The Wasserstein gradient flow of the Fisher information and the Quantum drift-diffusion equation

We prove the global existence of non-negative variational solutions to the “drift diffusion” evolution equation under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, non-negative 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 behavior of the solutions, proving their exponential decay to the equilibrium state when the potential is uniformly convex.