  We consider the effect of replacing in stochastic differential equations
leading to the dynamical collapse of the statevector, white noise stochastic
processes with non white ones. We prove that such a modification can be
consistently performed without altering the most interesting features of the
previous models. One of the reasons to discuss this matter derives from the
desire of being allowed to deal with physical stochastic fields, such as the
gravitational one, which cannot give rise to white noises. From our point of
view the most relevant motivation for the approach we propose here derives from
the fact that in relativistic models the occurrence of white noises is the main
responsible for the appearance of untractable divergences. Therefore, one can
hope that resorting to non white noises one can overcome such a difficulty. We
investigate stochastic equations with non white noises, we discuss their
reduction properties and their physical implications. Our analysis has a
precise interest not only for the above mentioned subject but also for the
general study of dissipative systems and decoherence.
