  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.
