Dynamics of stochastic Klein-Gordon-Schrödinger equations in unbounded domains. (English) Zbl 1240.60187

Summary: 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.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q55 NLS equations (nonlinear Schrödinger equations)
35R60 PDEs with randomness, stochastic partial differential equations