Convergence to Equilibrium in the Free Fokker-Planck Equation With a Double-Well Potential

Abstract

We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu \_t}{\partial t}=\frac{\partial }{\partial x}\left[\mu \_t\left(\frac{1}{2}{V}^{\prime }-H\mu \_t\right)\right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x)=\frac{1}{4}{x}^4+\frac{c}{2}{x}^2$, 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.