On global solutions to some non-Markovian quantum kinetic models of Fokker-Planck type

Abstract: In this paper, global well-posedness of the non-Markovian Unruh-Zurek and Hu-Paz-Zhang master equations with nonlinear electrostatic coupling is demonstrated. They both consist of a Wigner-Poisson like equation subjected to a dissipative Fokker-Planck mechanism with time-dependent coefficients of integral type, which makes necessary to take into account the full history of the open quantum system under consideration to describe its present state. From a mathematical viewpoint this feature makes particularly elaborated the calculation of the propagators that take part of the corresponding mild formulations, as well as produces rather strong decays near the initial time ($t=0$) of the magnitudes involved, which would be reflected in the subsequent derivation of a priori estimates and a significant lack of Sobolev regularity when compared with their Markovian counterparts. The existence of local-in-time solutions is deduced from a Banach fixed point argument, while global solvability follows from appropriate kinetic energy estimates.