The Wasserstein gradient flow of the Fisher information and the Quantum Drift-Diffusion equation

We prove the global existence of nonnegative variational solutions to the fourth order quantum ``drift diffusion\u27\u27 evolution equation under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, nonnegative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance between probability measures. We also study long time behaviour of the solutions, proving their exponential decay to the equilibrium state