Invariance of 0-currents under diffusions

Abstract

We study two actions of a stochastic flow ${\phi }_t$ 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.