Circulation of lifted processes on trivial principal bundles

摘要

A Brownian motion|xt|t≥0 on a compact Riemannian manifold M with a drift vector field X can be lifted to a diffusion process |xt|t≥0 on M×Tk corresponding to an Pk valued smooth differential one-form A on M.The circulations (rotation numbers) of the lifted process |xt|t≥0 around the k circles of Tk are studied.By choosing a certain Pk -valued differential one-form A,these circulations give the hidden circulation of |xt|t≥0 in M and the rotation numbers of i|xt|t≥0 around some closed curves in M which generalize the first homology group H1(M,Z) of M.