  Recent models formulated by Kafri, Taylor, and Milburn and by Tilloy and
Diosi describe the gravitational interaction through a continuous measurement
and feedback protocol. In such a way, although gravity is ultimately treated as
classical, they can reconstruct the proper quantum gravitational interaction at
the level of the master equation for the statistical operator. Following this
procedure, the price to pay is the presence of decoherence effects leading to
an asymptotic energy divergence. One does not expect the latter in isolated
systems. Here, we propose a dissipative generalization of these models. We show
that, in these generalizations, in the long time limit, the system thermalizes
to an effective finite temperature.
