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Brownian occupation measures, compactness and large deviations. (English) Zbl 1364.60037

In large deviation principles, the lower bound for open sets and the upper bound for compact sets are essentially local estimates. However, the upper bound for closed sets is global. Establishing the large deviation upper bound for closed sets usually requires compactness of the space or an exponential tightness estimate. The purpose of this paper is to study large deviations of the occupation measure of Brownian motion in \(\mathbb{R}^d\), for which there is no hope of getting exponential tightness. The space of probability measures \(\mathcal{M}_1(\mathbb{R}^d)\) on \(\mathbb{R}^d\) can be compactified by replacing the topology of weak convergence with the vague topology. This compactification ignores the underlying translation invariance. However, there are problems for which the underlying translation invariance is very important. In this paper, the authors present a compactification of the quotient space \(\widetilde{\mathcal{M}}_1\) of \(\mathcal{M}_1(\mathbb{R}^d)\) under the action of the translation group in \(\mathbb{R}^d\) and establish a large deviation principle for the Brownian occupation measure on this quotient space.
The compactification given in this paper should be useful in other problems, too.

MSC:

60F10 Large deviations
60J65 Brownian motion
60J35 Transition functions, generators and resolvents
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