Existence and regularitystudy for two-dimensional Kac equation without cutoff by a probabilistic approach

We consider a two-dimensional Kac equation without cutoff,which we relate to a stochastic differential equation.We prove the existence of a solution for this SDE, and we use the Malliavin calculus (or stochastic calculus of variations) to prove that the law of this solution admits a smooth density with respect to the Lebesgue measure on R2.This density satisfies the Kac equation.