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

Abstract

We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(B{B}_t{)}_{0\le t\le 1}$ 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.