We study the long time behavior of stochastic currents associated to
diffusion processes on compact Riemannian manifolds. In the first part
of the talk, sharp results about existence and tightness of stochastic
currents will be discussed.
In the second part, some problems related to random homologies
(homology class associated to the paths of diffusion processes) will
be addressed. In particular, we give a full geometric characterization
of manifolds such that the associated random homology has a gaussian
asymptotic. Some simpler related problems (Gallavotti-Cohen symmetry,
relation with the Riemannian metric).