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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold
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by Bruce K. Driver PDF
Trans. Amer. Math. Soc. 342 (1994), 375-395 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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