Invariance of 0-currents under diffusions

TL;DR

This paper investigates the invariance properties of 0-currents under stochastic flows on manifolds, providing conditions for invariance and applying results to harmonic measures related to foliations on homogeneous spaces.

Contribution

It introduces conditions for 0-current invariance under stochastic flows and connects these to harmonic measures on foliated homogeneous spaces.

Findings

Conditions for 0-current invariance under stochastic flows

Application to harmonic measures on foliated homogeneous spaces

Characterization of invariant 0-currents

Abstract

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.