Stochastic Characterization of Harmonic Sections and a Liouville Theorem

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G, P)$ be an associate fiber bundle. Our interest is to study harmonic sections of the projection $\pi_E$ of $E$ into $M$. Our first purpose is to give a stochastic characterization of harmonic section from $M$ into $E$ and a geometric characterization of harmonic sections with respect to its equivariant lift. The second purpose is to show a version of Liouville theorem for harmonic sections and to prove that section $M$ into $E$ is a harmonic section if and only if it is parallel.