  We investigate a system of Brownian particles weakly bound by attractive
parity-symmetric potentials that grow at large distances as $V(x) \sim
|x|^\alpha$, with $0 < \alpha < 1$. The probability density function $P(x,t)$
at long times reaches the Boltzmann-Gibbs equilibrium state, with all moments
finite. However, the system's relaxation is not exponential, as is usual for a
confining system with a well-defined equilibrium, but instead follows a
stretched exponential $e^{- \mathrm{const} \, t^\nu}$ with exponent
$\nu=\alpha/(2+\alpha)$. This problem is studied from three perspectives.
First, we propose a straightforward and general scaling rate-function solution
for $P(x,t)$. This rate-function, which is an important tool from large
deviation theory, also displays anomalous time scaling and a dynamical phase
transition. Second, through the eigenfunctions of the Fokker-Planck operator,
we obtain, using the WKB method, more complete solutions that reproduce the
rate function approach. Finally, we show how the alternative path-integral
formalism allows us to recover the same results, with the above rate-function
being the solution of the classical Hamilton-Jacobi equation describing the
most probable path. Properties such as parity, the role of initial conditions,
and the dynamical phase transition are thoroughly studied in all three
approaches.
