Integration by parts on the law of the modulus of the Brownian bridge

TL;DR

This paper establishes an infinite-dimensional integration by parts formula for the law of the modulus of the Brownian bridge, incorporating advanced white noise analysis and distribution theory to account for reflection at zero.

Contribution

It introduces a novel integration by parts formula involving Hida distributions and Wick products, specifically addressing the reflection phenomenon of the modulus of the Brownian bridge.

Findings

Derived an integration by parts formula for the modulus of the Brownian bridge.

Connected the additional distribution to the local time at zero of the process.

Utilized white noise analysis and Dirichlet form methods for the proof.

Abstract

We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(BB_t)_{0 \leq t \leq 1}$ from $0$ to $0$ in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker's delta (which correlates with the local time of $|BB|$ at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.