Brownian intersection local times: Exponential moments and law of large masses

Consider $p$ independent Brownian motions in $\mathbb{R} ^d$, each running up to its first exit time from an open domain $B$, and their intersection local time $\ell$ as a measure on $B$. We give a sharp criterion for the finiteness of exponential moments,

where $\varphi_1, \dots, \varphi_n$ are nonnegative, bounded functions with compact support in $B$. We also derive a law of large numbers for intersection local time conditioned to have large total mass.