Let $N$ denote a manifold equipped with a finite Borel measure $\gamma$. A vector field $Z$ on $N$ is said to be admissible with respect to $\gamma$ if $Z$ admits an integration by parts formula. The measure $\gamma$ is said to be quasi-invariant under $Z$ if the class of null sets of $\gamma$ is preserved by the flow generated by $Z$. In this talk we study the law $\gamma$ of an elliptic diffusion process with values in a closed compact manifold. We construct a class of admissible vector fields for $\gamma$, show that $\gamma$ is quasi-invariant under these vector fields, and give a formula for the associated family of Radon-Nikodym derivatives $d\gamma_s\over d\gamma$.