I present recent results obtained in collaboration with J.
Froehlich and W. de Roeck on quantum Brownian motion. We consider a quantum
particle on the lattice weakly coupled to a spatial array of independent
non-interacting reservoirs in thermal states (heat baths). We prove that the
motion of the particle is diffusive at large times. If in addition the
particle is driven by a weak external force field, we show that the motion
of the particle is diffusive around a mean ballistic motion with constant
velocity proportional to the external force field. Moreover, we prove that
the Einstein relation (or Green-Kubo formula) holds, linking the mobility of
the particle with the diffusion constant at vanishing external force.