Convergence of martingales with jumps on submanifolds of Euclidean spaces and its applications to harmonic maps

Abstract

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.