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A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold


Author: Bruce K. Driver
Journal: Trans. Amer. Math. Soc. 342 (1994), 375-395
MSC: Primary 60J65; Secondary 58G32
DOI: https://doi.org/10.1090/S0002-9947-1994-1154540-2
MathSciNet review: 1154540
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Abstract: The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let $ h:[0,1] \to {T_0}M$ be a $ {C^1}$ function where M is a compact Riemannian manifold, $ o \in M$ is a base point, and $ {T_o}M$ is the tangent space to M at $ o \in M$. Let $ W(M)$ be the space of continuous paths from [0,1] into M, $ \nu $ be Wiener measure on $ W(M)$ concentrated on paths starting at $ o \in M$, and $ {H_s}(\omega )$ denote the stochastic-parallel translation operator along a path $ \omega \in W(M)$ up to "time" s. (Note: $ {H_s}(\omega )$ is only well defined up to $ \nu $-equivalence.) For $ \omega \in W(M)$ let $ {X^h}(\omega )$ denote the vector field along $ \omega $ given by $ X_s^h(\omega ) \equiv {H_s}(\omega )h(s)$ for each $ s \in [0,1]$. One should interpret $ {X^h}$ as a vector field on $ W(M)$. The vector field $ {X^h}$ induces a flow $ {S^h}(t, \bullet ):W(M) \to W(M)$ which leaves Wiener measure $ (\nu )$ quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if $ h(1) = 0$ and the Wiener measure $ (\nu )$ is replaced by a pinned Wiener measure $ ({\nu _e})$. (The measure $ {\nu _e}$ is proportional to the measure $ \nu $ conditioned on the set of paths which start at $ o \in M$ and end at a fixed end point $ e \in M$.) Also as in [13], one gets an integration by parts formula for the vector-fields $ {X^h}$ defined above.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1154540-2
Article copyright: © Copyright 1994 American Mathematical Society

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