Weak differentiability of Wiener functionals and occupation times

TL;DR

This paper characterizes the non-martingale parts of weakly differentiable Wiener functionals using variational methods, linking them to occupation times and path-dependent transformations of Brownian motion.

Contribution

It introduces a universal variational framework for Wiener functionals, connecting weak differentiability, Dirichlet processes, and occupation time-based limits.

Findings

Dirichlet processes are differential forms with respect to Brownian motion.

Drift components are characterized via limits of integral functionals and occupation times.

Connections established between weak differentiability and local time integrals under regularity conditions.

Abstract

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Le\~ao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite $p$-variation ($p\ge 2$) regularity is also discussed. Under stronger regularity conditions in the sense of finite $(p,q)$-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.