  Studying the subexponential convergence towards equilibrium of a strong
Markov process, we exhibit an intermediate Lyapunov condition equivalent to the
control of some moment of a hitting time. This provides a link, similar
(although more intricate) to the one existing in the exponential case, between
the coupling method and the approach based on the existence of a Lyapunov
function for the generator, in the context of the subexponential rates found by
Fort-Roberts (2005), Douc-Fort-Guillin (2009) and Hairer (2016).
