×

SLE and \(\alpha \)-SLE driven by Lévy processes. (English) Zbl 1151.60025

From the authors’ abstract: Stochastic Loewner evolutions (SLE) with a multiple \(\sqrt {\kappa}B\) of Brownian motion \(B\) as driving process are random planar curves (if \(\kappa \leq 4\)) or growing compact sets generated by a curve (if \(\kappa >4\)). Now, consider more general Lévy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. The authors show that when the driving force is of the form \(\sqrt {\kappa} B + \theta^{1/\alpha}S\) for a symmetric \(\alpha \)-stable Lévy process \(S\), the cluster has zero or positive Lebesgue measure according to whether \(\kappa \leq 4\) or \(\kappa >4\). They also give mathematical evidence that a further phase transition at \(\alpha =1\) is attributable to the recurrence/transience dichotomy of the driving Lévy process. The authors introduce a new class of evolutions that we call \(\alpha \)-SLE. They have \(\alpha \)-self-similarity properties for \(\alpha \)-stable Lévy driving processes. The authors show the phase transition at a critical coefficient \(\theta =\theta _{0}(\alpha )\) analogous to the \(\kappa =4\) phase transition.

MSC:

60G52 Stable stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus . Cambridge Univ. Press. · Zbl 1073.60002
[2] Beliaev, D. and Smirnov, S. (2005). Harmonic measure on fractal sets. In European Congress of Mathematics 41-59. Eur. Math. Soc., Zürich. · Zbl 1079.30026
[3] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press. · Zbl 0861.60003
[4] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261-288. · Zbl 1023.92018
[5] Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127 89-152. · Zbl 1032.60047
[6] Cardy, J. (2005). SLE for theoretical physicists. Ann. Physics 318 81-118. · Zbl 1073.81068
[7] Guan, Q.-Y. (2006). Integration by parts formula for regional fractional Laplacian. Comm. Math. Phys. 266 289-329. · Zbl 1121.60051
[8] Guan, Q.-Y. and Ma, Z.-M. (2006). Reflected symmetric \alpha -stable processes and regional fractional Laplacian. Probab. Theory Related Fields 134 649-694. · Zbl 1089.60030
[9] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2 79-95. · Zbl 0118.13401
[10] Itô, K. (2004). Stochastic Processes. Lectures Given at Aarhus University . Springer, Berlin. · Zbl 1068.60002
[11] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane . Amer. Math. Soc., Providence, RI. · Zbl 1074.60002
[12] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237-273. · Zbl 1005.60097
[13] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011
[14] Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications : A Jubilee of Benoît Mandelbrot , Part 2. Proc. Sympos. Pure Math. 72 339-364. Amer. Math. Soc., Providence, RI. · Zbl 1069.60089
[15] Löwner, K. (1923). Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89 103-121. · JFM 49.0714.01
[16] Marshall, D. E. and Rohde, S. (2005). The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18 763-778. · Zbl 1078.30005
[17] Millar, P. W. (1973). Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc. 178 459-479. · Zbl 0268.60065
[18] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. ( 2 ) 161 883-924. · Zbl 1081.60069
[19] Rushkin, I., Oikonomou, P., Kadanoff, L. P. and Gruzberg, I. A. (2006). Stochastic Loewner evolution driven by Lévy processes. J. Stat. Mech. Theory Exp. 1 P01001 1-21.
[20] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093
[21] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244. · Zbl 0985.60090
[22] Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions . Princeton Univ. Press. · Zbl 0207.13501
[23] Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 65 243-256. · Zbl 1016.60054
[24] Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 107-195. Springer, Berlin. · Zbl 1057.60078
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.