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2020 Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles
Antoine Diez
Electron. J. Probab. 25: 1-38 (2020). DOI: 10.1214/20-EJP496

Abstract

In this article we study a system of $N$ particles, each of them being defined by the couple of a position (in $\mathbb {R}^{d}$) and a so-called orientation which is an element of a compact Riemannian manifold. This orientation can be seen as a generalisation of the velocity in Vicsek-type models such as [20, 16]. We will assume that the orientation of each particle follows a jump process whereas its position evolves deterministically between two jumps. The law of the jump depends on the position of the particle and the orientations of its neighbours. In the limit $N\to +\infty $, we first prove a propagation of chaos result which can be seen as an adaptation of the classical result on McKean-Vlasov systems [53] to Piecewise Deterministic Markov Processes (PDMP). As in [38], we then prove that under a proper rescaling with respect to $N$ of the interaction radius between the agents (moderate interaction), the law of the limiting mean-field system satisfies a BGK equation with localised interactions which has been studied as a model of collective behaviour in [14]. Finally, in the spatially homogeneous case, we give an alternative approach based on martingale arguments.

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Antoine Diez. "Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles." Electron. J. Probab. 25 1 - 38, 2020. https://doi.org/10.1214/20-EJP496

Information

Received: 1 August 2019; Accepted: 4 July 2020; Published: 2020
First available in Project Euclid: 6 August 2020

zbMATH: 1448.35507
MathSciNet: MR4136470
Digital Object Identifier: 10.1214/20-EJP496

Subjects:
Primary: 35Q70, 58J6, 60J25, 60J75, 60K35, 82C22

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