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