Characterization of BV functions on open domains: the Gaussian case and the general case

We provide three different characterizations of the space $BV(O,\gamma)$ of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure $\gamma$ on open domains $O$ in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition for belonging to $BV(O,\gamma)$ 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 technique to Fomin differentiable probability measures $\nu$ on a Hilbert space $X$, inferring a characterization of the space $BV(O,\nu)$ of the functions of bounded variation with respect to $\nu$ on open domains $O\subseteq X$.