Sample path large deviations for a class of random currents

We study long-time asymptotic behavior of the current-valued processes on compact Riemannian manifolds determined by the stochastic line integrals. Sample path large deviation estimates are proved, which induce the law of the iterated logarithm as a corollary. As their application, we give a probabilistic approach to the analysis on noncompact Abelian covering manifolds.