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Consider the intersection measure $\ell^{\mathrm{IS}}_t$ of $p$ independentBrownian motions on $\mathbb{R}^d$. In this article, we prove the largedeviation 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. Thisis an extension of [W. K\"onig and C. Mukherjee: Communications on Pure andApplied Mathematics, 66(2):263--306, 2013] which deals with the case $D$ isbounded. Our essential contribution is to prove the so-called super-exponentialestimate for the intersection measure of killed Brownian motions on such $D$ byan application of the Chapman-Kolmogorov relation.

Large deviation principle for the intersection measure of Brownian motions on unbounded domains