publication . Other literature type . Preprint . Article . 2004

The Brownian loop soup

Gregory F. Lawler; Wendelin Werner;
Open Access
  • Published: 01 Apr 2004
  • Publisher: Springer Science and Business Media LLC
We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to ‘‘chronologically add Brownian loops’’ to simple curves in the plane.
free text keywords: Mathematics - Probability, Statistics, Probability and Uncertainty, Statistics and Probability, Analysis, Mathematical analysis, Brownian motion, Invariant measure, Boundary (topology), Probability theory, A domain, Conformal symmetry, Mathematics, Mathematical finance, Schramm–Loewner evolution

[1] A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov (1984), Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241, 333-380. [OpenAIRE]

[2] P. Billingsley, Convergence of Probability Measures, Wiley, 1968.

[3] J.L. Cardy (1984), Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (FS12), 514-532. [OpenAIRE]

[4] J. Dub´edat (2003), SLE(κ, ρ) martingales and duality, math.PR/0303128, preprint.

[5] R. Friedrich, W. Werner (2002), Conformal fields, restriction properties, degenerate representations and SLE, C.R. Acad. Sci. Paris Ser. I. Math. 335, 947-952.

[6] R. Friedrich, W. Werner (2003), Conformal restriction, highest-weight representations and SLE, math-ph/0301018, preprint.

[7] G.F. Lawler, O. Schramm, W. Werner (2001), Conformal invariance of planar looperased random walks and uniform spanning trees, math.PR/0112234, Ann. Probab., to appear.

[8] G.F. Lawler, O. Schramm, W. Werner (2002), On the scaling limit of planar selfavoiding walks, math.PR/0204277, to appear in Fractal geometry and application, A jubilee of Benoit Mandelbrot, AMS Proc. Symp. Pure Math.

[9] G.F. Lawler, O. Schramm, W. Werner (2002), Conformal restriction. The chordal case, math.PR/0209343, J. Amer. Math. Soc., to appear

[10] G.F. Lawler, W. Werner (2000), Universality for conformally invariant intersection exponents, J. Europ. Math. Soc. 2, 291-328.

[11] O. Schramm (2000), Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118, 221-288.

[12] B. Vir´ag (2003), Brownian beads, preprint.

[13] W. Werner (2003), Girsanov's Theorem for SLE(κ,ρ) processes, intersection exponents and hiding exponents, math.PR/0302115, preprint.

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