# American Institute of Mathematical Sciences

doi: 10.3934/krm.2022009
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## Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium

 Vienna University of Technology, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria

Received  June 2021 Revised  September 2021 Early access March 2022

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. This analysis is complemented with numerical illustrations in 2D, and it includes a case study for time-dependent coefficient matrices.

Citation: Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, doi: 10.3934/krm.2022009
##### References:
 [1] F. Achleitner, A. Arnold and E. A. Carlen, On multi-dimensional hypocoercive BGK models,, Kinet. Relat. Models, 11 (2018), 953-1009.  doi: 10.3934/krm.2018038. [2] F. Achleitner, A. Arnold and B. Signorello, On optimal decay estimates for ODEs and PDEs with modal decomposition, Stochastic Dynamics out of Equilibrium, Springer Proceedings in Mathematics and Statistics, 282 (2019), 241–264. doi: 10.1007/978-3-030-15096-9_6. [3] A. Arnold, P. A. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246. [4] A. Arnold, C. Schmeiser and B. Signorello, Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift, Comm. Math. Sc. (2022). Available from: https://arXiv.org/abs/2003.01405. [5] A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, preprint, https://arXiv.org/abs/1409.5425. [6] P. Diaconis, The Markov chain Monte Carlo revolution, Bull. Amer. Math. Soc., 46 (2009), 179-205.  doi: 10.1090/S0273-0979-08-01238-X. [7] H. Dietert and J. Evans, Finding the jump rate for fastest decay in the Goldstein-Taylor model, preprint, https://arXiv.org/abs/2103.10064. [8] A. Guillin and P. Monmarché, Optimal linear drift for the speed of convergence of an hypoelliptic diffusion, Electron. Commun. Probab., 21 (2016), Paper No. 74, 14 pp. doi: 10.1214/16-ECP25. [9] L. Miclo and P. Monmarché, Étude spectrale minutieuse de processus moins indécis que les autres, In Séminaire de Probabilités XLV, 2078 (2013), 459–481. English summary available from: https://www.ljll.math.upmc.fr/ monmarche. doi: 10.1007/978-3-319-00321-4_18. [10] R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, Cambridge University Press 2013. [11] T. Lelièvre, F. Nier and G. A. Pavliotis, Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, J. Stat. Phys., 152 (2013), 237-274.  doi: 10.1007/s10955-013-0769-x. [12] J. Snyders and M. Zakai, On nonnegative solutions of the equation $AD+DA' = -C$, SIAM J. Appl. Math., 18 (1970), 704-714.  doi: 10.1137/0118063. [13] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

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##### References:
 [1] F. Achleitner, A. Arnold and E. A. Carlen, On multi-dimensional hypocoercive BGK models,, Kinet. Relat. Models, 11 (2018), 953-1009.  doi: 10.3934/krm.2018038. [2] F. Achleitner, A. Arnold and B. Signorello, On optimal decay estimates for ODEs and PDEs with modal decomposition, Stochastic Dynamics out of Equilibrium, Springer Proceedings in Mathematics and Statistics, 282 (2019), 241–264. doi: 10.1007/978-3-030-15096-9_6. [3] A. Arnold, P. A. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246. [4] A. Arnold, C. Schmeiser and B. Signorello, Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift, Comm. Math. Sc. (2022). Available from: https://arXiv.org/abs/2003.01405. [5] A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, preprint, https://arXiv.org/abs/1409.5425. [6] P. Diaconis, The Markov chain Monte Carlo revolution, Bull. Amer. Math. Soc., 46 (2009), 179-205.  doi: 10.1090/S0273-0979-08-01238-X. [7] H. Dietert and J. Evans, Finding the jump rate for fastest decay in the Goldstein-Taylor model, preprint, https://arXiv.org/abs/2103.10064. [8] A. Guillin and P. Monmarché, Optimal linear drift for the speed of convergence of an hypoelliptic diffusion, Electron. Commun. Probab., 21 (2016), Paper No. 74, 14 pp. doi: 10.1214/16-ECP25. [9] L. Miclo and P. Monmarché, Étude spectrale minutieuse de processus moins indécis que les autres, In Séminaire de Probabilités XLV, 2078 (2013), 459–481. English summary available from: https://www.ljll.math.upmc.fr/ monmarche. doi: 10.1007/978-3-319-00321-4_18. [10] R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, Cambridge University Press 2013. [11] T. Lelièvre, F. Nier and G. A. Pavliotis, Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, J. Stat. Phys., 152 (2013), 237-274.  doi: 10.1007/s10955-013-0769-x. [12] J. Snyders and M. Zakai, On nonnegative solutions of the equation $AD+DA' = -C$, SIAM J. Appl. Math., 18 (1970), 704-714.  doi: 10.1137/0118063. [13] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.
The solid curves show the FP- and ODE-propagator norms as functions of $t$ for 3 values of the multiplicative parameter: $c = 3,\,2,\,1.5$ (top to bottom). The dashed curves give the corresponding (sharp) exponential bound of the form $c e^{-\lambda_{opt}t}$ for the 3 cases. The dashed black curve shows the exponential bound in the high-rotational limit, i.e. for $c\searrow1$. Colors only online
Left: For $c = \sqrt2$, the solid blue and red curves show the FP- and ODE-propagator norms as functions of $t$ for the hypocoercive FP-equations constructed, respectively, in Theorem 3.1(a) here and Theorem 2.2 in [8]. The dashed blue curve gives the corresponding exponential bound $\sqrt2\, e^{-t}$; it is sharp for Theorem 3.1(a). The solid green and black curves show the FP- and ODE-propagator norms for the symmetric FP-equations in (2.9) and (4.8), respectively

Right: a zoom of the plot, close to $t=0$. Colors only online

Left: For $c = \sqrt{4/3}$, the FP- and ODE-propagator norms are given for hypocoercive FP-equations with piecewise constant coefficients, using 5 different values on $0\le t\le0.1$: The solid red curve corresponds to the optimal, constant matrices from Theorem 3.1(a) as reference case, and the dashed red curve is the corresponding decay estimate (3.1). The initially symmetric FP-equations from (4.8) and (2.9) are given by the black and green solid curves, respectively. Hypocoercive FP-equations with slower and faster rotational drift are represented, respectively, by the blue and magenta solid curves

Right: a zoom of the plot, close to $t=0$. Colors only online

For $c = \sqrt{4/3}$, the FP- and ODE-propagator norms are given for hypocoercive (h.c.) FP-equations with piecewise constant coefficients, using 3 different values on $0\le t\le t_0$: The solid red curve corresponds to the optimal, constant matrices from Theorem 3.1(a) as reference case. Hypocoercive FP-equations with the faster rotational drift matrices (FP5), (FP6) are represented by the magenta and blue solid curves, respectively. The dashed curves are the corresponding decay estimate (3.1). The discontinuity points $t_0$ of the coefficient matrices are marked with black dots
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