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Title: Brownian occupation measures, compactness and large deviations
Authors: Mukherjee, ChiranjibVaradhan, S. R. Srinivasa
Issue Date: 2015
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2193, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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 $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence 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 $mathcal M_1(R^d)$ 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 $mathcal M_le 1(R^d)$ 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 $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. 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.
Keywords: Brownian occupation measures; shift compactness; large deviations
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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Appears in Collections:Mathematik



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