BV functions on open domains: The wiener case and a fomin differentiable case

We provide three different characterizations of the space BV (O, γ) of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure γ on open domains O in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to BV (O, γ) by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures ν on a Hilbert space X, and we infer a characterization of the space BV (O, ν) of the functions of bounded variation with respect to ν on open domains O ⊆ X.