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Title: Large deviations for Brownian intersection measures
Authors: König, WolfgangMukherjee, Chiranjib
Issue Date: 2011
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 1610, ISSN 0946-8633
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: We consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)<d$. Let $ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $R^d$ that assigns to any measurable set $Asubset R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen citeCh09 derived the logarithmic asymptotics of the upper tails of the total mass $ell_t(R^d)$ as $ttoinfty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^-pell_t$ on the set of positive measures on some open bounded set $BsubsetR^d$ as $ttoinfty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $Usubset B$. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.
Keywords: Intersection of Brownian paths; intersection local time; intersection measure; exponential approximation; large deviations
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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