wiener space. The second one is related to some restrictions of f to subspaces of dimension 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 is replaced by a measure with a regular density.