Brownian intersection local time and large deviations
Max Planck Institute for Mathematics in Sciences, Leipzig.
We consider a number of independent Brownian motions running in the $d$- dimensional Euclidean space until they exit a fixed ball. We look at the spatial intersection of the paths. Pioneering works of Le Gall and others say that this set can be equipped with an object measuring the intensity of the intersections of the Brownian paths. Keeping track of the notion of local time pertaining to a single path, this object is called the ``Brownian intersection local time''. Koenig and Moerters recently studied the upper tails of this random object, sending the amount of intersection to infinity. The resulting variational formula admits minimizer(s) with certain probabilistic interpretation. We study these minimizers and link them to a large deviation principle for the intersection local time (as a measure) normalized on a compact set, conditional on the total amount of intersection being very large.