The Lévy–Fokker–Planck equation: Φ-entropies and convergence to equilibrium

In this paper, we study a Fokker–Planck equation of the form u_t=ℐ[u]+div(xu), where the operator ℐ, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the steady state of the associated stationary equation.