On sequences of martingales with jumps on Riemannian submanifolds

TL;DR

This paper studies sequences of jump martingales on Riemannian submanifolds, showing their convergence properties and relation to harmonic maps, with implications for stochastic analysis on manifolds.

Contribution

It establishes the equivalence of semimartingale topology and local uniform convergence for discontinuous martingales on manifolds, and characterizes their limits.

Findings

Semimartingale topology equals local uniform convergence in probability.

Limits of discontinuous martingales on compact manifolds are martingales.

Results apply to harmonic maps with non-local Dirichlet forms.

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. 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.