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October 2002 Brownian intersection local times: Upper tail asymptotics and thick points
Wolfgang König, Peter Mörters
Ann. Probab. 30(4): 1605-1656 (October 2002). DOI: 10.1214/aop/1039548368

Abstract

We equip the intersection of p independent Brownian paths in $\mathbb{R}^d$, $d\ge 2$, with the natural measure $\ell$ defined by projecting the intersection local time measure via one of the Brownian motions onto the set of intersection points. Given a bounded domain $U\subset\mathbb{R}^d$ we show that, as $a\uparrow\infty$, the probability of the event $\{\ell(U)>a\}$ decays with an exponential rate of $a^{1/p}\theta$, where $\theta$ is described in terms of a variational problem. In the important special case when U is the unit ball in $\mathbb{R}^3$ and $p=2$, we characterize $\theta$ in terms of an ordinary differential equation. We apply our results to the problem of finding the Hausdorff dimension spectrum for the thick points of the intersection of two independent Brownian paths in $\mathbb{R}^3$.

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Wolfgang König. Peter Mörters. "Brownian intersection local times: Upper tail asymptotics and thick points." Ann. Probab. 30 (4) 1605 - 1656, October 2002. https://doi.org/10.1214/aop/1039548368

Information

Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1032.60073
MathSciNet: MR1944002
Digital Object Identifier: 10.1214/aop/1039548368

Subjects:
Primary: 60G17, 60J55, 60J65

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 4 • October 2002
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