Existence and Regularity of a Solution of a Kac Equation Without Cutoff Using the Stochastic Calculus of Variations

Abstract

We consider a generalized Kac equation without cutoff, with which we associate a non-standard nonlinear stochastic differential equation. We adapt the techniques in Bichteler and Jacod [2] to prove that the law of a solution of the stochastic differential equation has a density, which is a solution of the Kac equation. The initial law is very general: we only assume it has second order moments and is not the Dirac mass at 0. We thus generalize the analytical results of existence of a solution of this equation. If we furthermore assume existence of all moments for the initial law, we obtain as a corollary using the proof in Desvillettes [6] that the density is smooth. We prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations, adapting the methods in [1].