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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 thecase of the wave equation). Here $K$ represents both the $3$-dimensional torusor a bounded domain with smooth boundary in $\Bbb R^3.$ That allows to deducesome corollaries on the long time behaviour of the flow of the equation in aprobabilistic sense. We also establish qualitative properties of theconstructed measure. This work extends the Fluctuation-Dissipation-Limit (FDL)approach to PDEs having only one (coercive) conservation law.

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