</script> ">
Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 12 (2015) open journal systems 


Conformal restriction and Brownian motion

Hao Wu, Mathematics Department of MIT


Abstract
This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014.

We describe and characterize all random subsets \(K\) of simply connected domain which satisfy the "conformal restriction" property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set \(K\) in the upper half-plane \(\mathbb{H}\) connects two fixed boundary points, say 0 and \(\infty\), and given that \(K\) stays in a simply connected open subset \(H\) of \(\mathbb{H}\), the conditional law of \(\Phi(K)\) is identical to that of \(K\), where \(\Phi\) is any conformal map from \(H\) onto \(\mathbb{H}\) fixing 0 and \(\infty \). In the radial case, the random set \(K\) in the upper half-plane \(\mathbb{H}\) connects one fixed boundary points, say 0, and one fixed interior point, say \(i\), and given that \(K\) stays in a simply connected open subset \(H\) of \(\mathbb{H}\), the conditional law of \(\Phi(K)\) is identical to that of \(K\), where \(\Phi\) is the conformal map from \(H\) onto \(\mathbb{H}\) fixing 0 and \(i\).

It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter \(\kappa=8/3\) and Poisson point processes of Brownian excursions and Brownian loops.

AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60J69.

Keywords: Conformal invariance, restriction property, Brownian excursion, Brownian loop, Schramm Loewner evolution.

Creative Common LOGO

Full Text: PDF
Wu, Hao, Conformal restriction and Brownian motion, Probability Surveys, 12, (2015), 55-103 (electronic). DOI: 10.1214/15-PS259.

References

[Bil99]     Patrick Billingsley. Convergence of Probability Measures. 1999. MR1700749

[BL90]     K Burdzy and G Lawler. Non-intersection exponents for random walk and brownian motion. part i: Existence and an invariance principle. probab. th. and rel. fields 84 393-410. Math. Review 91g, 60096, 1990. MR1035664

[DK88]     Bertrand Duplantier and Kyung-Hoon Kwon. Conformal invariance and intersections of random walks. Phys. Rev. Lett., 61:2514–2517, Nov 1988.

[DLLGL93]     Bertrand Duplantier, Gregory F Lawler, J-F Le Gall, and Terence J Lyons. The geometry of the brownian curve. Bulletin des sciences mathématiques, 117(1):91–106, 1993. MR1205413

[Law96a]     Gregory F Lawler. The dimension of the frontier of planar brownian motion. Electronic Communications in Probability, 1:29–47, 1996. MR1386292

[Law96b]     Gregory F. Lawler. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab., 1:no. 2, approx. 20 pp. (electronic), 1996. MR1386294

[Law05]     Gregory F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. MR2129588

[LP97]     Gregory F Lawler and Emily E Puckette. The disconnection exponent for simple random walk. Israel Journal of Mathematics, 99(1):109–121, 1997. MR1469089

[LP00]     Gregory F Lawler and Emily E Puckette. The intersection exponent for simple random walk. Combinatorics, Probability and Computing, 9(05):441–464, 2000. MR1810151

[LSW01a]     Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187(2):237–273, 2001. MR1879850

[LSW01b]     Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187(2):275–308, 2001. MR1879851

[LSW02a]     Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Analyticity of intersection exponents for planar Brownian motion. Acta Math., 189(2):179–201, 2002. MR1961197

[LSW02b]     Gregory F Lawler, Oded Schramm, and Wendelin Werner. On the scaling limit of planar self-avoiding walk. arXiv preprint math/0204277, 2002. MR2112127

[LSW02c]     Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., 38(1):109–123, 2002. MR1899232

[LSW03]     Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc., 16(4):917–955 (electronic), 2003. MR1992830

[LW99]     Gregory F. Lawler and Wendelin Werner. Intersection exponent for planar brownian motion. The Annals of Probability, 27(4):1601–1642, 1999. MR1742883

[LW00a]     Gregory F. Lawler and Wendelin Werner. Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. (JEMS), 2(4):291–328, 2000. MR1796962

[LW00b]     Gregory F Lawler and Wendelin Werner. Universality for conformally invariant intersection exponents. Journal of the European Mathematical Society, 2(4):291–328, 2000. MR1796962

[LW04]     Gregory F. Lawler and Wendelin Werner. The Brownian loop soup. Probab. Theory Related Fields, 128(4):565–588, 2004. MR2045953

[Man83]     Benoit B Mandelbrot. The fractal geometry of nature, volume 173. Macmillan, 1983.

[RS05]     Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883–924, 2005. MR2153402

[Sch00]     Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000. MR1776084

[Smi01]     Stanislav Smirnov. 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. MR1851632

[SW05]     Oded Schramm and David B. Wilson. SLE coordinate changes. New York J. Math., 11:659–669 (electronic), 2005. MR2188260

[SW12]     Scott Sheffield and Wendelin Werner. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. (2), 176(3):1827–1917, 2012. MR2979861

[Wer04]     Wendelin Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pages 107–195. Springer, Berlin, 2004. MR2079672

[Wer05]     Wendelin Werner. Conformal restriction and related questions. Probab. Surv., 2:145–190, 2005. MR2178043

[Wer07]     Wendelin Werner. Lectures on two-dimensional critical percolation. 2007.

[Wer08]     Wendelin Werner. The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc., 21(1):137–169, 2008. MR2350053

[Wu15]     Hao Wu. Conformal restriction: the radial case. Stochastic Process. Appl., 125(2):552–570, 2015. MR3293294
Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787