Brownian occupation measures, compactness and large deviations

Abstract

In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_{t}(A)=\frac{1}{t}\int_{0}^{t}{\mathbf{1}}_{A}(W_{s})\,\mathrm{d}s$ of the $d$-dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal{M}_{1}(\mathbb{R}^{d})$ can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^{d}$ by adding a point at $\infty$ that results in the compactification of $\mathcal{M}_{1}(\mathbb{R}^{d})$ by allowing some mass to escape to the point at $\infty$. If one were to use only test functions that are continuous and vanish at $\infty$, then the compactification results in the space of sub-probability distributions $\mathcal{M}_{\le1}(\mathbb{R}^{d})$ by ignoring the mass at $\infty$.

The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $\tilde{\mathcal{M}}_{1}=\tilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ under the action of the translation group $\mathbb{R}^{d}$ on $\mathcal{M}_{1}(\mathbb{R}^{d})$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.