Weak homogenization of point processes by space deformations

We study the transformation of a non-stationary point process xi on R-n int o a weakly stationary point process <(<xi>)over tilde>, with <(<xi>)over ti lde>(B) = xi(Phi (-1) (B)), where B is a Borel set, via a deformation Phi o f the space R-n. When the second-order measure is regular, Phi is uniquely determined by the homogenization equations of the second-order measure. In contrast, the first-order homogenization transformation is not unique. Seve ral examples of point processes and transformations are investigated with a particular interest to Poisson processes.