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

In this article, the authors investigate the system of Schrodinger and Klein-Gordon equations with Yukawa coupling. They first prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satisfies 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. (C) 2021 Elsevier Inc. All rights reserved.