Probability Theory and Related Fields

, Volume 129, Issue 3, pp 340–380 | Cite as

Stochastic Loewner evolution in doubly connected domains

Article

Abstract.

This paper introduces the annulus SLEκ processes in doubly connected domains. Annulus SLE6 has the same law as stopped radial SLE6, up to a time-change. For κ ≠ 6, some weak equivalence relation exists between annulus SLEκ and radial SLEκ. Annulus SLE2 is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE2 satisfies the reversibility property. We also consider the disc SLEκ process defined as a limiting case of the annulus SLE’s. Disc SLE6 has the same law as stopped full plane SLE6, up to a time-change. Disc SLE2 is the scaling limit of loop-erased random walk, and is the reversal of radial SLE2.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlfors, L.V.: Conformal invariants: topics in geometric function theory. New York: McGraw-Hill Book Co., 1973Google Scholar
  2. 2.
    Chandrasekharan, K.: Elliptic functions. Springer-Verlag Berlin Heidelberg, 1985Google Scholar
  3. 3.
    Dubédat, J.: Critical percolation in annuli and SLE6, arXiv:math.PR/0306056Google Scholar
  4. 4.
    Lawler, G.F.: Intersection of random walks. Boston: Birkhäuser, 1991Google Scholar
  5. 5.
    Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Mathematica 187, 237–273 (2001)MathSciNetMATHGoogle Scholar
  6. 6.
    Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents II: Plane exponents. Acta Mathematica 187, 275–308 (2001)MathSciNetMATHGoogle Scholar
  7. 7.
    Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents III: two-sided exponents. Ann. Int. Henri Poincaré 38, 109–123, (2002)CrossRefMATHGoogle Scholar
  8. 8.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees, arXiv: math.PR/012234Google Scholar
  9. 9.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Amer. Math. Soc. 16 (4), 917–955 (2003)CrossRefMATHGoogle Scholar
  10. 10.
    Oksendal, B.: Stochastic Differential Equations: an introduction with applications. Berlin-Heidelberg-New York: Springer-Verlag, 1995Google Scholar
  11. 11.
    Pommerenke, C.: On the Löwner differential equation. Michigan Math. J. 13, 435–443 (1968)MATHGoogle Scholar
  12. 12.
    Pommerenke, C.: Univalent functions. Göttingen: Vandenhoeck & Ruprecht, 1975Google Scholar
  13. 13.
    Pommerenke, C.: Boundary behaviour of conformal maps. Berlin-Heidelberg-New York: Springer-Verlag, 1991Google Scholar
  14. 14.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer-Verlag, 1991Google Scholar
  15. 15.
    Rohde, S., Schramm, O.: Basic properties of SLE, arXiv:math.PR/ 0106036Google Scholar
  16. 16.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–228 (2000)MathSciNetMATHGoogle Scholar
  17. 17.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (3), 239–244 (2001)MATHGoogle Scholar
  18. 18.
    Villat, H.: Le problème de Dirichlet dans une aire annulaire. Rend. circ. mat. Palermo 134–175 (1912)Google Scholar
  19. 19.
    Werner, W.: Critical exponents, conformal invariance and planar Brownian motion. Proceedings of the 3rd Europ. Congress Math., Prog. Math. 202, 87–103 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations