This thesis is devoted to the study of the asymptotic of grazing collisions for Kac's and Boltzmann's equations and to the study of the chaos propagation for some sub-critical Keller-Segel equation with non-linear Stochastic Differentials Equations. The first chapter is devoted to the Kac equation with a Maxwellian potential. We start by giving an explicit rate of convergence (than we believe to be optimal) for the asymptotic of grazing collisions. Then, we approximate the solution of Kac's equation in the general case, which allows us to show the chaos propagation for some particle system to this last one in a quantitative way. In the second chapter, we study the asymptotic of grazing collisions for the Boltzmann equation with soft and Coulomb potentials. We also give explicit rates of convergence (which are not optimal).Finally in the third and last chapter, we show the chaos propagation for some sub-critical Keller-Segel equation. To this aim, we use compactness arguments (tightness of the particle system)
Contribution to the study of Boltzmann's, Kac's and Keller-Segel's equations with non-linear stochastic differentials equations