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

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.