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 X =X + h with h smooth deterministic function and >0 and we differentiate the law of X at = 0. This infinitesimal perturbation changes drastically the set of zeros of X for any >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. © 2004 Elsevier Inc. All rights reserved.