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A characterization of the unbounded stochastic generators of quantumcompletely positive flows is given. This suggests the general form of quantumstochastic adapted evolutions with respect to the Wiener (diffusion), Poisson(jumps), or general Quantum Noise. The corresponding irreversible Heisenbergevolution 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 quantumstochastic dilation of the subfiltering and contractive flows with unboundedgenerators. A unitary quantum stochastic cocycle, dilating the subfiltering CPflows over L(H), is reconstructed.Comment: 33 page

Quantum Stochastic Positive Evolutions:¶Characterization, Construction, Dilation