In this paper we construct an invariant probability measure concentrated on \(H^2(K)\times H^1(K)\) for a general cubic Klein–Gordon equation (including the case of the wave equation). Here K represents both the 3-dimensional torus or a bounded domain with smooth boundary in \({\mathbb {R}}^3\). That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the fluctuation–dissipation-limit approach to PDEs having only one (coercive) conservation law.

中文翻译：

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三维Klein-Gordon方程在3D中的不变测度和大时间动力学

在本文中，我们针对一般三次Klein-Gordon方程（包括波动方程的情况）构造了集中于\（H ^ 2（K）\乘以H ^ 1（K）\）的不变概率测度。在此，K表示3维圆环或\（{\ mathbb {R}} ^ 3 \）中具有平滑边界的有界域。这样就可以从概率意义上推导出方程流的长期行为的某些推论。我们还建立了所构建度量的定性性质。这项工作将波动-耗散极限方法扩展到仅具有一个（强制性）守恒定律的PDE。