  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 $\Bbb R^3.$ 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 (FDL)
approach to PDEs having only one (coercive) conservation law.

Analysis of PDEs