Diffusion processes on principal bundles and differential operators on the associated bundles (Q1201606)

The authors study the lifting of diffusion processes and differential operators on a Riemannian base space \(M\) to diffusion processes on a principal bundle \(P\) and differential operators on the associated bundle. They prove that the infinitesimal generator of the lifted process can be seen as a second-order differential operator on the section space \(\Gamma\) of the associated bundle, which is just the lifted operator of the infinitesimal generator of the diffusion process on the base space. They give the covariant Feynman-Kac formula on a non-trivial principal bundle, generalizing a formula by \textit{S. Albeverio} [A covariant Feynman-Kac formula for unitary bundles over Euclidean spaces, preprint 1989]. As an application, they give a geometric proof of the GCM-theorem on the Riemannian manifolds (for an analytical proof cf. \textit{K. D. Elworthy} [Some geometric aspects of diffusion on manifolds, Probability theory and applications, Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986, Vol. 1, 501-513 (1987; Zbl 0683.58049)].