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Title: Brownian occupation measures, compactness and large deviations: Pair interaction
Authors: Mukherjee, Chiranjib
Issue Date: 2015
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2197, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: Continuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.
Keywords: Compactness; large deviations; Gibbs measures on Brownian paths; Parabolic Anderson model
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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