Collective behavior models with vision geometrical constraints: Truncated noises and propagation of chaos

Abstract

We consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains. We rigorously derive a Vlasov-Fokker-Planck type of kinetic mean-field equation from the corresponding stochastic integral inclusion system. More specifically, we construct a global-in-time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1-Wasserstein distance.where R ∈ C 2 (R d ) is a truncation function compactly supported in B d Vm := {x ∈ R d : |x| < V m }. Here X i t and V i t are position and velocity of i-th particle at time t ≥ 0, respectively, and F [µ] denotes

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

“…The propagation of chaos and its consequence, the rigorous proof of the mean-field limit, are questions of nowadays importance in many other problems; for instance, in kinetic models of collective behavior [1,4,6,9,10,11] with or without noise in velocity, see also [12,28] for more general types of equations. However, in most of these applications the singularity of the kernels is much better behaved.…”

“…For the 2D Navier-Stokes equation in vortex formulation, the propagation of chaos is obtained in [14] by compactness, thanks to result of [20]. Similar techniques have been applied to the case of the diffusion dominated 2D Keller-Segel equations in [16,32] and geometrical constraints interactions with reflecting diffusions in [9].…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…Different approaches to the derivation of the Vlasov-like kinetic equations with alignments/interaction terms or the aggregation equations have been taken leading to a very lively interaction between different communities of researchers in analysis and probability. We refer to [3,4,10,20,30,31,35,44,47,50,[54][55][56]64,67] for the classical references and non-Lipschitz regularity velocity fields in kinetic cases, to [48,49] for very related incompressible fluid problems, and to [7,9,16,17,37,43,45,51,52,61,63,65,66] for results with more emphasis on the singular interaction kernels both at the kinetic and the aggregation-diffusion equation cases.…”

Mean-field limits: from particle descriptions to macroscopic equations

“…Different approaches to the derivation of the Vlasov-like kinetic equations with alignments/interaction terms or the aggregation equations have been taken leading to a very lively interaction between different communities of researchers in analysis and probability. We refer to [3,31,57,60,39,42,17,2,7,26,27,45,49,50,51] for the classical references and non-Lipschitz regularity velocity fields in kinetic cases, to [44,43] for very related incompressible fluid problems, and to [46,14,6,38,47,33,40,56,58,13,59,4] for results with more emphasis on the singular interaction kernels both at the kinetic and the aggregation-diffusion equation cases.…”