Document - Mean-field limits beyond ordinary differential equations

2016

Conference article Open Access

Mean-field limits beyond ordinary differential equations

Bortolussi L., Gast N.

Markov chain [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] Differential inclusions Dif-ferential inclusions [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI] [MATH]Mathematics [math] [INFO.INFO-PF]Computer Science [cs]/Performance [cs.PF] Hybrid systems Population models Mean-field limits [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]

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.

Source: 16th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2016, pp. 61–82, Bertinoro, Italy, 20-24 June, 2016

Publisher: Springer, Berlin , Germania

Citations

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[6] L. Bortolussi and N. Gast. \Mean Field Approximation of Imprecise Population Processes". In: QUANTICOL Technical Report TR-QC-07-2015 (2015).

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BibTeX entry

@inproceedings{oai:it.cnr:prodotti:424322,
	title = {Mean-field limits beyond ordinary differential equations},
	author = {Bortolussi L. and Gast N.},
	publisher = {Springer, Berlin , Germania},
	doi = {10.1007/978-3-319-34096-8_3},
	booktitle = {16th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2016, pp. 61–82, Bertinoro, Italy, 20-24 June, 2016},
	year = {2016}
}