The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.
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Anton Arnold, José L. López, Peter A. Markowich, Juan Soler, An analysis of quantum Fokker–Planck models: A Wigner function approach. Rev. Mat. Iberoam. 20 (2004), no. 3, pp. 771–814