  The influence of the noise on the long-time ageing dynamics of a quenched
ferromagnetic spin system with a non-conserved order parameter and described
through a Langevin equation with a thermal noise term and a disordered initial
state is studied. If the noiseless part of the system is Galilei-invariant and
scale-invariant with dynamical exponent z=2, the two-time linear response
function is independent of the noise and therefore has exactly the form
predicted from the local scale-invariance of the noiseless part. The two-time
correlation function is exactly given in terms of certain noiseless three- and
four-point response functions. An explicit scaling form of the two-time
autocorrelation function follows. For disordered initial states, local
scale-invariance is sufficient for the equality of the autocorrelation and
autoresponse exponents in phase-ordering kinetics. The results for the scaling
functions are confirmed through tests in the kinetic spherical model, the
spin-wave approximation of the XY model, the critical voter model and the free
random walk.
