Integration by parts on the law of the reflecting Brownian motion

We prove an integration by parts formula on the law of the reflecting Brownian motion X≔|B| in the positive half line, where B is a standard Brownian motion. In other terms, we consider a perturbation of X of the form Xε=X+εh with h smooth deterministic function and ε>0 and we differentiate the law of Xε at ε=0. This infinitesimal perturbation changes drastically the set of zeros of X for any ε>0. As a consequence, the formula we obtain contains an infinite-dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squared derivative of B and in terms of the local time of X at 0. We also compute the divergence on the Wiener space of a class of vector fields not taking values in the Cameron-Martin space.