Consider the intersection measure $\ell^{ \mathrm{ IS}}_t$ of $p$ independent Brownian motions on $\mathbb{ R}^d$. In this article, we prove the large deviation principle for the normalized intersection measure $t^{ -p}\ell^{ \mathrm{ IS}}_t$ as $t\rightarrow \infty$, before exiting a (possibly unbounded) domain $D\subset\mathbb{ R}^d$ with smooth boundary. This is an extension of [W. K\"onig and C. Mukherjee: Communications on Pure and Applied Mathematics, 66(2):263--306, 2013] which deals with the case $D$ is bounded. Our essential contribution is to prove the so-called super-exponential estimate for the intersection measure of killed Brownian motions on such $D$ by an application of the Chapman-Kolmogorov relation.
Categories: math.PR