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### Open Access Journals

NHM

Differential equation approximations of stochastic network processes: An operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting
from the differential equation point of view the stochastic model is identified by its master
equation, which is a system of linear ODEs with large state space size ($N$). We derive a single
non-linear ODE (called mean-field approximation) for the expected value that yields a good
approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof
holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for
Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving
convergence to the mean-field limit is proposed.

NHM

We study a transport equation in a network and control it in a
single vertex. We describe all possible reachable states and prove a
criterion of Kalman type for those vertices in which the problem is
maximally controllable. The results are then applied to concrete
networks to show the complexity of the problem.

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