Integration by parts on the law of the reflecting Brownian motion

Details Integration by parts on the law of the reflecting Brownian motion Integration by parts on the law of the reflecting Brownian motion

2004-04-27

arxiv.org/abs/math/0404489v1

Mathematics

Probability

32 pages

Scientific paper

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 $X^\epsilon = X+\epsilon h$ with $h$ smooth deterministic function and $\epsilon>0$ and we differentiate the law of $X^\epsilon$ at $\epsilon=0$. This infinitesimal perturbation changes drastically the set of zeros of $X$ for any $\epsilon>0$. 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.

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