Propagation of chaos for topological interactions by a coupling technique
Pierre Degond
Université de Toulouse, FranceMario Pulvirenti
Università di Roma ‘La Sapienza’, ItalyStefano Rossi
Università di Roma ‘La Sapienza’, Italy
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Abstract
We consider a system of particles that interact through a jump process. The jump intensities are functions of the proximity rank of the particles, a type of interaction referred to as topological in the literature. Such interactions have been shown relevant for the modelling of bird flocks. We show that, in the large number of particles limit and under minimal smoothness assumptions on the data, the model converges to a kinetic equation which was derived in earlier works both formally and rigorously under more stringent regularity assumptions. The proof relies on the coupling method which assigns to the particle and limiting processes a joint process posed on the cartesian product of the two configuration spaces of the former processes. By appropriate estimates in a suitable Wasserstein metric, we show that the distance between the two processes tends to zero as the number of particles tends to infinity, with an error typical of the law of large numbers.
Cite this article
Pierre Degond, Mario Pulvirenti, Stefano Rossi, Propagation of chaos for topological interactions by a coupling technique. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 34 (2023), no. 3, pp. 641–655
DOI 10.4171/RLM/1022