The Semigeostrophic Equations Discretized in reference and dual variables

We study the evolution of a system of n particles {(xi, vi)} n i=1 in IR . That system is a conservative system with a Hamiltonian of the form H[μ] = W 2 2 (μ, ν n), where W2 is the Wasserstein distance and μ is a discrete measure concentrated on the set {(xi, vi)} n i=1. Typically, μ(0) is a discrete measure approximating an initial L density and can be chosen randomly. When d = 1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to Lebesgue measure. When {ν}n=1 converges to a measure concentrated on a special d–dimensional sets, we obtain the VlasovMonge-Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov-Poisson system.