Beschreibung
Birth-and-death processes with killing are a special class of Markov processes; they allow stochastic modelling of the size of “populations” in view of the respective probabilities that an individuum is born after a certain time – corresponding to a birth rate – and that an individuum dies – corresponding to a death rate. The additional structure of killing accounts for the possibility of immediate extinction of the population at hand – corresponding to a so-called killing rate; all the rates solely depend on the population size, not on the time of evolution. Many populations and systems may be modelled in this way. The fundamental guideline in this thesis is a spectral representation for the transition probabilities of such population processes: they can be related explicitly to a system of orthogonal polynomials (OP) with an appropriate spectral measure. In principle, given that the OP and their measure can be computed, one can derive quite explicit results on the expected behavior of the corresponding population; e.g. for processes with linear rates, being important in several genetic models for instance. Problematic is the following aspect: as soon as the rates are sufficiently complicated, the OP and their spectral measure may no more be computed explicitly. The main tool utilized here is regular perturbation theory for corresponding linear Jacobi operators, being especially useful under a certain domination of killing. In fact, such “killing dominated” birth-and-death processes allow to derive good approximations and criteria for their spectra.