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This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix the sets from which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for: the mean field game of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.
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Charles Bertucci, Fokker-Planck equations of jumping particles and mean field games of impulse control. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 5, pp. 1211–1244DOI 10.1016/J.ANIHPC.2020.04.006