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

Davide Addona; Giorgio Menegatti; Michele Miranda jr.

doi:10.3934/cpaa.2020117

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2020, Volume 19, Issue 5: 2679-2711. Doi: 10.3934/cpaa.2020117

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

1.
Department of Mathematics and applications, University of Milano Bicocca, via Cozzi 55, 20125 Milano, Italy
2.
Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 30, 44121 Ferrara, Italy

^* Corresponding author
^* Corresponding author

Received: February 2019

Revised: October 2019

Published: March 2020

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Abstract

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 $.

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

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