The Annals of Probability

No triple point of planar Brownian motion is accessible

Krzysztof Burdzy and Wendelin Werner

Full-text: Open access

Abstract

We show that the boundary of a connected component of the complement of a planar Brownian path on a fixed time interval contains almost surely no triple point of this Brownian path.

Article information

Source
Ann. Probab., Volume 24, Number 1 (1996), 125-147.

Dates
First available in Project Euclid: 15 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1042644710

Digital Object Identifier
doi:10.1214/aop/1042644710

Mathematical Reviews number (MathSciNet)
MR1387629

Zentralblatt MATH identifier
0860.60063

Subjects
Primary: 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion triple points

Citation

Burdzy, Krzysztof; Werner, Wendelin. No triple point of planar Brownian motion is accessible. Ann. Probab. 24 (1996), no. 1, 125--147. doi:10.1214/aop/1042644710. https://projecteuclid.org/euclid.aop/1042644710


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References

  • ADELMAN, O. and DVORETZKY, A. 1985. Plane Brownian motion has strictly n-multiple points. Israel J. Math. 52 361 364. Z.
  • AHLFORS, L. V. 1973. Conformal Invariants: Topics in Geometrical Function Theory. McGrawHill, New York. Z.
  • BANUELOS, R., BASS, R. and BURDZY, K. 1991. Holder domains and the boundary Harnack ¨ principle. Duke Math. J. 64 195 200. Z.
  • BASS, R. 1995. Probabilistic Techniques in Analy sis. Springer, New York. Z.
  • BASS, R. and BURDZY, K. 1991. A boundary Harnack principle in twisted Holder domains. Ann. ¨ Math. 134 253 276.
  • BASS, R., BURDZY, K. and KHOSHNEVISAN, D. 1994. Intersection local time for points of infinite multiplicity. Ann. Probab. 22 566 625. Z.
  • BURDZY, K. 1989a. Geometric properties of two-dimensional Brownian paths. Probab. Theory Related Fields 81 485 505. Z.
  • BURDZY, K. 1989b. Cut points on Brownian paths. Ann. Probab. 17 1012 1036. Z.
  • BURDZY, K. 1995. Laby rinth dimension of Brownian trace. Probab. Math. Statist. Volume in honor of Jerzy Ney man. To appear. Z.
  • BURDZY, K. and LAWLER, G. F. 1990a. Non-intersection exponents for Brownian paths, Part I. Existence and an invariance principle. Probab. Theory Related Fields 84 393 410. Z.
  • BURDZY, K. and LAWLER, G. F. 1990b. Non-intersection exponents for Brownian paths. Part II. Estimates and application to a random fractal. Ann. Probab. 18 981 1009. Z.
  • DOOB, J. L. 1984. Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York. Z.
  • DVORETZKY, A., ERDOS, P. and KAKUTANI, S. 1954. Multiple points of Brownian motion in the plane. Bull. Res. Council Israel Sect. F 3 364 371. Z.
  • DVORETZKY, A., ERDOS, P. and KAKUTANI, S. 1958. Points of multiplicity c of plane Brownian paths. Bull. Res. Council Israel Sect. F 7 175 180. Z.
  • LAWLER, G. F. 1989. Intersection of random walks with random sets. Israel J. Math. 65 113 132. Z.
  • LAWLER, G. F. 1991. Intersection of Random Walks. Birkhauser, Boston. ¨ Z.
  • LE GALL, J. F. 1987a. Le comportement du mouvement Brownien entre les deux instants ou il passe par un point double. J. Funct. Anal. 71 246 262. Z.
  • LE GALL, J. F. 1987b. The exact Hausdorff measure of Brownian multiple points. In Seminar on Stochastic Processes 1986 107 137. Birkhauser, Boston. ¨ Z.
  • LE GALL, J. F. 1991. Some properties of planar Brownian motion. Ecole d'Ete de Probabilites de ´ ´ St.-Flour XX. Lecture Notes in Math. 1527 111 235. Springer, New York. Z.
  • LE GALL, J. F. and MEy RE, T. 1992. Points cones du mouvement Brownien plan, le cas critique. Probab. Theory Related Fields 93 231 247. Z.
  • LEVY, P. 1965. Process Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. ´ Z.
  • MANDELBROT, B. B. 1982. The Fractal Geometry of Nature. W. H. Freeman, New York. Z.
  • MEy ER, P. A., SMy THE, R. T. and WALSH, J. B. 1972. Birth and death of Markov processes. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 3 295 305. Univ. California Press, Berkeley. Z.
  • OHTSUKA, M. 1970. Dirichlet Problem, Extremal Length and Prime Ends. Van Nostrand, New York. Z.
  • OKSENDAL, B. 1983. Projection estimates for harmonic measure. Ark. Mat. 21 191 203. Z.
  • POMMERENKE, CH. 1992. Boundary Behavior of Conformal Maps. Springer, Berlin. Z.
  • REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. Z.
  • WERNER, W. 1994. Sur la forme des composantes connexes du complementaire de la courbe ´ brownienne plane. Prob. Theory Related Fields 98 307 337. Z.
  • WERNER, W. 1995. On Brownian disconnection exponents. Bernoulli 1 371 380. Z.
  • WILLIAMS, D. 1974. Path decomposition and continuity of local time for one-dimensional () diffusions. Proc. London Math. Soc. 3 28 738 768.
  • STATISTICAL LABORATORY, D.P.M.M.S. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CAMBRIDGE GN-50 16 MILL LANE UNIVERSITY OF WASHINGTON CAMBRIDGE CB2 1SB
  • SEATTLE, WASHINGTON 98195 UNITED KINGDOM E-mail: burdzy@math.washington.edu E-mail: wwerner@dmi.ens.fr