Calculus of Variations and Geometric Measure Theory

U. Gianazza - G. Savaré - G. Toscani

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

created by savare on 01 Sep 2006
modified on 25 Feb 2009


Accepted Paper

Inserted: 1 sep 2006
Last Updated: 25 feb 2009

Journal: Arch. Ration. Mech. Anal.
Pages: 1-68
Year: 2006


We prove the global existence of nonnegative variational solutions to the ``drift diffusion'' 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 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 in many important cases.

Keywords: Optimal transport, Fisher information, quantum drift diffusion, Wasserstein distance, nonnegative solutions, fourth order evolution equations, entropy, log-concave measures