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Author: Mukherjee, Chiranjib × Subject: Coulomb functional ×

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    Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) König, Wolfgang; Mukherjee, Chiranjib
    In this paper, we study a transformed path measure that arises from a mean-field type interaction of a three dimensional Brownian motion in a Coulomb potential. Under the influence of such a transformed measure, the large-t behavior of the normalized occupation measures, denoted by Lt, is of high interest. This is intimately connected to the well-known polaron problem from statistical mechanics and a full understanding of the behavior of Lt under the aforementioned transformation is crucial for the analysis of the polaron path measure under ‘strong coupling’ , its effective mass and justification of mean-field approximations. For physical relevance of this model, we refer to [S86]. Some mathematically rigorous research in this direction began in the 1980s with the analysis of the partition function of Donsker and Varadhan ([DV83-P]), but it was not until recently that a new technique was developed [MV14] for handling the actual path measures, and the main results the present paper, besides being interesting on their own, make determinant contribution towards a deeper analysis and a full identification of the limiting distribution of Lt under the transformed path measure.
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    Mean-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, Chiranjib
    We 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]