Large Deviations for Brownian Intersection Measures

Abstract

AbstractWe consider p independent Brownian motions in \input amssym ${\Bbb R}^d$. We assume that p ≥ 2 and p (d − 2) < d. Let ℓt denote the intersection measure of the p paths by time t, i.e., the random measure on \input amssym ${\Bbb R}^d$ that assigns to any measurable set \input amssym $A \subset {\Bbb R}^d$ the amount of intersection local time of the motions spent in A by time t. Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass \input amssym $\ell _t \left({{\Bbb R}^d } \right)$ as t → ∞. In this paper, we derive a large‐deviation principle for the normalized intersection measure t−pℓt on the set of positive measures on some open bounded set \input amssym $B \subset {\Bbb R}^d$ as t → ∞ 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 normalized 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 $U \subset B$. This extends earlier studies on the intersection measure by König and Mörters. © 2012 Wiley Periodicals, Inc.