  We characterize weakly harmonic maps with respect to non-local Dirichlet
forms by Markov processes and martingales. In particular, we can obtain
discontinuous martingales on Riemannian manifolds from the image of symmetric
stable processes under fractional harmonic maps in a weak sense. Based on this
characterization, we also consider the continuity of weakly harmonic maps along
the paths of Markov processes and describe the condition for the continuity of
harmonic maps by quadratic variations of martingales in some situations
containing cases of energy minimizing maps.
