  Martingales with jumps on Riemannian manifolds and harmonic maps with respect
to Markov processes are discussed in this paper. Discontinuous martingales on
manifolds were introduced in Picard (1991). We obtain results about the
convergence of martingales with finite quadratic variations on Riemannian
submanifolds of higher dimensional Euclidean space as $t\to \infty$ and $t\to
0$. Furthermore we apply the result about martingales with jumps on
submanifolds to harmonic maps with respect to Markov processes such as
fractional harmonic maps.
