Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium

This paper is concerned with finding Fokker-Planck equations in R d with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. Such an “optimal” Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. In an L2–analysis, we find that the maximum decay rate equals the maximum eigenvalue of the inverse covariance matrix, and that the infimum of the attainable multiplicative constant is 1, corresponding to the high-rotational limit in the Fokker-Planck drift. This analysis is complemented with numerical illustrations in 2D, and it includes a case study for time-dependent coefficient matrices.