Electronic Journal of Probability

Strict Concavity of the Half Plane Intersection Exponent for Planar Brownian Motion

Gregory Lawler

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The intersection exponents for planar Brownian motion measure the exponential decay of probabilities of nonintersection of paths. We study the intersection exponent $\xi(\lambda_1,\lambda_2)$ for Brownian motion restricted to a half plane which by conformal invariance is the same as Brownian motion restricted to an infinite strip. We show that $\xi$ is a strictly concave function. This result is used in another paper to establish a universality result for conformally invariant intersection exponents.

Article information

Electron. J. Probab., Volume 5 (2000), paper no. 8, 33 pp.

Accepted: 3 March 2000
First available in Project Euclid: 7 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Brownian motion intersection exponent

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Lawler, Gregory. Strict Concavity of the Half Plane Intersection Exponent for Planar Brownian Motion. Electron. J. Probab. 5 (2000), paper no. 8, 33 pp. doi:10.1214/EJP.v5-64. https://projecteuclid.org/euclid.ejp/1457376443

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  • L. V. Ahlfors (1973). Conformal Invariants, Topics in Geometric Function Theory McGraw-Hill
  • R. Bass (1995). Probabilistic Techniques in Analysis Springer-Verlag
  • P. Berg and J. McGregor (1966). Elementary Partial Differential Equations Holden-Day
  • X. Bressaud, R. Fernandez, A. Galves (1999). Decay of correlations for non-Holderian dynamics: a coupling approach Electron. J. Probab. 4, paper no. 3
  • B. Duplantier (1999). Two-dimensional copolymers and exact conformal multifractality, Phys. Rev. Lett. 82, 880–883.
  • G. F. Lawler (1995). Hausdorff dimension of cut points for Brownian motion, Electron. J. Probab. 1, paper no.2.
  • G. F. Lawler (1996). The dimension of the frontier of planar Brownian motion, Electron. Comm. Prob. 1, paper no 5.
  • G. F. Lawler (1997). The frontier of a Brownian path is multifractal, preprint.
  • G. F. Lawler (1998). Strict concavity of the intersection exponent for Brownian motion in two and three dimensions, Math. Phys. Electron. J. 4, paper no. 5
  • G. F. Lawler, W. Werner (1999). Intersection exponents for planar Brownian motion, Ann. Probab. 27, 1601–1642.
  • G. F. Lawler, W. Werner (1999). Universality for conformally invariant intersection exponents, preprint.