Geodesic
Boundary behavior of SLE
Kang, Nam-Gyu1
1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 185-210

Voir la notice de l'article provenant de la source American Mathematical Society

We show that the normalized (pre-)Schwarzian derivative of SLE, after we subtract a negligible term, is a complex BMO martingale. Its BMO norm gives strong evidence for Duplantier’s duality conjecture. We also show that it has correlations that decay exponentially in the hyperbolic distance. We reexamine S. Rohde and O. Schramm’s derivative expectation to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.
DOI : 10.1090/S0894-0347-06-00547-9

Kang, Nam-Gyu 1

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 
@article{10_1090_S0894_0347_06_00547_9,
     author = {Kang, Nam-Gyu},
     title = {Boundary behavior of {SLE}},
     journal = {Journal of the American Mathematical Society},
     pages = {185--210},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2007},
     doi = {10.1090/S0894-0347-06-00547-9},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00547-9/}
}
TY  - JOUR
AU  - Kang, Nam-Gyu
TI  - Boundary behavior of SLE
JO  - Journal of the American Mathematical Society
PY  - 2007
SP  - 185
EP  - 210
VL  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00547-9/
DO  - 10.1090/S0894-0347-06-00547-9
ID  - 10_1090_S0894_0347_06_00547_9
ER  - 
%0 Journal Article
%A Kang, Nam-Gyu
%T Boundary behavior of SLE
%J Journal of the American Mathematical Society
%D 2007
%P 185-210
%V 20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00547-9/
%R 10.1090/S0894-0347-06-00547-9
%F 10_1090_S0894_0347_06_00547_9
Kang, Nam-Gyu. Boundary behavior of SLE. Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 185-210. doi: 10.1090/S0894-0347-06-00547-9

[1] Bishop, Christopher J., Jones, Peter W. Harmonic measure, 𝐿² estimates and the Schwarzian derivative J. Anal. Math. 1994 77 113

[2] Bishop, Christopher J., Jones, Peter W. Wiggly sets and limit sets Ark. Mat. 1997 201 224

[3] Bishop, Christopher J., Jones, Peter W., Pemantle, Robin, Peres, Yuval The dimension of the Brownian frontier is greater than 1 J. Funct. Anal. 1997 309 336

[4] De Branges, Louis A proof of the Bieberbach conjecture Acta Math. 1985 137 152

[5] Duplantier, Bertrand Conformally invariant fractals and potential theory Phys. Rev. Lett. 2000 1363 1367

[6] Duplantier, Bertrand Conformal fractal geometry & boundary quantum gravity 2004 365 482

[7] Durrett, Richard Brownian motion and martingales in analysis 1984

[8] Erdã©Lyi, Arthur, Magnus, Wilhelm, Oberhettinger, Fritz, Tricomi, Francesco G. Higher transcendental functions. Vol. I 1981

[9] Folland, Gerald B. Fourier analysis and its applications 1992

[10] Garnett, John B., Marshall, Donald E. Harmonic measure 2005

[11] Goluzin, G. M. Geometric theory of functions of a complex variable 1969

[12] Graczyk, Jacek, Jones, Peter Dimension of the boundary of quasiconformal Siegel disks Invent. Math. 2002 465 493

[13] John, F., Nirenberg, L. On functions of bounded mean oscillation Comm. Pure Appl. Math. 1961 415 426

[14] Jones, Peter W. Rectifiable sets and the traveling salesman problem Invent. Math. 1990 1 15

[15] Karatzas, Ioannis, Shreve, Steven E. Brownian motion and stochastic calculus 1991

[16] Lawler, G. The dimension of the frontier of planar Brownian motion Electron. Comm. Probab. 1996

[17] Lawler, Gregory F. Conformally invariant processes in the plane 2005

[18] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin The dimension of the planar Brownian frontier is 4/3 Math. Res. Lett. 2001 13 23

[19] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Values of Brownian intersection exponents. I. Half-plane exponents Acta Math. 2001 237 273

[20] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Values of Brownian intersection exponents. I. Half-plane exponents Acta Math. 2001 237 273

[21] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Analyticity of intersection exponents for planar Brownian motion Acta Math. 2002 179 201

[22] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Values of Brownian intersection exponents. III. Two-sided exponents Ann. Inst. H. Poincaré Probab. Statist. 2002 109 123

[23] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Conformal invariance of planar loop-erased random walks and uniform spanning trees Ann. Probab. 2004 939 995

[24] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin On the scaling limit of planar self-avoiding walk 2004 339 364

[25] Pommerenke, Ch. Boundary behaviour of conformal maps 1992

[26] Ratcliffe, John G. Foundations of hyperbolic manifolds 1994

[27] Rohde, Steffen, Schramm, Oded Basic properties of SLE Ann. of Math. (2) 2005 883 924

[28] Schmuland, Byron, Sun, Wei A central limit theorem and law of the iterated logarithm for a random field with exponential decay of correlations Canad. J. Math. 2004 209 224

[29] Schramm, Oded Scaling limits of loop-erased random walks and uniform spanning trees Israel J. Math. 2000 221 288

[30] Schramm, Oded, Sheffield, Scott Harmonic explorer and its convergence to 𝑆𝐿𝐸₄ Ann. Probab. 2005 2127 2148

[31] Smirnov, Stanislav Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits C. R. Acad. Sci. Paris Sér. I Math. 2001 239 244

[32] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals 1993

[33] Werner, Wendelin Random planar curves and Schramm-Loewner evolutions 2004 107 195

Cité par Sources :