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This paper is concerned with finding Fokker-Planck equations in$\mathbb{R}^d$ with the fastest exponential decay towards a given equilibrium.For a prescribed, anisotropic Gaussian we determine a non-symmetricFokker-Planck equation with linear drift that shows the highest exponentialdecay rate for the convergence of its solutions towards equilibrium. At thesame time it has to allow for a decay estimate with a multiplicative constantarbitrarily close to its infimum. Such an optimal Fokker-Planck equation isconstructed explicitly with a diffusion matrix of rank one, hence beinghypocoercive. In an $L^2$-analysis, we find that the maximum decay rate equalsthe maximum eigenvalue of the inverse covariance matrix, and that the infimumof the attainable multiplicative constant is 1, corresponding to thehigh-rotational limit in the Fokker-Planck drift.

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