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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.

MSC:

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