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Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

  • *Corresponding author: Antoine Diez

    *Corresponding author: Antoine Diez

The work of AD is supported by an EPSRC-Roth scholarship cofunded by the Engineering and Physical Sciences Research Council and the Department of Mathematics at Imperial College London

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  • The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

    Mathematics Subject Classification: Primary: 82C22, 82C40, 35Q70; Secondary: 65C35, 92-10.


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  • Figure 1.  An interaction graph. The vertical axis represents time. Each particle is represented by a vertical line parallel to the time axis. The index of a given particle is written on the horizontal axis. The construction is done backward in time starting from time $ t $ where only particle $ i $ is present. At each time $ t_\ell $, if $ i_\ell $ does not already belong to the graph, it is added on the right (with a vertical line which starts at $ t_\ell $). The couple $ r_\ell = (i_\ell,j_\ell) $ of interacting particles at time $ t_\ell $ is depicted by an horizontal line joining two big black dots on the vertical line representing the particles $ i_\ell $ and $ j_\ell $. for instance, on the depicted graph, $ r_2 = (i_2,i) $. Note that at time $ t_3 $, $ r_3 = (i_1,i_2) $ (or indifferently $ r_3 = (i_2,i_1) $) where $ i_1 $ and $ i_2 $ were already in the system. Index $ i_3 $ is skipped and at time $ t_4 $, the route is $ r_4 = (i_4,i_1) $. The recollision occurring at time $ t_3 $ is depicted in red

    Figure 2.  A route of size $ q $ between $ i $ and $ j $. The chain of interactions which links $ i $ and $ j $ are depicted by horizontal lines as explained in Section 2.3.5

    Figure 3.  At time $ t_1^+ $, there is only one particle, at the same position for the BBGKY (in white) and Boltzmann (in red) pseudo-trajectories. At time $ t_1^+ $, a particle (dotted) is added next to the white particle in a pre-collisional way. At time $ t_2^+ $, a particle (dashed) is added next to the dotted particle in a post-collisional way. Due to a recollision at time $ \tau\in(t_3,t_2) $, the Boltzmann and BBGKY pseudo-trajectories of the white/red particle are no longer close to each other at time $ t_3 $

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