QUANTUM STOCHASTIC, POSITIVE EVOLUTIONS - CHARACTERIZATION, CONSTRUCTION, DILATION

A characterization of the unbounded stochastic generators of quantum c ompletely positive flows is given. This suggests the general form of q uantum stochastic adapted evolutions with respect to the Wiener (diffu sion), Poisson (jumps), or general Quantum Noise. The corresponding ir reversible Heisenberg evolution in terms of stochastic completely posi tive (CP) maps is constructed. The general form and the dilation of th e stochastic completely dissipative (CD) equation over the algebra L ( H) is discovered, as well as the unitary quantum stochastic dilation o f the subfiltering and contractive flows with unbounded generators. A unitary quantum stochastic cocycle,dilating the subfiltering CP flows over L (H), is reconstructed.