Dynamics of stochastic Klein--Gordon--Schrödinger equations in unbounded domains

Abstract

The long-time behavior in the sense of distributions for stochastic Klein--Gordon--Schrödinger equations in the whole space ${\mathbb R}^n$, $1\leq n\leq 3$, is studied. First the existence of one stationary measure from any moment-finite initial data in the space $H^1({\mathbb R}^n)\times H^1({\mathbb R}^n) \times L^2({\mathbb R}^n)$ is proved and then a global measure attractor is constructed in the space consisting of probability measures supported on $H^2({\mathbb R}^n)\times H^2({\mathbb R}^n)\times H^1({\mathbb R}^n)$. Because of the lack of compact embedding, some a priori estimates and a split of solutions play important roles in the approach.