Journal of Statistical Physics

, Volume 139, Issue 1, pp 108–121 | Cite as

Stationarity of SLE

  • Antti KemppainenEmail author


A new method to study a stopped hull of SLE κ (ρ) is presented. In this approach, the law of the conformal map associated to the hull is invariant under a SLE induced flow. The full trace of a chordal SLE κ can be studied using this approach. Some example calculations are presented.


Schramm-Loewner evolutions Stationarity Reversibility Random curves 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer, M., Bernard, D., Houdayer, J.: Dipolar stochastic Loewner evolutions. J. Stat. Mech. Theory Exp. 2005(03), P03,001 (2005) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Durrett, R.: Stochastic Calculus. Probability and Stochastics Series. CRC, Boca Raton (1996) zbMATHGoogle Scholar
  3. 3.
    Kytölä, K.: Conformal field theory methods for variants of Schramm-Loewner evolutions. Doctoral dissertation, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics (2006) Google Scholar
  4. 4.
    Kytölä, K., Kemppainen, A.: SLE local martingales, reversibility and duality. J. Phys. A 39(46), L657–L666 (2006) zbMATHCrossRefADSGoogle Scholar
  5. 5.
    Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187(2), 237–273 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005) (electronic) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich (2006) Google Scholar
  11. 11.
    Yor, M.: Exponential Functionals of Brownian Motion and Related Processes. Springer Finance. Springer, Berlin (2001) zbMATHGoogle Scholar
  12. 12.
    Zhan, D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations