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Piecewise deterministic Markov processes: an analytic approach

Promotor: S.M. Verduyn Lunel, Co-Promotores: S.C.Hille, O.W. van Gaans

Taleb Alkurdi
29 augustus 2013
Thesis in Leiden Repository

The subject of this thesis, piecewise deterministic Markov processes, an analytic approach, is on the border between analysis and probability theory. Such processes can either be viewed as random perturbations of deterministic dynamical systems in an impulsive fashion, or as a particular kind of stochastic process in continuous time in which parts of the sample trajectories are deterministic. Accordingly, questions concerning theses processes may be approached starting from either side. The approach we used is an analytic one flavoured by probability, as the title says. The topic we study in this thesis is the long--term behaviour of a system that results from a particular type of stochastic perturbation of a deterministic dynamical system. As examples of such systems as they appear naturally in the modelling of particular real world phenomena, one can think of harvesting from a growing bacterial colony, and a ma rine--ecosystem from which fish are harvested with a net. In experimental set--ups where one grows bacteria or fungi like baker's yeast in a fermenter, or plant cells in a cell suspension culture, one regularly draws small samples for inspection as part of the experimental protocol. These interventions will interfer with further growth of the population. The question is to quantify how small a sample can be and how `much' this sampling process will disturb the dynamics of the population.

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