  We study the dynamics of a strongly interacting bosonic quantum gas in an
optical lattice potential under the effect of a dissipative environment. We
show that the interplay between the dissipative process and the Hamiltonian
evolution leads to an unconventional dynamical behavior of local number
fluctuations. In particular we show, both analytically and numerically, the
emergence of an anomalous diffusive evolution in configuration space at short
times and, at long times, an unconventional dynamics dominated by rare events.
Such rare events, common in disordered and frustrated systems, are due here to
strong interactions. This complex two-stage dynamics reveals information on the
level structure of the strongly interacting gas.
