  We analyse biased ensembles of trajectories for the random-field Ising model
on a fully-connected lattice, which is described exactly by mean-field theory.
By coupling the activity of the system to a dynamical biasing field, we find a
range of dynamical phase transitions, including spontaneous symmetry breaking
into ordered states. For weak bias, the phase behaviour is controlled by
extrema of the free energy, which may be local minima or saddle points. For
large bias, the system tends to states of extremal activity, which may differ
strongly from free energy minima. We discuss connections of these results to
random first-order transition theory of glasses, which motivates an extension
of the analysis to random-field Ising models where the dynamical activity is
not symmetric under magnetisation reversal.
