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 partial derivative(t)u + div(uD(2 Delta root u/root u - f)) = 0 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 F-f (u) := 1/2 integral vertical bar D logu vertical bar(2) u dx + integral fu dx 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 g = e(-V) characterized by -Delta V + 1/2 vertical bar DV vertical bar(2) = f, integral e(-V) dx = integral u(0) dx, when the potential V is uniformly convex: in this case the functional F-f coincides with the relative Fisher information F-f (u) = 1/2 F(u vertical bar g) = integral vertical bar D log(u/g)vertical bar(2) u dx.