  Stretched-exponential relaxation is a widely observed phenomenon found in
glassy systems. It was previously modeled with non-Markovian dynamics
reflecting a memory effect. Here, we study a Brownian particle under the
influence of a confining, albeit weak, potential field that grows with distance
as a sub-linear power law. We find that for this memoryless model, observables
display stretched-exponential relaxation. The probability density function of
the system is studied using a rate function ansatz. We obtain analytically the
stretched-exponential exponent along with an anomalous power-law scaling of
length with time. The rate function exhibits a point of nonanalyticity,
indicating a dynamical phase transition. In particular, the rate function is
double-valued both to the left and right of this point, leading to four
different rate functions, depending on the choice of initial conditions and
symmetry.

Statistical Mechanics