  We investigate the problem of the rate of convergence to equilibrium for
ergodic stochastic differential equations driven by fractional Brownian motion
with Hurst parameter $H\textgreater{}1/2$ and multiplicative noise component
$\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in
\cite{hairer} that, under some mean-reverting assumptions, such a process
converges to its equilibrium at a rate of order $t^{-\alpha}$ where
$\alpha\in(0,1)$ (depending on $H$). The aim of this paper is to extend such
types of results to some multiplicative noise setting. More precisely, we show
that we can recover such convergence rates when $H\textgreater{}1/2$ and the
inverse of the diffusion coefficient $\sigma$ is a Jacobian matrix. The main
novelty of this work is a type of extension of Foster-Lyapunov like techniques
to this non-Markovian setting, which allows us to put in place an asymptotic
coupling scheme such as in \cite{hairer} without resorting to deterministic
contracting properties.
