## A deterministic displacement theorem for Poisson processes.(English)Zbl 0887.58018

Summary: 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 $\dot{P} + y \cdot \nabla_x P-\nabla_y \cdot E(P)- c \Delta P=0.$

### MSC:

 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 82C22 Interacting particle systems in time-dependent statistical mechanics 70H05 Hamilton’s equations 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J60 Diffusion processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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### References:

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