Martingales on Principal Fiber Bundles

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