Divergence theorems in path space III: hypoelliptic diffusions and beyond∗

Let x denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the space of paths associated to x, under the assumption that x is elliptic. In this article, this method is extended to yield similar results for degenerate diffusion processes. In particular, these results apply to non-elliptic diffusions satisfying Hörmander’s condition. ∗This paper is a modification of the one published in JFA, containing an expanded version of the original Section 3.D. 1Research partially supported by NSF grant DMS-0451194.