Title: Diffusion in quantum system-reservoir models.

Abstract: 
We treat the following question: Consider a quantum particle on a
lattice coupled to a thermal reservoir consisting of free bosons. 
Common wisdom suggests that, if the temperature of the reservoir is
strictly positive, the particle diffuses. 
Up to now, we can prove this in two cases (both assuming that the
coupling between particle and reservoir is weak)
1) If we replace the thermal reservoir  by independent reservoirs at 
each lattice site.
2) If the space dimension is higher than 4, the particle is heavy and it
has some internal degrees of freedom (e.g. spin) which provide
additional coupling to the field.