Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations
The authors study a Kac-type equation without cutoff with which they associate a nonlinear stochastic differential equation. The stochastic calculus of variations allows them to prove, inspired in a previous paper by Jacod and Bichteler, that the law of the solution to the aforementioned stochastic differential equation has a density with respect to Lebesgue measure, which is a solution to the Kac equation. They allow the initial law to be rather general by demanding only the existence of its second moment. This leads to a significant improvement of previous results on the existence of solutions to Kac-type equations. Moreover, they show that better results on the smoothness of the density are obtained if all the moments exist for the initial law.
scientific article; zbMATH DE number 1379887