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

TL;DR

This paper introduces a capacity on Wiener spaces, extends calculus rules for BV functions, and analyzes the second derivative measure of the Wiener process maximum, revealing its singularity.

Contribution

It defines a new capacity on Wiener spaces and demonstrates its use in extending calculus rules for BV functions and analyzing the second derivative of the Wiener process maximum.

Findings

Total variation measure |Du| is absolutely continuous w.r.t. capacity C.

The maximum of the Wiener process admits a second derivative measure.

The second derivative measure of the Wiener maximum is singular w.r.t. Wiener measure.

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.