  The linearization of semiclassical theories of gravity is investigated in a
toy model, consisting of a quantum scalar field in interaction with a second
classical scalar field which plays the role of a classical background. This toy
model mimics also the evolution induced by semiclassical Einstein equations,
such as the one which describes the early universe in the cosmological case.
The equations governing the dynamics of linear perturbations around simple
exact solutions of this toy model are analyzed by constructing the
corresponding retarded fundamental solutions, and by discussing the
corresponding initial value problem. It is shown that, if the quantum field
which drives the back-reaction to the classical background is massive, then
there are choices of the renormalization parameters for which the linear
perturbations with compact spatial support decay polynomially in time for large
times, thus indicating stability of the underlying semiclassical solution.
