Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations
This work provides rigorous convergence guarantees for discretizations of Fokker-Planck equations, which is important for numerical simulations in statistical physics and related fields, but the results are incremental as they extend existing continuous theory to discrete settings.
The authors propose discrete versions of Fokker-Planck equations and prove exponential convergence to equilibrium for both time-continuous and time-discrete cases, using new discrete Poincaré inequalities and adapted hypocoercive methods.
In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a "two-direction" discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.