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- ItemBrownian occupation measures, compactness and large deviations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, Chiranjib; Varadhan, S.R. SrinivasaIn 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.
- ItemBrownian occupation measures, compactness and large deviations: Pair interaction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, ChiranjibContinuing 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 $0$.
- ItemMean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bolthausen, Erwin; König, Wolfgang; Mukherjee, ChiranjibWe consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]
- ItemLarge deviations for Brownian intersection measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) König, Wolfgang; Mukherjee, ChiranjibWe consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)