Archive for Rational Mechanics and Analysis

, Volume 194, Issue 1, pp 133–220 | Cite as

The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation

  • Ugo Gianazza
  • Giuseppe SavaréEmail author
  • Giuseppe Toscani


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 = eV 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\({\fancyscript F^f(u)=\frac12\fancyscript I(u|g)= \int \left|{\mathrm{D}}\log(u/g)\right|^2u dx}\).


Fisher Information Lower Semicontinuity Relative Entropy Planck Equation Logarithmic Sobolev Inequality 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Ugo Gianazza
    • 1
  • Giuseppe Savaré
    • 1
    Email author
  • Giuseppe Toscani
    • 1
  1. 1.Dipartimento di Matematica “F. Casorati”Universita di PaviaPaviaItaly

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