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We study the transformed path measure arising from the self-interaction of athree-dimensional Brownian motion via an exponential tilt with the Coulombenergy of the occupation measures of the motion by time $t$. The logarithmicasymptotics of the partition function were identified in the 1980s by Donskerand Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a newtechnique for studying the path measure itself was introduced, which allows forproving that the normalized occupation measure asymptotically concentratesaround the set of all maximizers of the formula. In the present paper, we showthat likewise the Coulomb functional of the occupation measure concentratesaround the set of corresponding Coulomb functionals of the maximizers in theuniform topology. This is a decisive step on the way to a rigorous proof of theconvergence of the normalized occupation measures towards an explicit mixtureof the maximizers, derived in [BKM15]. Our methods rely on derivingH{\"o}lder-continuity of the Coulomb functional of the occupation measure withexponentially small deviation probabilities and invoking the large deviationtheory developed in [MV14] to a certain shift-invariant functional of theoccupation measures.

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