Open AccessOptimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium
Author(s) -
Anton Arnold,
Beatrice Signorello
Publication year - 2022
Publication title -
kinetic and related models
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.987
H-Index - 28
eISSN - 1937-5093
pISSN - 1937-5077
DOI - 10.3934/krm.2022009
Subject(s) - infimum and supremum , fokker–planck equation , multiplicative function , eigenvalues and eigenvectors , mathematics , dimension (graph theory) , inverse , exponential decay , diffusion , matrix (chemical analysis) , exponential function , physics , mathematical physics , mathematical analysis , combinatorics , quantum mechanics , geometry , differential equation , materials science , composite material
This paper is concerned with finding Fokker-Planck equations in \begin{document}$ \mathbb{R}^d $\end{document} with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. Such an "optimal" Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. In an \begin{document}$ L^2 $\end{document} –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.