Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional

Abstract

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.