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Mean-field limits beyond ordinary differential equations. (English) Zbl 1346.68217
Bernardo, Marco (ed.) et al., Formal methods for the quantitative evaluation of collective adaptive systems. 16th international school on formal methods for the design of computer, communication, and software systems, SFM 2016, Bertinoro, Italy, June 20–24, 2016. Advanced lectures. Cham: Springer (ISBN 978-3-319-34095-1/pbk; 978-3-319-34096-8/ebook). Lecture Notes in Computer Science 9700, 61-82 (2016).
Summary: We study the limiting behaviour of stochastic models of populations of interacting agents, as the number of agents goes to infinity. Classical mean-field results have established that this limiting behaviour is described by an ordinary differential equation (ODE) under two conditions: (1) that the dynamics is smooth; and (2) that the population is composed of a finite number of homogeneous sub-populations, each containing a large number of agents. This paper reviews recent work showing what happens if these conditions do not hold. In these cases, it is still possible to exhibit a limiting regime at the price of replacing the ODE by a more complex dynamical system. In the case of non-smooth or uncertain dynamics, the limiting regime is given by a differential inclusion. In the case of multiple population scales, the ODE is replaced by a stochastic hybrid automaton.
For the entire collection see [Zbl 1337.68006].

68T42 Agent technology and artificial intelligence
34A60 Ordinary differential inclusions
34F05 Ordinary differential equations and systems with randomness
60J28 Applications of continuous-time Markov processes on discrete state spaces
68Q45 Formal languages and automata
68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
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