The lie bracket of adapted vector fields on Wiener spaces

Let W(M) be the based (at o is an element of M) path space of a compact Riemannian manifold M equipped with Wiener measure nu. This paper is devoted to considering vector fields on W(M) of the form X-s(h)(sigma) = P-s(sigma)h(s)(sigma) where P-s(sigma) denotes stochastic parallel translation up to time s along a Wiener path sigma is an element of W(M) and {h(s)}(s is an element of[0,1]) is an adapted ToM-valued process on W(M). It is shown that there is a large class of processes it (called adapted vector fields) for which we may view X-h as first-order differential operators acting on functions on W(M). Moreover, if h and k are two such processes, then the commutator of X-h with X-k is again a vector field on W(M) of the same form.