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

TL;DR

This paper studies the convergence behavior of jump martingales on submanifolds of Euclidean spaces and explores their applications to harmonic maps related to Markov processes, including fractional harmonic maps.

Contribution

It establishes convergence results for jump martingales on Riemannian submanifolds and applies these findings to harmonic maps influenced by Markov processes.

Findings

Martingales with jumps converge as time approaches zero and infinity.

Results extend to harmonic maps with respect to various Markov processes.

Applications include fractional harmonic maps on manifolds.

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.