Limits for Stochastic Reaction

"Reaction systems are mathematical models used in biochemistry and in a number of other fields. They model the evolution of a biochemical mechanism either deterministically, by means of a system of ODEs, or stochastically, by means of a continuous-time Markov chain. My main concern is the stochastic modelling regime: I study asymptotic results for the associated Markov chain when some parameters of the
model tend to infinity, in order to approximate the model by a simpler one or
to recover some features from its deterministic counterpart. I also study the
stationary distributions of the model. Specifically, in the first paper collected in this manuscript the original stochastic model is approximated by a lower dimensional one, where particular chemical species have been eliminated.
We study different kind of convergence of the reduced model to the original
one, and a similar result for the deterministic model is also proved. In the
second paper connections between the form of the stationary distribution of a
stochastic system and structural conditions of the underlying chemical
reactions are unveiled. Finally, in the third paper it is proved that if a
chemical species has always the same value for any positive steady state of a deterministic reaction system, then the counts of that species in the stochastic model
are, on average and up to a finite time T, near to that value. This results holds when the counts of the other species in the initial condition tend to infinity: in such a
situation the production and degradation rates tend to infinity as well, and the evolution of the different species counts is not clear a priori."

Supervisor: Prof. Carsten Wiuf, Math, University of Copenhagen,

Assessment committee:

Prof. Niels Richard (chairman), MATH, University of Copenhagen

Prof. János Tóth, Budapost University of Technology and Economics

Prof.  Peter Phaffelhuber, Universität Freiburg