Constructing some measures related to diffusions in Riemannian manifolds

A new approach to constructing diffusions on Riemannian manifolds based On the notion of surface measure is considered. One of motivations of this approach is to develop simple methods of obtaining Feynman-Kac type formulas for evolution equations on Riemannian manifolds and Cameron-Martin-Girsanov-Maruyama-Ramer type formulas for measures, on some spaces of functions taking values in Riemannian manifolds, generated by diffusions. Some applications to defining Onsager-Machlup functionals for such measures are also possible.