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

Abstract

Consider the intersection measure ${\ell }_t^{\mathrm{IS}}$ of $p$ independent Brownian motions on ${\mathbb{R}}^d$. In this article, we prove the large deviation principle for the normalized intersection measure $t^{-p}{\ell }_t^{\mathrm{IS}}$ as $t\to \infty$, before exiting a (possibly unbounded) domain $D\subset {\mathbb{R}}^d$ with smooth boundary. This is an extension of [W. König 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.

19 pages; minor corrections

References in corpus (1)

• Large Deviations for Brownian Intersection Measures