October 1997
·
2 Reads
Science in China Series A Mathematics
A Brownian motion {x t }t⩾0 on a compact Riemannian manifold M with a drift vector field X can be lifted to a diffusion process on M × Tk corresponding to an ℝk valued smooth differential one-form A on M. The circulations (rotation numbers) of the lifted process around the k circles of Tk are studied. By choosing a certain ℝk -valued differential one-form A, these circulations give the hidden circulation of {x t }t⩾0 in M and the rotation numbers of {x t }t⩾0 around some closed curves in M which generalize the first homology group H1(M,ℤ) of M.