Mean-Field Limits Beyond Ordinary Differential Equations

Abstract : 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.
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Submitted on : Monday, June 20, 2016 - 6:04:28 PM
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Luca Bortolussi, Nicolas Gast. Mean-Field Limits Beyond Ordinary Differential Equations. Bernardo, Marco; De Nicola, Rocco; Hillston, Jane. Formal Methods for the Quantitative Evaluation of Collective Adaptive Systems, Programming and Software Engineering, 2016, 978-3-319-34095-1. ⟨10.1007/978-3-319-34096-8_3⟩. ⟨hal-01334358⟩

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