On sequences of martingales with jumps on Riemannian submanifolds

Abstract

In this article, we investigate sequences of discontinuous martingales on submanifolds of higher-dimensional Euclidean space. Those sequences naturally arise when we deal with a sequence of harmonic maps with respect to non-local Dirichlet forms, such as fractional harmonic maps. We prove that the semimartingale topology is equivalent to the topology of locally uniform convergence in probability on the space of discontinuous martingales on manifolds with bounded jumps. In particular, we show that the limit of any sequence of discontinuous martingales on a compact Riemannian manifold, with respect to the topology of locally uniform convergence in probability, is a martingale on the manifold. As an application, we prove that for sequences of quasi-harmonic maps on an open set with respect to a non-local Dirichlet form, local $L^{\infty}$-convergence in the open set with pointwise convergence q.e. on the whole space implies strong local convergence in the Dirichlet space and the limit preserves harmonicity.