  We study the dynamics of thermalization resulting from a time-dependent noise
in a Quantum Ising Chain subject to a sudden quench of the transverse magnetic
field. For weak noise the dynamics shows a pre-thermalized state at
intermediate time scales, eventually drifting towards an asymptotic infinite
temperature steady state characterized by diffusive behavior. By computing
analytically the density of kinks, as well as the transverse and longitudinal
magnetic field correlators, we characterize these two regimes, their
observability and their signatures in the various physical quantities.
