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 $${{\partial_t} u+ div \left(u{\mathrm{D}}\left(2 \frac{\Delta \sqrt u}{\sqrt u}-{f}\right)\right)=0}$$ 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 $${\fancyscript F^f(u):=\frac 12\int \left|{\mathrm{D}} \log u\right|^2 {u} dx+\int fu dx}$$ 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 $${-\Delta V+\frac12 \left|{\mathrm{D}} V\right|^2=f,\quad \int {\rm e}^{-V} dx=\int u_{0}dx,}$$ when the potential V is uniformly convex: in this case the functional $${\fancyscript F^f}$$ coincides with the relative Fisher information$${\fancyscript F^f(u)=\frac12\fancyscript I(u|g)= \int \left|{\mathrm{D}}\log(u/g)\right|^2u dx}$$.