  This paper is devoted to the study of the dynamics of two weakly-coupled
Bose-Einstein condensates confined in a double-well trap and perturbed by
random external forces. Energy diffusion due to random forcing allows the
system to visit symmetry-breaking states when the number of atoms exceeds a
threshold value. The energy distribution evolves to a stationary distribution
which depends on the initial state of the condensate only through the total
number of atoms. This loss of memory of the initial conditions allows a simple
and complete description of the stationary dynamics of the condensate which
randomly visits symmetric and symmetry-breaking states.
