Characterization of Brownian motion on manifolds through integration by parts

Inspired by Stein’s theory in mathematical statistics, we show that the Wiener measure on the pinned path space over a compact Riemannian manifold is uniquely characterized by its integration by parts formula among the set of probability measures on the path space for which the coordinate process is a semimartingale. Because of the presence of the curvature, the usual proof will not be readily extended to this infinite dimensional setting. Instead, we show that the integration by parts formula implies that the stochastic anti-development of the coordinate process satisfies Lévy’s criterion.