Abstracts:

Wolfgang Koenig
Brownian intersection local times: upper tails and large deviations.
(joint work with Peter Moerters, Bath)

We consider the intersection of several independent Brownian paths in
dimension $d\geq 2$. The motions are not considered until a fixed time, but until
their exit time from a fixed open set $B$, in the case $B=\R^d$ up to time infinity. The so-called Brownian intersection local time, $\ell$, is a natural random measure on the intersection of the paths.

We fix an open bounded subset $U$ of $B$ and derive the asymptotics of the upper tails of the random variable $\ell(U)$ in terms of a variational formula and additionally in terms of a certain non-linear eigenvalue of the Laplace operator. Furthermore, we discuss the asymptotic shape of the normalized intersection local time, $\ell/\ell(U)$ as $\ell(U)$ diverges to infinity, in terms of a large-deviation principle, which is work in progress.

Although this principle may be well interpreted in terms of the well-known Donsker-Varadhan theory, our proofs do not use this theory, but rely on a careful analysis of the high mixed moments of the intersection local time in various sets.