Chiranjib Mukherjee

Large Deviations for Brownian Intersection Measures

We considerp independent Brownian motions in R d . We assume thatp 2 and p.d 2/ < d . Let‘t denote the intersection measure of thep paths by timet, i.e., the random measure on R d that assigns to any measurable setA 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‘t.R d / ast!1. In this paper, we derive a large-deviation principle for the normalized intersection measure t p ‘t on the set of positive measures on some open bounded set B R d as t ! 1 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 fromB, conditional on a large total mass in some compact set U B. This extends earlier studies on the intersection measure by Konig and Morters.

Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time

Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional

t

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in [BKM15]. Our methods rely on deriving H{o}lder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.