Skip to main content
SpringerLink
Log in

Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

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 (Séminaire de Probabilités de Strasbourg 25:196–219, 1991). We obtain results about the convergence of martingales with finite quadratic variations on Riemannian submanifolds of higher-dimensional Euclidean space as \(t\rightarrow \infty \) and as \(t\rightarrow 0\). Furthermore, we apply the result about martingales with jumps on submanifolds to harmonic maps with respect to Markov processes such as fractional harmonic maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Atsuji, A.: Parabolicity, the divergence theorem for \(\delta \)-subharmonic functions and applications to the Liouville theorems for harmonic maps. Tohoku Math. J. 57, 353–373 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.-Y. Calais and M. Génin, Sur les martingales locales continues indexées par \( ]0,\infty [ \), Séminaire de probabilités 17 (1983), 162.178

  3. Chen, Z.-Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.-S.: Stochastic calculus for symmetric Markov processes. Ann. Probab. 36, 931–970 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z.-Q., Fukushima, M.: Symmetric Markov processes, time change, and boundary theory. Princeton University Press, Princeton (2011)

    Book  MATH  Google Scholar 

  5. Çinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 54, 161–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen, S.: Géométrie différentielle stochastique avec sauts 1. Stoch. Int. J. Probab. Stoch. Process 56, 179–203 (1996)

    MATH  Google Scholar 

  7. Cohen, S.: Géométrie différentielle stochastique avec sauts 2: discrétisation et applications des eds avec sacutes. Stoch. Int. J. Probab. Stoch. Process 56, 205–225 (1996)

    MATH  Google Scholar 

  8. Da Lio, F., Riviére, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\) -harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Da Lio, F., Riviére, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227, 1300–1348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Darling, R.W.R.: Martingales on manifolds and geometric Itô calculus, Ph.D. Thesis, University of Warwick (1982)

  11. Darling, R.W.R.: Convergence of martingales on a Riemannian Manifold. Publ. Res. Inst. Math. Sci. 19(2), 753–763 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Émery, M.: Stochastic caluculus in manifolds. Universitext, Springer-Verlag, Berlin (1989)

    Book  Google Scholar 

  13. Émery, M.: Convergence des martingales dans les variétś. Colloque en l’honneur de Laurent Schwartz 2(132), 47–63 (1985)

  14. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric markov processes, 2nd ed., de Gruyter Stud. Math., vol. 19, Walter de Gruyter & Co., Berlin, (2010)

  15. Fukushima, M.: A decomposition of additive functionals of finite energy. Nagoya Math. J. 74, 137–168 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. He, S.-W., Yan, J.-A., Zheng, W.-A.: Sur la convergence des semimartingales continues dans \({\mathbb{R} }^{n}\) et des martingales dans une variétés. Séminaire de probabilités 17, 179–184 (1983)

    Google Scholar 

  17. He, S.-W., Zheng, W.-A.: Remarques sur la convergence des martingales dans les variétés. Séminaire de probabilités 18, 174–178 (1984)

    MATH  Google Scholar 

  18. Hsu, E.P.: Stochastic analysis on manifolds, graduate studies in mathematics, vol. 38, American Mathematical Society, (2002)

  19. Kendall, W.S.: Martingales on manifolds and harmonic maps. Contemp. Math. 73, 121–157 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kuwae, K.: Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38, 1532–1569 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maillard-Teyssier, L.: Stochastic Covariant Calculus with Jumps and Stochastic Calculus with Covariant Jumps, Séminaire de Probabilités XXXIX, In: Memoriam Paul-André Meyer. Lecture notes in math. 1874, 381–417 (2006)

  22. Meyer, P.-A., Le théoréme de convergence des martingales dans les variétés riemanniennes d’aprés R.W. Darling et W.A. Zheng. Séminaire de probabilités de Strasbourg textbf17 (1983), 187–193

  23. Nakao, S.: Stochastic calculus for continuous additive functionals of zero energy. Z. Wahrsch. Verw. Gebiete 68, 557–578 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Picard, J.: Calcul stochastique avec sauts sur une variété. Séminaire de Probabilités de Strasbourg 25, 196–219 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Picard, J.: Barycentres et martingales sur une variété. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 30(4), 647–702 (1994)

  26. Picard, J.: Smoothness of harmonic maps for hypoelliptic diffusions. Ann. Probab. 28(2), 643–666 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Picard, J.: The manifold-valued dirichlet problem for symmetric diffusions. Potential Anal. 14, 53–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Protter, P.: Stochastic integration and differential equations. In: Stochastic modelling and applied probability, Springer, Berlin (2005)

  29. Sharpe, M.J.: Local times and singularities of continuous local martingales. Séminaire de probabilités 14, 76–101 (1980)

    MathSciNet  MATH  Google Scholar 

  30. Silverstein, M.L.: The reflected Dirichlet space. Ill. J. Math. 18, 310–355 (1974)

    MathSciNet  MATH  Google Scholar 

  31. Walsh, J.B.: A property of conformal martingales. Séminaire de probabilités 581, 490–492 (1977)

Download references

Acknowledgements

The author is grateful to Professor Hariya, his supervisor, for his careful reading of the previous version 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.

Author information

Authors and Affiliations

  1. Graduate School of Science, Mathematical Institute, Tohoku University, Sendai, Japan

    Fumiya Okazaki

Authors
  1. Fumiya Okazaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fumiya Okazaki.

Ethics declarations

Conflict 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

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Okazaki, F. Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps. J Theor Probab (2023). https://doi.org/10.1007/s10959-023-01273-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10959-023-01273-6

Keywords

Mathematics Subject Classification (2020)

Search

Navigation