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

TL;DR

This paper establishes a large deviation principle for the normalized intersection measure of multiple Brownian motions in unbounded domains, extending previous results to more general settings with smooth boundaries.

Contribution

It extends the large deviation principle for intersection measures to unbounded domains with smooth boundaries, using super-exponential estimates and Chapman-Kolmogorov relations.

Findings

Proves large deviation principle for intersection measures in unbounded domains.

Develops super-exponential estimates for killed Brownian motions.

Extends prior bounded domain results to more general unbounded domains.

Abstract

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.