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

Takahiro Mori

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arXiv · 2005.09219

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

Takahiro Mori

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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.

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Takahiro Mori. 2020-05-19. Large deviation principle for the intersection measure of Brownian motions on unbounded domains. https://arxiv.org/abs/2005.09219

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