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

We consider the one-dimensional free Fokker-Planck equation partial derivative mu(/)(t)partial derivative t = partial derivative/partial derivative x [mu(t).(1/2V' - H mu(t) where H denotes the Hilbert transform and V is a particular double-well quartic potential, namely V (x) = 1/4x(4) + c/2x(2), with c >= - 2. We prove that the solution (mu(t))(t)>= 0 of this PDE converges in Wasserstein distance of any order p >= 1 to the equilibrium measure mu(V) as t goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques.