Stochastic Cucker-Smale flocking dynamics of jump-type

Martin Friesen; Oleksandr Kutoviy

doi:10.3934/krm.2020008

Article Contents

2020, Volume 13, Issue 2: 211-247. Doi: 10.3934/krm.2020008

This issue Previous Article Next Article On Fokker-Planck equations with In- and Outflow of Mass

Stochastic Cucker-Smale flocking dynamics of jump-type

Martin Friesen^1, , and
Oleksandr Kutoviy^2,

1.
School of Mathematics and Natural Sciences, University of Wuppertal, Gauẞstraẞe 20, 42119 Wuppertal, Germany
2.
Department of Mathematics, Bielefeld University, Universitätsstraẞe 25, 33615 Bielefeld, Germany

^* Corresponding author: Martin Friesen
^* Corresponding author: Martin Friesen

Received: May 31, 2018

Revised: June 30, 2019

Published: January 2020

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Abstract

We present a stochastic version of the Cucker-Smale flocking dynamics described by a system of $ N $ interacting particles. The velocity aligment of particles is purely discontinuous with unbounded and, in general, non-Lipschitz continuous interaction rates. Performing the mean-field limit as $ N \to \infty $ we identify the limiting process with a solution to a nonlinear martingale problem associated with a McKean-Vlasov stochastic equation with jumps. Moreover, we show uniqueness and stability for the kinetic equation by estimating its solutions in the total variation and Wasserstein distance. Finally, we prove uniqueness in law for the McKean-Vlasov equation, i.e. we establish propagation of chaos.

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Mathematics Subject Classification: Primary: 35Q83, 60F05; Secondary: 60K35.

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Figure 1. Change of velocities with $ \eta = 3/4 $

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