Skip to main content
SpringerLink
Log in

On the slit motion obeying chordal Komatu–Loewner equation with finite explosion time

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

This paper studies the behavior of solutions near the explosion time to the chordal Komatu–Loewner equation for slits, motivated by the preceding studies by Bauer and Friedrich (Math Z 258:241–265, 2008) and by Chen and Fukushima (Stoch Process Appl 128:545–594, 2018). The solution to this equation represents moving slits in the upper half-plane. We show that the distance between the slits and driving function converges to zero at its explosion time. We also prove a probabilistic version of this asymptotic behavior for stochastic Komatu–Loewner evolutions under some natural assumptions.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer monographs in mathematics, Springer-Verlag, London, 2001.

    Book  Google Scholar 

  2. R. O. Bauer and R. M. Friedrich, Stochastic Loewner evolution in multiply connected domains, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 579–584.

    Article  MathSciNet  Google Scholar 

  3. R. O. Bauer and R. M. Friedrich, On radial stochastic Loewner evolution in multiply connected domains, J. Funct. Anal. 237 (2006), 565–588.

    Article  MathSciNet  Google Scholar 

  4. R. O. Bauer and R. M. Friedrich, On chordal and bilateral SLE in multiply connected domains, Math. Z. 258 (2008), 241–265.

    Article  MathSciNet  Google Scholar 

  5. Z.-Q. Chen and M. Fukushima, Stochastic Komatu–Loewner evolutions and BMD domain constant, Stochastic Process. Appl. 128 (2018), 545–594.

    Article  MathSciNet  Google Scholar 

  6. Z.-Q. Chen, M. Fukushima and S. Rohde, Chordal Komatu–Loewner equation and Brownian motion with darning in multiply connected domains, Trans. Amer. Math. Soc. 368 (2016), 4065–4114.

    Article  MathSciNet  Google Scholar 

  7. Z.-Q. Chen, M. Fukushima and H. Suzuki, Stochastic Komatu–Loewner evolutions and SLEs, Stochastic Process. Appl. 127 (2017), 2068–2087.

    Article  MathSciNet  Google Scholar 

  8. J. B. Conway, Functions of One Complex Variable II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995.

    Book  Google Scholar 

  9. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland/Kodansha, 1989.

    MATH  Google Scholar 

  10. M. Katori, Bessel Processes, Schramm–Loewner Evolution, and the Dyson Model, SpringerBriefs in Mathematical Physics, vol. 11, Springer, 2015.

  11. Y. Komatu, Untersuchungen über konforme Abbildung von zweifach zusammenhängenden Gebieten, Proc. Phys. Math. Soc. Japan (3) 25 (1943), 1–42.

    MathSciNet  MATH  Google Scholar 

  12. Y. Komatu, On conformal slit mapping of multiply-connected domains, Proc. Japan Acad. 26 (1950), 26–31.

    Article  MathSciNet  Google Scholar 

  13. G. F. Lawler, Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005.

    MATH  Google Scholar 

  14. G. F. Lawler, The Laplacian-\(b\) random walk and the Schramm–Loewner evolution, Illinois J. Math. 50 (2006), 701–746.

    Article  MathSciNet  Google Scholar 

  15. G. F. Lawler, O. Schramm and W. Werner, Values of Brownian intersection exponents, I: Half-plane exponents, Acta Math. 187 (2001), 237–273.

    Article  MathSciNet  Google Scholar 

  16. T. Murayama, Chordal Komatu–Loewner equation for a family of continuously growing hulls, Stochastic Process. Appl. (2018), https://doi.org/10.1016/j.spa.2018.08.012.

    Article  MATH  Google Scholar 

  17. T. Murayama, Reformulation of Laplacian-\(b\) motion in terms of stochastic Komatu–Loewner evolution in the chordal case, preprint, available at arXiv:1902.06392 [math.PR].

  18. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

    MATH  Google Scholar 

  19. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I wish to express my gratitude to Professor Roland M. Friedrich for pointing out a lack of references in Section 1 and to the anonymous referee for his or her suggestions very helpful in making the proof transparent.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Takuya Murayama

Authors
  1. Takuya Murayama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takuya Murayama.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Murayama, T. On the slit motion obeying chordal Komatu–Loewner equation with finite explosion time. J. Evol. Equ. 20, 233–255 (2020). https://doi.org/10.1007/s00028-019-00519-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-019-00519-3

Keywords

Mathematics Subject Classification

Search

Navigation