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arXiv · 2106.15742

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

Anton Arnold·Beatrice Signorello
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Abstract

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-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 $L^2$-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.

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Anton Arnold, Beatrice Signorello. 2021-06-29. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. https://arxiv.org/abs/2106.15742

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