  We analyze the stability and decay of supercurrents of strongly interacting
bosons on optical lattices. At the mean field level, the system undergoes an
irreversible dynamic phase transition, whereby the current decays beyond a
critical phase gradient that depends on the interaction strength. At
commensurate filling the transition line smoothly interpolates between the
classical modulational instability of weakly interacting bosons and the
equilibrium Mott transition at zero current. Below the mean field instability,
the current can decay due to quantum and thermal phase slips. We derive
asymptotic expressions of the decay rate near the critical current. In a three
dimensional optical lattice this leads to very weak broadening of the
transition. In one and two dimensions the broadening leads to significant
current decay well below the mean field critical current. We show that the
temperature scale below which quantum phase slips dominate the decay of
supercurrents, is easily within experimental reach.
