We rigorously show the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. For instance, these models include nonlocal repulsive-attractive forces locally averaged over sharp vision cones and Cucker-Smale interactions with discontinuous communication weights. We construct global-in-time defined notion of solutions through a differential inclusion system corresponding to the particle descriptions. We estimate the error between the solutions to the differential inclusion system and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory. Quantitative bounds on the expansion of the 1-Wasserstein distance along flows based on a weak-strong stability estimate are obtained. We also provide different examples of realistic sensitivity sets satisfying the assumptions of our main results.

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocitydependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.

We consider a N -particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like N −δ with δ < 1/d in the force, we provide a quantitative error estimate between the empirical measure associated to that N -particle system and the solutions of the d-dimensional Vlasov-Poisson-Fokker-Planck system. We also study the propagation of chaos for the Vlasov-Fokker-Planck equation with less singular interaction forces than the Newtonian one.

We consider an interacting N -particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuitytype of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles N goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman [J. Funct. Anal., 56, (1984), 311-336] to that one with discontinuous kernels. 14 4. Propagation of chaos 15 4.1. Law of large numbers like estimates 15 4.2. Proof of Theorem 1.3 21 Appendix A. Gronwall-type inequalities 21 Appendix B. A representation for solutions to the PDE (1.4) 22 Acknowledgments 24 References 24 1. IntroductionMathematical modelling of collective behaviors, such as flocks of birds, schools of fish, or aggregation of bacteria, etc, has received a bulk of attention because of its possible applications in the field of engineering, biology, industry, and sociology [2,19,23,30,31]. These models are usually based on incorporating different mechanisms of interactions between individuals, for instance, a short-range repulsion, a longrange attraction, and an alignment in certain spatial regions. We refer the reader to [5,7,9] and the references therein for recent surveys of collective behavior models. In this current work, we consider Date: August 27, 2018.