Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations

Abstract

In this article, the authors investigate the system of Schr odinger and Klein-Gordon equations with Yukawa coupling. They rst prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satis es a Liouville type theorem and is indeed a statistical solution for the coupling equations. Further, they reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem.