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This work focusses on applications of classical probability theory, especially point process theory, to quantum stochastics. We consider a class of quantum Markov chains in the sense of ACCARDI on the basis of beam splitters and generalized splitting procedures. Time evolutions of boson systems described by these procedures are constructed. Furthermore, the following questions are discussed: Which initial states allow an explicit description of later states, especially of the corresponding position distribution? Which states are invariant under the considered dynamics? For which initial states one obtains convergence to an invariant state? The considered boson systems are described using a representation of the symmetric Fock space as £2-space with a not necessarily atomless measure. So, the case of bosons on a lattice is also included.