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

Abstract:

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, \[ \mathbb {E}\Big [\exp \Big (\sum _{i=1}^n \langle \varphi _i, \ell \rangle ^{1/p}\Big ) \Big ],\] 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.