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

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.(1/3,1) and multiplicative noise component .. When . is constant and for every H.(0,1), it was proved in [Ann. Probab. 33 (2005) 703.758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order t.. where ..(0,1) (depending on H). In [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503.538], this result has been extended to the multiplicative case when H>1/2. In this paper, we obtain these types of results in the rough setting H.(1/3,1/2). Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503.538] by deleting the gradient assumption on the noise coefficient .. The main theorems include some existence and uniqueness results for the invariant distribution.