Local Stationarity for the Klein—Gordon Equations

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