Quantum stochastic, positive evolutions: Characterization, construction, dilation

A characterization of the unbounded stochastic generators of quantum completely positive flows is given. This suggests the general form of quantum stochastic adapted evolutions with respect to the Wiener (diffusion), Poisson (jumps), or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is constructed. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L (H) is discovered, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators. A unitary quantum stochastic cocycle,dilating the subfiltering CP flows over L (H), is reconstructed.