Existence of multi-point boundary Green’s function for chordal Schramm-Loewner evolution (SLE). (English) Zbl 1517.60107

Fakhry, Rami; Zhan, Dapeng

Electron. J. Probab. 28, Paper No. 46, 29 p. (2023).

Summary: In the paper we prove that, for \(\kappa \in (0,8)\), the \(n\)-point boundary Green’s function of exponent \(\frac{8}{\kappa }-1\) for chordal SLE\(_\kappa\) exists. We also prove that the convergence is uniform over compact sets and the Green’s function is continuous. We also give up-to-constant bounds for the Green’s function.

MSC:

60J67	Stochastic (Schramm-)Loewner evolution (SLE)
60J45	Probabilistic potential theory
31C15	Potentials and capacities on other spaces

Keywords:

Green’s function; SLE; Schramm-Loewner evolution

References:

[1]	Lars V. Ahlfors. Conformal invariants: topics in geometric function theory. McGraw-Hill Book Co., New York, 1973. · Zbl 0272.30012
[2]	Tom Alberts and Michael Kozdron. Intersection probabilities for a chordal SLE path and a semicircle, Electron. Comm. Probab., 13:448-460, 2008. · Zbl 1187.82034
[3]	Tom Alberts and Scott Sheffield. Hausdorff dimension of the SLE curve intersected with the real line, Electron J. Probab., 40:1166-1188, 2008. · Zbl 1192.60025
[4]	Vincent Beffara. The dimension of SLE curves, Ann. Probab., 36:1421-1452, 2008. · Zbl 1165.60007
[5]	Alex Karrila, Kalle Kytölä and Eveliina Peltola. Boundary Correlations in Planar LERW and UST, Commu. Math. Phys., 376:2065-2145, 2020. · Zbl 1441.82023
[6]	Gregory Lawler. Schramm-Loewner evolution, in statistical mechanics, S. Sheffield and T. Spencer, ed., IAS/Park City Mathematical Series, AMS, 231-295, 2009. · Zbl 1180.82002
[7]	Gregory Lawler. Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line, J. Math. Soc. Japan., 67:1631-1669, 2015. · Zbl 1362.60069
[8]	Gregory Lawler. Conformally invariant processes in the plane, Amer. Math. Soc., 2005. · Zbl 1074.60002
[9]	Gregory Lawler and Mohammad Rezaei. Minkowski content and natural parametrization for the Schramm-Loewner evolution. Ann. Probab., 43(3):1082-1120, 2015. · Zbl 1331.60165
[10]	Gregory Lawler, Oded Schramm and Wendelin Werner. Values of Brownian intersection exponents I: half-plane exponents. Acta Math., 187(2):237-273, 2001. · Zbl 1005.60097
[11]	Gregory Lawler, Oded Schramm and Wendelin Werner. Conformal restriction: the chordal case, J. Amer. Math. Soc., 16(4): 917-955, 2003. · Zbl 1030.60096
[12]	Gregory Lawler and Scott Sheffield. A natural parametrization for the Schramm-Loewner evolution. Annals of Probab., 39(5):1896-1937, 2011. · Zbl 1234.60087
[13]	Greg Lawler and Brent Werness. Multi-point Green’s function for SLE and an estimate of Beffara, Annals of Prob. 41:1513-1555, 2013. · Zbl 1277.60134
[14]	Greg Lawler and Wang Zhou. SLE curves and natural parametrization. Ann. Probab., 41(3A):1556-1584, 2013. · Zbl 1288.60098
[15]	Benjamin Mackey and Dapeng Zhan. Multipoint Estimates for Radial and Whole-plane SLE. J. Stat. Phys., 175:879-903, 2019. · Zbl 1421.30010
[16]	Jason Miller and Scott Sheffield. Imaginary geometry I: intersecting SLEs. Probab. Theory Relat. Fields, 164(3):553-705, 2016. · Zbl 1336.60162
[17]	Mohammad Rezaei and Dapeng Zhan. Green’s function for chordal SLE curves. Probab. Theory Rel., 171:1093-1155, 2018. · Zbl 1428.60118
[18]	Mohammad Rezaei and Dapeng Zhan. Higher moments of the natural parameterization for SLE curves, Ann. IHP. 53(1):182-199, 2017. · Zbl 1361.60074
[19]	Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. · Zbl 0731.60002
[20]	Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. Math., 161:879-920, 2005. · Zbl 1081.60069
[21]	Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221-288, 2000. · Zbl 0968.60093
[22]	Dapeng Zhan. The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36:467-529, 2008 · Zbl 1153.60057