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Abstract
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Our talk consists of two parts. In the first half we talk about a generalization to Malliavin's lifting approach. Simply speaking, given a vector field $X$ on $\mathbb{R}^d$, Malliavin's lifting approach produces a Cameron-Martin process $\tilde{X}$ on Wiener space $W$ and there is an integration by parts formula associated to $\tilde{X}$ under certain conditions. This technique is a crucial tool in Malliavin calculus. We generalize this lifting approach as follows: Firstly, we consider vector field $X$ on a manifold $M$ and try to lift it onto the curved Wiener space $W(M)$. Secondly, other than the usual $H^1$ metric used in the Cameron-Martin space, we lift $X$ according to a Ricci-damped metric. In the second half we talk about a finite dimensional approximation to a path integral representation of heat kernel on some symmetric spaces where the generalization of Malliavin's approach plays an important role.
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