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Probabilistic characterization of weakly harmonic maps with respect to non-local Dirichlet forms

Fumiya Okazaki

Abstract

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.

Probabilistic characterization of weakly harmonic maps with respect to non-local Dirichlet forms

Abstract

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.
Paper Structure (4 sections, 10 theorems, 137 equations)

This paper contains 4 sections, 10 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Markov processes
  3. Proof of Theorem \ref{['harmonicmartingale']}
  4. Fine continuity of weakly harmonic maps

Key Result

Theorem 1.1

Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^d$. We assume that a Borel measurable map $u:E\to M$ is in $\mathcal{F}_{loc}^D(M)$ and quasi-continuous on $D$. Then $u$ is weakly harmonic on $D$ if and only if $u$ is quasi-harmonic on $D$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 3.1
  • ...and 28 more