  We study two actions of a stochastic flow $\varphi_t$ on the space of
$0-$currents $T$ of a differentiable manifold $M$. In particular, we give
conditions on a current $T$ to be invariant under these actions. Also, we apply
our results to the case of $0-$currents that come from harmonic measures
associated to certain type of foliations on homogeneous spaces.
