Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

Abstract: We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in (0,1)$, it was proved in [19] 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 $H$). In [11], this result has been extended to the multiplicative case when $H\textgreater{}1/2$. In this paper, we obtain these types of results in the rough setting $H\in (1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [11] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.