Special It\^o maps and an $L^2$ Hodge theory for one forms on path spaces

In `Stochastic processes, physics and geometry: new interplays, I' (Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2000 We prove a Kodaira-Hodge decomposition on differential 1-forms on the space of non-smooth paths over a Riemannian manifold, allowing us to define the corresponding first cohomology group. This uses the It\^o map of a Brownian system and damped stochastic parallel translation.