Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles

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 [Degond et al. 2017; Degond, Motsch 2008]. 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 [Sznitman 1991] to Piecewise Deterministic Markov Processes (PDMP). As in [Jourdain, M\'el\'eard 1998], 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 [Degond et al. 2018]. Finally, in the spatially homogeneous case, we give an alternative approach based on martingale arguments.