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Duality of chordal SLE. II. (English) Zbl 1200.60071
Summary: We improve the geometric properties of SLE$$(\kappa ; \vec{\rho })$$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $$\kappa \in (4, 8)$$, the boundary of a standard chordal SLE$$(\kappa )$$ hull stopped on swallowing a fixed $$x\in \mathbb R \setminus \{0\}$$ is the image of some SLE$$(16/\kappa ; \vec{\rho })$$ trace started from a random point. Using this fact together with a similar proposition in the case that $$\kappa \geq 8$$, we obtain a description of the boundary of a standard chordal SLE$$(\kappa )$$ hull for $$\kappa >4$$, at a finite stopping time. Finally, we prove that for $$\kappa >4$$, in many cases, a chordal or strip SLE$$(\kappa ; \vec{\rho })$$ trace a.s. ends at a single point.
For part I, cf. Invent. Math. 174, No. 2, 309–353 (2008; Zbl 1158.60047).

MSC:
 60J67 Stochastic (Schramm-)Loewner evolution (SLE) 30C20 Conformal mappings of special domains 60H05 Stochastic integrals
Keywords:
SLE; duality; coupling technique
Full Text:
References:
 [1] L. V. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory . McGraw-Hill, New York, 1973. · Zbl 0272.30012 [2] V. Beffara. Hausdorff dimensions for SLE 6 . Ann. Probab. 32 (2004) 2606-2629. · Zbl 1055.60036 [3] V. Beffara. The dimension of the SLE curves. Ann. Probab. 36 (2008) 1421-1452. · Zbl 1165.60007 [4] J. Dubédat. Duality of Schramm-Loewner evolutions. Ann. Sci. École Norm. Sup. (4) 42 (2009) 697-724. · Zbl 1205.60147 [5] G. F. Lawler, O. Schramm and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004) 939-995. · Zbl 1126.82011 [6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion . Springer, Berlin, 1991. · Zbl 0731.60002 [7] S. Rohde and O. Schramm. Basic properties of SLE. Ann. Math. 161 (2005) 883-924. · Zbl 1081.60069 [8] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000) 221-288. · Zbl 0968.60093 [9] D. Zhan. Duality of chordal SLE. Inven. Math. 174 (2008) 309-353. · Zbl 1158.60047 [10] D. Zhan. The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36 (2008) 467-529. · Zbl 1153.60057 [11] D. Zhan. Reversibility of chordal SLE. Ann. Probab. 36 (2008) 1472-1494. · Zbl 1157.60051
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