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$ BV $ functions on open domains: the Wiener case and a Fomin differentiable case

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  • 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 in order to belong 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 techniques to Fomin differentiable probability measures $ \nu $ on a Hilbert space $ X $, and we infer a characterization of the space $ BV(O, \nu) $ of the functions of bounded variation with respect to $ \nu $ on open domains $ O\subseteq X $.

    Mathematics Subject Classification: Primary: 49Q20, 58E99, 28C20; Secondary: 26B30, 60H07.


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