• Wiener space
• Malliavin calculus
• BV functions
• Probabilty

The aim of this master thesis is to study the theory of functions of bounded variation, on the classical Wiener space setting.

Our exposition relies on the fundamental results recently established by L. Ambrosio, but we assume a different point of view. Under the supervision of M. Pratelli, we focused mostly on the classical Wiener
space setting, using tools from stochastic analysis.

We worked on many explicit examples, some of them elementary, some rather complicated and possibly useful for further developments. They all helped to better visualize the general problems that we were considering. In particular, we first investigated the possibility of an explicit chain-rule for the measure-derivative of a BV function, in a simple but useful case. Then, the central role played by the
approximation with regular functions led us to strengthen some known results. Finally, we showed an extension to the BV case of the famous Clark-Ocone-Karatzas formula, which gives an explicit representation of a random variable on the classical Wiener space, in terms of a stochastic integral of its derivative.