Integration by parts on the law of the reflecting Brownian motion |
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Authors: | Lorenzo Zambotti |
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Institution: | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy |
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Abstract: | We prove an integration by parts formula on the law of the reflecting Brownian motion 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. |
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Keywords: | 60H07 60J65 |
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