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Multiple Intersection Exponents for Planar Brownian Motion

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Abstract

Let p≥2, n 1≤⋅⋅⋅≤n p be positive integers and \(B_{1}^{1},\ldots,B_{n_{1}}^{1};\ldots;B_{1}^{p},\ldots,B_{n_{p}}^{p}\) be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ς p (n 1,…,n p ), as the exponential rate of decay of the probability that the packets \(\bigcup_{j=1}^{n_{i}}B_{j}^{i}[0,t^{2}]\) , i=1,…,p, have no joint intersection. The case p=2 is well-known and, following two decades of numerical and mathematical activity, Lawler et al. (Acta Math. 187:275–308, 2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for p>2. We present an extensive mathematical and numerical study, leading to an exact formula in the case n 1=1, n 2=2, and several interesting conjectures for other cases.

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Authors and Affiliations

  1. Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Achim Klenke

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Peter Mörters

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  1. Achim Klenke
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  2. Peter Mörters
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Correspondence to Achim Klenke.

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Klenke, A., Mörters, P. Multiple Intersection Exponents for Planar Brownian Motion. J Stat Phys 136, 373–397 (2009). https://doi.org/10.1007/s10955-009-9780-7

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  • DOI: https://doi.org/10.1007/s10955-009-9780-7

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