Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

  • Mouhamadou SyEmail author


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.


Klein–Gordon equation Wave equation Invariant measure Fluctuation–dissipation Inviscid limit 

Mathematics Subject Classification

28D05 60H30 35B40 35L05 35L71 



I thank Armen Shirikyan, Nikolay Tzvetkov and Laurent Thomann for useful discussions and valuable remarks. I thank referees as well for remarks that have improved the text. This research was supported by the program DIM RDMath of FSMP and Région Ile-de-France.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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