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

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Description:	Anton Arnold, Technische Universität Wien 10 January 2022 – 13:30 to 14:30


Created:	2022-01-12 13:33
Collection:	Frontiers in analysis of kinetic equations
Publisher:	Isaac Newton Institute
Copyright:	Anton Arnold
Language:	eng (English)


Abstract:	We are concerned with finding Fokker-Planck equations in whole space 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. This infimum is 1, corresponding to the high-rotational limit in the Fokker-Planck drift. Such an “optimal” Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the -propagator norms of the Fokker-Planck equation and of its drift-ODE coincide. Finally we give an outlook onto using Fokker-Planck equations with t-dependent coefficients. References: * A. Arnold, B. Signorello: Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, preprint 2021. * A. Arnold, C. Schmeiser, B. Signorello. Sharp decay estimates and -propagator norm for Fokker-Planck equations with linear drift, preprint 2020.

Abstract:

We are concerned with finding Fokker-Planck equations in whole space 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. This infimum is 1, corresponding to the high-rotational limit in the Fokker-Planck drift.

Such an “optimal” Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the -propagator norms of the Fokker-Planck equation and of its drift-ODE coincide.

Finally we give an outlook onto using Fokker-Planck equations with t-dependent coefficients.

References:
* A. Arnold, B. Signorello: Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, preprint 2021.
* A. Arnold, C. Schmeiser, B. Signorello. Sharp decay estimates and -propagator norm for Fokker-Planck equations with linear drift, preprint 2020.

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