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We consider the GNS Hilbert space $\mathcal{H}$ of a uniformly hyper-finite$C^*$- algebra and study a class of unbounded Lindbladian arises fromcommutators. Exploring the local structure of UHF algebra, we have shown thatthe associated Hudson-Parthasarathy type quantum stochastic differentialequation admits a unitary solution. The vacuum expectation of homomorphicco-cycle, implemented by the Hudson-Parthasarathy flow, is conservative andgives the minimal semi-group associated with the formal Lindbladian. We alsoassociate conservative minimal semi-groups to another class of Lindbladian bysolving the corresponding Evan-Hudson equation.

Some Quantum Dynamical Semi-groups with Quantum Stochastic Dilation