Existence and regularity study 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 R-2. This density satisfies the Ka c equation.