  We study the transformed path measure arising from the self-interaction of a
three-dimensional Brownian 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 [DV83-P] in terms of a variational formula. Recently [MV14] 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 [BKM15]. Our methods rely on deriving
H{\"o}lder-continuity of the Coulomb functional of the occupation measure with
exponentially small deviation probabilities and invoking the large deviation
theory developed in [MV14] to a certain shift-invariant functional of the
occupation measures.
