Local Stationarity for the Klein—Gordon Equations

We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε at large distance. For any τ ≠ 0 $${r\in {\mathbb{R}}}^{d}$$ we distributions solution time moments t = τ/ε points close r/ε. study asymptotics these as → derive energy transport equation.