BV-capacities on Wiener Spaces and Regularity of the Maximum of the Wiener Process

Abstract

We define a capacity C on abstract Wiener spaces and prove that, for any u with bounded variation, the total variation measure |Du| is absolutely continuous with respect to C: this enables us to extend the usual rules of calculus in many cases dealing with BV functions. As an application, we show that, on the classical Wiener space, the random variable sup_{0\leqt\leqT} W_t admits a measure as second derivative, whose total variation measure is singular w.r.t. the Wiener measure.