Generic quantum Markov semigroups: the Fock case

We introduce the class of generic quantum Markov semigroups. Within this class we study the class corresponding to the Fock case which is further split into four sub-classes each of which contains both bounded and unbounded generators, depending on some global characteristics of the intensities of jumps. For the first two of these classes we find an explicit solution which reduces the problem of finding the quantum semigroup to the calculation of two classical semigroups, one of which is diagonal (in suitable basis) and the other one is triangular (in the same basis). In the bounded case our formula gives the unique solution. In the unbounded case it gives one solution, which we conjecture to be the minimal one.