A deterministic displacement theorem for Poisson processes

We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an $n$-body evolution, the mean measure $P$ satisfies a nonlinear Vlasov type equation $\dot {P} + y \cdot \nabla _x P - \nabla _y \cdot E(P) = 0$. Combined with Bartlett’s theorem, the result generalizes to interacting Brownian particles, where the mean measure satisfies a McKean-Vlasov type diffusion equation $\dot {P} + y \cdot \nabla _x P-\nabla _y \cdot E(P)- c \Delta P=0$.