Martingales on Principal Fiber Bundles

Let $P(M,G)$ be a principal fiber bundle, $\omega$ be a connection form on $P(M,G)$ and $\nabla^{P}$ be a projectable connection on $P(M,G)$. The aim of this work is determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.