Invariant measure for the Klein-Gordon equation in a non periodic setting

In this paper, we build a Gibbs measure for the 1d cubic Klein-Gordon equation on $\mathbb R$ with a decreasing non linearity, in the sense that the non linearity $f^3$ is multiplied by $\chi$ where $\chi$ is a sufficiently integrable non negative function. We prove that this equation is almost surely globally well-posed in $H^{1/2-}_{\textrm{loc}}$ with respect to this measure and that the measure is invariant under the flow of the equation.