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arXiv · 2103.11146

On a class of Fokker-Planck equations with subcritical confinement

G. Toscani·M. Zanella
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Abstract

We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x) $, a polynomial rate of convergence to equilibrium.Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.

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G. Toscani, M. Zanella. 2021-03-20. On a class of Fokker-Planck equations with subcritical confinement. https://arxiv.org/abs/2103.11146

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