The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

We consider the linear Wigner-Fokker-Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker-Planck type operators in certain weighted $L^2$-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.Comment: This manuscript essentially replaces (and corrects a mistake found in) the submission arXiv:0707.2445, by establishing a new functional framework and new spectral estimate

Instability and bifurcation in a trend depending price formation model

A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and representing price oscillations; and traveling waves, that represent either inflationary or deflationary behavior.Postprint (author's final draft