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MiS Preprint
23/2011
Large Deviations for Brownian Intersection Measures
Wolfgang König and Chiranjib Mukherjee
Abstract
We consider $p$ independent Brownian motions in $\mathbb{R}^d$. We assume that $p\geq 2$ 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 $\mathbb{R}^d$ that assigns to any measurable set $A\subset \mathbb{R}^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\mathbb{R}^d)$ as $t\to\infty$.
In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\mathbb{R}^d$ as $t\to\infty$ 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 $U\subset B$. This extends earlier studies on the intersection measure by König and Mörters.
X. Chen, Random Walk Intersections: Large Deviations and Related Topics, Mathematical Surveys and Monographs, AMS (2009) Vol. 157, Providence, RI.
W. König and P. Mörters, Brownian intersection local times: upper tail asymptotics and thick points, Ann. Probab.30, 1605--1656 (2002).
W. König and P. Mörters, Brownian intersection local times: Exponential moments and law of large masses, Trans. Amer. Math. Soc.358, 1223-1255 (2006).
