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

We study the transformed path measure arising from the self-interaction of a three-dimensional rownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time t. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan (Comm. Pure Appl. Math. 505 (1983) 505–528) in terms of a variational formula. Recently (Brownian occupations measures, compactness and large deviations (2014) Preprint) a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in (Brownian occupations measures, compactness and large deviations (2014) Preprint) to a certain shift-invariant functional of the occupation measures