  We investigate the overdamped Langevin motion for particles in a potential
well that is asymptotically flat. When the potential well is deep compared to
temperature, physical observables like the mean square displacement are
essentially time-independent over a long time interval, the stagnation epoch.
However the standard Boltzmann-Gibbs (BG) distribution is non-normalizable,
given that the usual partition function is divergent. For this regime, we have
previously shown that a regularization of BG statistics allows the prediction
of the values of dynamical and thermodynamical observables in the
non-normalizable quasi-equilibrium state. In this work, based on the
eigenfunction expansion of the time-dependent solution of the associated
Fokker-Planck equation with free boundary conditions, we obtain an approximate
time-independent solution of the BG form, valid for times which are long, but
still short compared to the exponentially large escape time. The escaped
particles follow a general free-particle statistics, where the solution is a an
error function, shifted due to the initial struggle to overcome the potential
well. With the eigenfunction solution of the Fokker-Planck equation in hand, we
show the validity of the regularized BG statistics and how it perfectly
describes the time-independent regime though the quasi-stationary state is
non-normalizable.
