  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.
