Entropy decay of discretized fokker-planck equations I—Temporal semidiscretization

In this paper, we study the large time behavior of a fully implicit semidiscretization (in time) of parabolic Fokker-Planck type equations. Using logarithmic Sobolev inequalities exponential decay of the relative entropy (w.r.t. the steady state) is proved which yields convergence of the discrete scheme towards the unique steady state. The exponential decay rate recovers as At J 0 the decay rate of the original Fokker-Planck type equations.