Large deviations for Brownian intersection measures

Abstract

We consider p independent Brownian motions in Rd. We assume that pgeq2 and p(d−2)<d. Let ellt denote the intersection measure of the p paths by time t, i.e., the random measure on Rd that assigns to any measurable set AsubsetRd 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 ellt(Rd) as ttoinfty. In this paper, we derive a large-deviation principle for the normalised intersection measure t−pellt on the set of positive measures on some open bounded set BsubsetRd 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 UsubsetB. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.