  We discuss the non-equilibrium dynamics of a Quantum Ising Chain (QIC)
following a quantum quench of the transverse field and in the presence of a
gaussian time dependent noise. We discuss the probability distribution of the
work done on the system both for static and dynamic noise. While the effect of
static noise is to smooth the low energy threshold of the statistic of the
work, appearing for sudden quenches, a dynamical noise protocol affects also
the spectral weight of such features. We also provide a detailed derivation of
the kinetic equation for the Green's functions on the Keldysh contour and the
time evolution of observables of physical interest, extending previously
reported results (J. Marino, A. Silva, Phys. Rev. B 86, 060408 (2012)), and
discussing the extension of the concept of prethermalization which can be used
to study noisy quantum many body hamiltonians driven out-of-equilibrium.
