Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation

We reckon the rate of exponential convergence to equilibrium both in relati ve entropy and in relative Fisher information, for the solution to the spat ially homogeneous Fokker-Planck equation. The result follows by lower bound s of the entropy production which are explicitly computable. Second, we sho w that the Gross's logarithmic Sobolev inequality is a direct consequence o f the lower bound for the entropy production relative to Fisher information . The entropy production arguments are finally applied to reckon the rate o f convergence of the solution to the heat equation towards the fundamental one in various norms.