  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 [6] in terms of the {\it{Pekar variational
formula}}, which coincides with the behavior of the partition function of the
{\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14]
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
{\it{Pekar 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 [11], its extension to the uniform strong metric for the singular
Coulomb interaction carried out in [8], as well as an idea inspired by a
{\it{partial path exchange}} argument appearing in [1].
