  We study long-range interacting systems driven by external stochastic forces
that act collectively on all the particles constituting the system. Such a
scenario is frequently encountered in the context of plasmas, self-gravitating
systems, two-dimensional turbulence, and also in a broad class of other
systems. Under the effect of stochastic driving, the system reaches a
stationary state where external forces balance dissipation on average. These
states have the invariant probability that does not respect detailed balance,
and are characterized by non-vanishing currents of conserved quantities. In
order to analyze spatially homogeneous stationary states, we develop a kinetic
approach that generalizes the one known for deterministic long-range systems;
we obtain a very good agreement between predictions from kinetic theory and
extensive numerical simulations. Our approach may also be generalized to
describe spatially inhomogeneous stationary states. We also report on numerical
simulations exhibiting a first-order nonequilibrium phase transition from
homogeneous to inhomogeneous states. Close to the phase transition, the system
shows bistable behavior between the two states, with a mean residence time that
diverges as an exponential in the inverse of the strength of the external
stochastic forces, in the limit of low values of such forces.
