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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London (2001)
Arnaudon, M., Li, X.-M., Thalmaier, A.: Manifold-valued martingales, changes of probabilities and smoothness of finely harmonic maps. Ann. Inst. H. Poincaré Probab. Statist. 35, 765–791 (1999)
Bass, R.F.: Probabilistic Techniques in Analysis, Probability and Its Applications. Springer, New York (1995)
Bethuel, F.: On the singular set of stationary harmonic maps. Manuscripta Math. 78, 417–443 (1993)
Chen, Z.-Q.: On notions of harmonicity. Proc. Am. Math Soc. 137, 3497–3510 (2009)
Chen, Z.Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.S.: Stochastic calculus for symmetric Markov processes. Ann. Probab. 36, 931–970 (2008)
Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton (2011)
Deny, J.: Un théorème sur les ensembles effilés. Ann. Univ. Grenoble Sect. Sci. Math. Phys. 23, 139–142 (1948)
Francesca, D.L., Riviere, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\)-harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)
Francesca, D.L., Riviere, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227, 1300–1348 (2011)
Evans, L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116, 101–113 (1991)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2nd ed., de Gruyter Stud. Math., Walter de Gruyter & Co., Berlin 19 (2010)
Fukushima, M.: A decomposition of additive functionals of finite energy. Nagoya Math. J. 74, 137–168 (1979)
Hong, M.C., Wang, C.: On the singular set of stable stationary harmonic maps. Calc. Var. Partial Differ. Equ. 9, 141–156 (1999)
Kuwae, K.: Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38(4), 1532–1569 (2010)
Millot, V., Pegon, M., Schikorra, A.: Partial regularity for fractional harmonic maps into spheres. Arch. Ration. Mech. Anal. 242, 747–825 (2021)
Millot, V., Sire, Y.: On a fractional Ginzburg-Landau equation and \(\frac{1}{2}\)-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215, 125–210 (2015)
Nakao, N.: Stochastic calculus for continuous additive functionals of zero energy. Z. Wahrsch. Verw. Gebiete 68, 557–578 (1985)
Okazaki, F.: Convergence of martingales with jumps on submanifolds of Euclidean spaces and its applications to harmonic maps, published online in Journal of Theoretical Probability (2023)
Picard, J.: Calcul stochastique avec sauts sur une variété. Séminaire de Probabilités de Strasbourg 25, 196–219 (1991)
Picard, J.: Barycentres et martingales sur une variété. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 30(4), 647–702 (1994)
Picard, J.: Smoothness of harmonic maps for hypoelliptic diffusions. Ann. Probab. 28(2), 643–666 (2000)
Picard, J.: The manifold-valued Dirichlet problem for symmetric diffusions. Potential Anal. 14, 53–72 (2001)
Riviére, T.: Everywhere discontinuous harmonic maps into spheres. Acta. Math. 175, 197–226 (1995)
Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Diff. Geom. 17, 307–335 (1982)
Schoen, R., Uhlenbeck, K.: Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78, 89–100 (1984)
Thalmaier, A.: Brownian motion and the formation of singularities in the heat flow for harmonic maps. Probab. Theory Relat. Fields 105, 335–367 (1996)
Thalmaier, A.: Martingales on Riemannian manifolds and the nonlinear heat equation. In: Davies, I. M., Truman, A., Elworthy, K. D. (eds.) Stochastic Analysis and Applications, Gregynog, 1995. Proc. of the Fifth Gregynog Symposium. Singapore: World Scientific Press, pp. 429–440 (1996)
Walsh, A.: Extended Itô calculus for symmetric Markov processes. Bernoulli 18(4), 1150–1171 (2012)
Acknowledgements
The author would like to express his gratitude to Professor Hariya, his supervisor, for his careful reading of the manuscript and for helpful comments. The author also thanks a referee for his/her constructive comments and corrections which have led to significant improvement of this paper.
Funding
This work was supported by JSPS KAKENHI Grant Number 22J11051.
Author information
Authors and Affiliations
Contributions
Okazaki wrote the main manuscript text.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Conflicts of interest
No conflict is related to this article, and the author has no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Okazaki, F. Probabilistic Characterization of Weakly Harmonic Maps with Respect to Non-Local Dirichlet Forms. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10129-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-024-10129-5