We consider the one-dimensional free Fokker-Planck equation $\frac{ \partial \mu\_t}{ \partial t} = \frac{ \partial}{ \partial x} \left[ \mu\_t \left( \frac12 V' - H\mu\_t \right) \right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x) = \frac14 x^4 + \frac{ c}{ 2} x^2$, with $-2 \le c < 0$. We prove that the solution $(\mu\_t)\_{ t \ge 0}$ of this PDE converges to the equilibrium measure $\mu\_V$ as $t$ goes to infinity, which provides a first result of convergence in a non-convex setting. The proof involves free probability and complex analysis techniques.
Categories: math.PR math.AP
